# Trace methods for coHochschild homology

Sarah Klanderman  and  Maximilien Péroux  Department of Mathematical and
Computational Sciences, Marian University, 3200 Cold Spring Road,
Indianapolis, IN, 46222, USA<EMAIL_ADDRESS>Department of
Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia,
PA 19104, USA<EMAIL_ADDRESS>
###### Abstract.

We provide traces for coHochschild homology, dualizing the Hattori-Stallings
trace. Considering the trace of the identity, we define traces between
coHochschild homology and certain $K$-theories of coalgebras. We employ
bicategorical methods of Ponto to show that coHochschild homology is a shadow.
Consequently, we obtain that coHochschild homology is Morita-Takeuchi
invariant.

###### Key words and phrases:

coalgebra, comodule, coHochschild homology, trace, bicategory, $K$-theory,
Morita-Takeuchi equivalence.

###### 2020 Mathematics Subject Classification:

Primary: 16E40, 16T15, 18N10, 19A99, 55P43, 55U15. Secondary: 16D90, 18F30,
18M70, 18N60, 57T30.

###### Contents

  1. 1 Introduction
  2. 2 The universal cotrace
  3. 3 Bicategories of bicomodules
  4. 4 CoHochschild homology as a shadow
  5. 5 Duality
  6. 6 Traces
  7. 7 Morita-Takeuchi invariance
  8. A Homotopy theory of connective bicomodules
  9. B CoHochschild homology in higher categories

##  1\. Introduction

### Background

The trace of a square matrix is a fundamental invariant in linear algebra. One
of its characteristic properties is that it is additive and cyclic:

$\mathrm{tr}(A+B)=\mathrm{tr}(A)+\mathrm{tr}(B),\quad\quad\mathrm{tr}(AB)=\mathrm{tr}(BA),$

which make the trace easily computable and tractable. The trace also records
valuable information on fixed points. For instance, given a finite CW-complex
$X$ and a continuous map $f\colon X\rightarrow X$, the alternating sum of the
traces of $f$ induced on rational homology of $X$ is the Lefschetz number,
$L(f)$. If $L(f)\neq 0$, then $f$ is has at least one fixed point [Lef26,
Lef37].

The definition of the trace can be extended from matrices to any endomorphism
on a dualizable object in a symmetric monoidal category [DP80, PS14]. A more
subtle situation arises in non-symmetric contexts. For instance, the relative
tensor product of bimodules over a non-commutative ring $R$ does not define a
symmetric monoidal structure. Nevertheless, in [Hat65, Sta65], the trace of an
$R$-linear endomorphism on a finitely generated projective $R$-module $M$ is
defined as a homomorphism $\mathsf{End}_{R}(M)\rightarrow R/[R,R]$, called the
Hattori-Stallings trace. Ponto generalized this approach and introduced the
notion of traces in shadowed bicategories [Pon10]. Essentially, a bicategory
is a generalization of a monoidal category in which we allow a change of base
ring, and a shadow is a choice of a symmetry in this monoidal structure.

Algebraic $K$-theory is a crucial invariant for rings. Computations in
$K$-theory can provide new insights in other areas of mathematics; for
instance, determining the $K$-theory of $\mathbb{Z}$ would resolve the
Vandiver conjecture [Kur92], an important problem in algebraic number theory.
However, just as with homotopy groups of spheres, $K$-theory is extremely
difficult to compute. Hochschild homology, denoted $\mathrm{HH}$, is another
invariant for rings which approximates $K$-theory via the Dennis trace
$K_{*}(R)\rightarrow\mathrm{HH}_{*}(R)$, and is easier to compute. At level
zero, this is precisely the Hattori-Stallings rank $K_{0}(R)\rightarrow
R/[R,R]$, i.e., the trace of the identity, since $\mathrm{HH}_{0}(R)=R/[R,R]$.
As shown by Bökstedt, calculations of Hochschild homology are simplified by
considering its homotopy coherent analogue, called topological Hochschild
homology (THH). The approximation can be further refined using cyclotomic
structures [NS18]. Most sources of computations of $K$-theory stem from
calculations of $\mathrm{HH}$, $\mathrm{THH}$, and cyclotomic refinements.
Algebraic $K$-theory is the natural home for additive invariants [BGT13],
while topological Hochschild homology is the natural home for traces and fixed
point invariants [CLM+20], which makes the study of Hochschild homology
valuable in its own right.

CoHochschild homology (coHH) is the invariant for coalgebras analogous to
Hochschild homology. First introduced by Doi [Doi81], the definition was
extended to chain complexes in [HPS09], then generalized to the stable
homotopy theory setting [HS21], which in this case is called topological
coHochschild homology (coTHH). Work of the second author shows that strictly
coassociative and counital coalgebras in any monoidal model category of
spectra do not model $\mathbb{E}_{1}$-coalgebras in spectra [PS19, Pér22b].
That is, coalgebras in spectra that are coassociative and counital up to
higher homotopy cannot be rigidified to a strictly coassociative and counital
coalgebra. The result remains true if we replace spectra by modules over $R$,
where $R$ is any $\mathbb{E}_{\infty}$-ring spectrum. Essentially, the only
possible (strictly coassociative and counital) coalgebras in spectra in the
current monoidal model categories are of the form $\Sigma^{\infty}_{+}X$, with
comultiplication induced by the diagonal $X\rightarrow X\times X$.
Consequently, only computations of $\mathrm{coTHH}(\Sigma^{\infty}_{+}X)$ are
considered in [HS21, Kla22]. Therefore topological coHochschild homology
requires an $\infty$-categorical setting in general. This is the approach of
[BP23, BGS22].

Recall that given a connected space $X$, we obtain an equivalence of spectra
$\mathrm{THH}(\Sigma^{\infty}_{+}\Omega
X)\simeq\Sigma^{\infty}_{+}\mathcal{L}X$, where $\mathcal{L}X$ is the free
loop space on $X$ [Lod98, 7.3.11]. Further conditions on $X$ induced by Koszul
duality show that we obtain an equivalence of spectra
$\mathrm{coTHH}(\Sigma^{\infty}_{+}X)\simeq\Sigma^{\infty}_{+}\mathcal{L}X$
[HS21, 3.7]. This leads to new computations in string topology [BGS22].
Moreover, together with the Dennis trace, Hess-Shipley obtained a new trace to
coTHH [HS21]

$K(\Sigma^{\infty}_{+}\Omega
X)\longrightarrow\mathrm{THH}(\Sigma^{\infty}_{+}\Omega
X)\simeq\mathrm{coTHH}(\Sigma^{\infty}_{+}X),$

which is a reformulation of the Euler characteristic on a simply connected
space $X$ [CP19]. Induced by Spanier-Whitehead duality, the Dennis trace also
determines a pairing [BP23, 4.4]

$K(C^{\vee})\wedge\mathrm{coTHH}(C)\longrightarrow\mathbb{S},$

for which $C$ is an $\mathbb{E}_{1}$-coalgebra in spectra with some finiteness
conditions, and $C^{\vee}$ is its Spanier-Whitehead dual, and thus an
$\mathbb{E}_{1}$-ring spectrum. However, all the above traces are unsatisfying
as they are directly induced by combining the usual Dennis trace with an
identification of coTHH with THH, and are not internal to the coHochschild
setting. Our paper aims to provide a Hattori-Stallings trace for coHochschild
homology internally, without restriction on the coalgebras.

### Results

Let $\Bbbk$ be a commutative ring with global dimension zero. The quotient
$R\rightarrow R/R[R,R]=\mathrm{HH}_{0}(R)$ is regarded as a universal trace
and can be extended to the Hattori-Stallings trace
$\mathrm{tr}\colon\mathsf{End}_{R}(M)\rightarrow\mathrm{HH}_{0}(R)$ defined on
any finitely generated projective right module $M$ over a ring $R$. Similarly,
Quillen introduce the dual notion of universal cotrace
$\mathrm{cotr}\colon\mathrm{coHH}_{0}(C)\hookrightarrow C$ in [Qui88]. He
employs this cotrace as a means to algebraically extend Chern-Weil theory on
Connes’ cyclic complex of a ring (see more details in Example 2.4 below). Just
as Hattori and Stallings, we extend the universal cotrace
$\mathrm{cotr}\colon\mathrm{coHH}_{0}(C)\hookrightarrow C$ as follows.

######  Theorem 1.1.

Let $C$ be a $\Bbbk$-coalgebra. Let $M$ be a finitely cogenerated injective
left $C$-comodule. Then there exists a universal cotrace on $M$ as a
$\Bbbk$-linear homomorphism

$\mathrm{cotr}\colon{}_{C}\mathsf{End}(M)\otimes_{\Bbbk}\mathrm{coHH}_{0}(C)\longrightarrow\Bbbk$

that is cyclic in the following sense. Given another finitely cogenerated left
$C$-comodule $N$ and $C$-colinear homomorphisms $f:M\rightarrow N$ and
$g:N\rightarrow M$, then for all $c\in\mathrm{coHH}_{0}(C)$ we have

$\mathrm{cotr}((f\circ g)\otimes c)=\mathrm{cotr}((g\circ f)\otimes c).$

In the case of a ring $R$, considering the trace of the identity defines a
homomorphism $K_{0}(R)\rightarrow\mathrm{HH}_{0}(R)$, the Hattori-Stallings
rank. This is the level zero of the Dennis trace
$K_{*}(R)\rightarrow\mathrm{HH}_{*}(R)$. Considering the cotrace of the
identity leads to the following.

######  Theorem 1.2.

Let $C$ be a $\Bbbk$-coalgebra. Denote by $K_{0}(C)$ the group completion on
the set of isomorphism classes of finitely cogenerated injective left
$C$-comodules with direct sums. The corank defines a $\mathbb{Z}$-linear
homomorphism
$K_{0}(C)\otimes_{\mathbb{Z}}\mathrm{coHH}_{0}(C)\longrightarrow\Bbbk$, or in
other words a $\Bbbk$-linear homomorphism

$\mathrm{coHH}_{0}(C)\longrightarrow K_{0}^{\vee}(C),$

where $K_{0}^{\vee}(C)=\mathsf{Hom}_{\mathbb{Z}}(K_{0}(C),\Bbbk)$.

This new $K$-theory captures coalgebraic structures. The Hattori-Stallings
corank relates to the usual rank in the following way.

  * •

If we let $C$ be the trivial $\Bbbk$-coalgebra $\Bbbk$, then
$\mathrm{coHH}_{0}(\Bbbk)=\Bbbk$ and $\Bbbk$-comodules are just
$\Bbbk$-modules. Consequently, the cotrace is simply a $\Bbbk$-linear
homomorphism

$\mathsf{End}_{\Bbbk}(M)\longrightarrow\Bbbk,$

which is the usual trace on $M$, a finitely generated and projective
$\Bbbk$-module. Moreover, $K_{0}(\Bbbk)$ equals the usual $K$-theory of
$\Bbbk$ as a ring, and thus $K_{0}(\Bbbk)=\mathbb{Z}$, and
$K^{\vee}_{0}(\Bbbk)=\Bbbk$, and the corank is the identity homomorphism on
$\Bbbk$.

  * •

Coalgebras that are finitely generated as $\Bbbk$-modules are equivalent to
algebras that are finitely generated as $\Bbbk$-modules. Therefore it is to be
expected that the corresponding trace methods are equivalent. Indeed, if $C$
is finitely generated as a $\Bbbk$-module, then
$\mathrm{coHH}_{0}(C)\cong\mathrm{HH}_{0}(C^{*})^{*}$. The usual rank on
$C^{*}$ provides a homomorphism
$K_{0}(C^{*})\rightarrow\mathrm{HH}_{0}(C^{*})$, which by taking duals
corresponds to a homomorphism
$\mathrm{coHH}_{0}(C)\rightarrow\mathsf{Hom}_{\mathbb{Z}}(K_{0}(C^{*}),\Bbbk)$,
and recovers our cotrace, and the results in [BP23, 4.4].

One can extend the definition of the Hattori-Stallings trace of an
endomorphism to a twisted endomorphism $M\rightarrow M\otimes_{R}P$ where $M$
is finitely generated projective right $R$-module and $P$ is an
$(R,R)$-bimodule. It defines a homomorphism
$\mathsf{Hom}_{R}(M,M\otimes_{R}P)\rightarrow\mathrm{HH}_{0}(P,R)$ (see
Definition 2.3 below). We show that if $P=C$ is a $\Bbbk$-coalgebra and $M$
has a right $C$-comodule structure, it can lift to a trace on coalgebras the
following way.

######  Theorem 1.3.

Let $C$ be a $\Bbbk$-coalgebra. Let $M$ be a right $C$-comodule that is
finitely generated as a $\Bbbk$-module. Then there exists a universal
$C$-trace on $M$ as a $\Bbbk$-linear homomorphism

$\mathrm{tr}^{C}\colon\mathsf{End}_{C}(M)\longrightarrow\mathrm{coHH}_{0}(C),$

that is cyclic in the following sense. Given another right $C$-comodule $N$
that is finitely generated as a $\Bbbk$-module, and $C$-colinear homomorphisms
$f:M\rightarrow N$ and $g:N\rightarrow M$, we have

$\mathrm{tr}^{C}(f\circ g)=\mathrm{tr}^{C}(g\circ f).$

In short, the $C$-trace of a $C$-colinear homomorphism $f\colon M\rightarrow
M$ will be defined as $\mathrm{tr}^{C}(f)=\mathrm{tr}(\rho\circ f)$ where
$\rho:M\rightarrow M\otimes_{\Bbbk}C$ is the coaction on $M$. Considering the
$C$-trace of the identity yields a rank on an another $K$-theory more closely
related to relative $K$-theory described below.

######  Theorem 1.4.

Let $C$ be a $\Bbbk$-coalgebra. Denote by $K^{C}_{0}(\Bbbk)$ the group
completion on the set of isomorphism classes of right $C$-comodules that are
finitely generated as $\Bbbk$-modules with direct sums. Considering the
$C$-trace of the identity yields a $\mathbb{Z}$-linear homomorphism

$K^{C}_{0}(\Bbbk)\longrightarrow\mathrm{coHH}_{0}(C).$

Given a topological space $X$, our $K$-theory $K^{C}_{0}(\Bbbk)$ has a similar
definition as the Waldhausen $K$-theory of homotopically finite pointed spaces
comodules over $X_{+}=X\coprod\\{*\\}$, which was shown to relate to the
$A$-theory of $X$ [HS16, 1.3,1.4].

Just as the Hattori-Stallings trace over perfect chain complex over $R$ can be
defined as the alternating sum of the traces at each level, one can carefully
extend the cotrace (Theorem 1.1) and $C$-trace (Theorem 1.3) above to chain
complexes. See more in section 6.

One of the defining properties of $K$-theory and Hochschild homology is
invariance under Morita equivalences – rings that have equivalent (possibly
derived) categories of left modules. The dual notion for coalgebras and
comodules is called Morita-Takeuchi equivalence. It has been more challenging
to show that coHochschild homology is invariant under Morita-Takeuchi
equivalence in the derived context due to the pathological nature of model
categories of comodules as we recall below. Nevertheless we obtain the
following theorem.

######  Theorem 1.5.

Let $C$ and $D$ be homotopically Morita-Takeuchi equivalent simply connected
differential graded $\Bbbk$-coalgebras. Then we obtain an isomorphism:

$\mathrm{coHH}(C)\cong\mathrm{coHH}(D).$

### Methods

All the theorems above follow from the fact that coHochschild homology is a
shadow on bicategories of bicomodules (Theorems 4.5 and 4.8). As mentioned
above, a shadowed bicategory is a generalization of a symmetric monoidal
category, and thus come equipped with a notion of trace. Morita-Takeuchi
invariance becomes a property not of coHH itself, but rather its categorical
context. It is the natural notion of an equivalence in these bicategories, so
coHH is Morita-Takeuchi invariant simply because it is a bicategorical
construct.

Theorems 1.1 and 1.3 are therefore consequences of Theorem 4.5 and Proposition
6.2. We first present them in Definition 2.11 and Definition 2.25. Theorems
1.2 and 1.4 are direct corollaries and are introduced in Definition 2.18 and
in Definition 2.28. Lastly, Theorem 1.5 is a consequence of Theorem 4.8 and
Proposition 7.2, see Example 7.4.

Philosophically, our approach dualize the arguments that show Hochschild
homology is a shadow [Pon10], using an adaptation of the Dennis-Waldhausen
Morita argument. However, one must be careful as not every result immediately
dualizes. Firstly, obtaining model structures on comodules and coalgebras is
challenging [BHK+15, HKRS17, GKR20]. Even in the case when they do exist, they
may not be well behaved. For instance, given a nice enough monoidal model
category, its associated module and algebra categories represent modules and
algebras that are associative and unital up to higher homotopy, see for
instance [Lur17, 4.1.8.4, 4.3.3.17]. While the dual result for comodule and
coalgebras might not be even true [PS19, Pér22b]. However, the second author
showed that a model structure on connective comodules over simply connected
differential graded $\Bbbk$-coalgebras up to quasi-isomorphism represent
homotopy coherent connective comodules in $H\Bbbk$-spectra [Pér20].

Secondly, a cornerstone requirement to show that HH is a shadow is the
existence of a derived tensor product, modeled by a two-sided bar
construction. For (bi)comodules, we should then need a two-sided cobar
construction to model a derived cotensor product. As we point out in Remark
3.11, topologists and algebraists have a different meaning for the cobar
construction, and obstacles emerge in both. The topological definition of
cobar is well-behaved homotopically, but usually does not preserve comodule
structures, while the algebraic definition of cobar does preserve these
structures, but is not invariant under quasi-isomorphisms. To circumvent these
issues, the second author in [Pér20] restricts to the context of simply
connected differential graded coalgebras, which is the setting for Koszul
duality, which thus allows us to mimic some of the arguments in algebra. Many
challenges are still faced, such as totalization not commuting with the tensor
product, which we address in this paper.

It is our hope that our results are the first stepping stones toward trace
methods for coTHH. As we just recalled, derived coalgebras and comodules need
the language of $\infty$-categories rather than model categories. Thus, in
stable homotopy theory, coTHH needs a notion of a shadow in an
$(\infty,2)$-categorical context rather than a bicategorical context, which
has yet to be developed [HR21a]. We also hope that these new $K$-theories will
share similar universal properties and that the traces we have obtained can be
recovered in a more abstract structure as in [BGT13].

### Organization

We finish this section with notation and definitions that are used throughout
this paper. In section 2, we introduce the definitions of the cotrace and
trace on coHochschild homology. This section does not require any knowledge of
bicategories nor homotopy theory. In order to be a self-contained paper, we
will recall all necessary bicategorical definitions in later sections. We
introduce the different bicategories of bicomodules in Section 3. Our key
result, that coHochschild homology defines a bicategorical shadow, is proved
in Section 4. We then explore the notion of duality in bicomodules in Section
5, which has been under-documented in the literature thus far. In Section 6,
we provide a bicategorical description of the traces that appear in Section 2
and Theorems 1.1 and 1.3. In Section 7, we show how Morita-Takeuchi
equivalences are part of the bicategorical structures. In Appendix A, we
carefully detail how to represent the derived cotensor product of bicomodules
and explore what fibrant bicomodules are. Finally, in Appendix B, we give an
interpretation of coHochschild homology with coefficients in higher
categories, as it was previously undone, and show how it relates to classical
definitions of coHochschild homology. We left the terminology of
$\infty$-categories in the appendices.

### Acknowledgements

We thank Jonathan Campbell and Cary Malkiewich for discussions that sparked
some of the initial ideas of this paper, as well as Teena Gerhardt and Gabe
Angelini-Knoll. We also thank Haldun Özgür Bayındır, Thomas Brazelton, Maxine
Calle, Andres Mejia, Svetlana Makarova, Mona Merling, and Manuel Rivera for
helpful and enlightening conversations.

### Notation

We begin by establishing notation that we use throughout this paper.

  1. (1)

The letter $\Bbbk$ shall always denote a commutative ring with global
dimension zero, i.e. a finite product of fields. Every $\Bbbk$-module is
projective and injective.

  2. (2)

Let $\mathsf{Mod}_{\Bbbk}$ denote the category of $\Bbbk$-modules. We write
the relative tensor product $\otimes_{\Bbbk}$ as $\otimes$.

  3. (3)

Given $\Bbbk$-modules $M$ and $C$, and $\Bbbk$-linear homomorphisms
$\Delta:C\rightarrow C\otimes C$ and $\rho:M\rightarrow M\otimes C$, we may
employ the Sweedler notation in coalgebraic contexts:

$\Delta(c)=\sum_{i=1}^{n}{c_{(1)}}_{i}\otimes{c_{(2)}}_{i}=:c_{(1)}\otimes
c_{(2)},\quad\text{and}\quad\rho(m)=\sum_{i=1}^{n}{m_{(0)}}_{i}\otimes{m_{(1)}}_{i}=:m_{(0)}\otimes
m_{(1)}.$

  4. (4)

Let ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$ be the category of non-negative chain
complexes of $\Bbbk$-modules (graded homologically). The category is endowed
with a symmetric monoidal structure. The tensor product of two chain complexes
$X$ and $Y$ is defined by

$(X\otimes Y)_{n}=\bigoplus_{i+j=n}X_{i}\otimes_{\Bbbk}Y_{j},$

with differential given on homogeneous elements by

$d(x\otimes y)=dx\otimes y+(-1)^{|x|}x\otimes dy.$

We denote the tensor simply as $\otimes$. The monoidal unit is denoted
$\Bbbk$, which is the chain complex $\Bbbk$ concentrated in degree zero.

  5. (5)

We write $\mathsf{Hom}_{\Bbbk}(-,-)$ for the internal hom for both
$\mathsf{Mod}_{\Bbbk}$ and ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$. We write
$M^{*}=\mathsf{Hom}_{\Bbbk}(M,\Bbbk)$, the linear dual.

  6. (6)

Given any symmetric monoidal category $(\mathsf{C},\otimes)$, we write
$\tau:X\otimes Y\stackrel{{\scriptstyle\cong}}{{\rightarrow}}Y\otimes X$, the
natural symmetric isomorphism.

### General definitions

We recall here the categorical definitions of coalgebras and comodules, and
their related concepts.

######  Definition 1.6.

Let $(\mathsf{C},\otimes,\mathbb{I})$ be a symmetric monoidal category. A
_coalgebra_ $(C,\Delta,\varepsilon)$ in $\mathsf{C}$ consists of an object $C$
in $\mathsf{C}$ together with a _coassociative_ comultiplication
$\Delta:C\rightarrow C\otimes C$, such that the following diagram commutes:

${C}$${C\otimes C}$${C\otimes C}$${C\otimes C\otimes
C,}$$\scriptstyle{\Delta}$$\scriptstyle{\Delta}$$\scriptstyle{\textup{id}_{C}\otimes\Delta}$$\scriptstyle{\Delta\otimes\mathsf{id}_{C}}$

and that admits a _counit_ morphism $\varepsilon:C\rightarrow\mathbb{I}$ such
that we have the following commutative diagram:

${C\otimes C}$${C\otimes\mathbb{I}\cong C\cong\mathbb{I}\otimes C}$${C\otimes
C}$${C.}$$\scriptstyle{\textup{id}_{C}\otimes\varepsilon}$$\scriptstyle{\varepsilon\otimes\textup{id}_{C}}$$\scriptstyle{\Delta}$$\scriptstyle{\Delta}$

The coalgebra is _cocommutative_ if the following diagram commutes:

${C\otimes C}$${C\otimes
C.}$${C}$$\scriptstyle{\tau}$$\scriptstyle{\cong}$$\scriptstyle{\Delta}$$\scriptstyle{\Delta}$

Given a coalgebra $(C,\Delta,\varepsilon)$, we can define its opposite
coalgebra $C^{\mathsf{op}}$ to be the coalgebra
$(C,\tau\circ\Delta,\varepsilon)$. If $C$ is cocommutative, then
$C=C^{\mathsf{op}}$.

######  Definition 1.7.

Let $(\mathsf{C},\otimes,\mathbb{I})$ be a symmetric monoidal category, and
let $(C,\Delta,\varepsilon)$ be a coalgebra in $\mathsf{C}$. A _right
$C$-comodule_ $(M,\rho)$ is an object $M$ in $\mathsf{C}$ together with a
_coassociative_ and _counital_ right coaction morphism $\rho:M\rightarrow
M\otimes C$ in $\mathsf{C}$, i.e., the following diagrams commute:

${M}$${M\otimes C}$${M}$${M\otimes C}$${M\otimes C}$${M\otimes C\otimes
C,}$${M\otimes\mathbb{I}}$${M.}$$\scriptstyle{\rho}$$\scriptstyle{\rho}$$\scriptstyle{\rho\otimes\mathsf{id}_{C}}$$\scriptstyle{\rho}$$\scriptstyle{\mathsf{id}_{M}\otimes\varepsilon}$$\scriptstyle{\mathsf{id}_{M}\otimes\Delta}$$\scriptstyle{\cong}$

We can similarly define _a left $C$-comodule_ $(M,\lambda)$. In fact, notice
that a left $C$-comodule is a right $C^{\mathsf{op}}$-comodule. Given two
coalgebras $C$ and $D$ in $\mathsf{C}$, a $(C,D)$-bicomodule
$(M,\lambda,\rho)$ is a left $C$-comodule $(M,\lambda)$ and a right
$D$-comodule $(M,\rho)$ such that the following diagram commutes:

${M}$${M\otimes D}$${C\otimes M}$${C\otimes M\otimes
D.}$$\scriptstyle{\rho}$$\scriptstyle{\lambda}$$\scriptstyle{\lambda\otimes\mathsf{id}_{D}}$$\scriptstyle{\mathsf{id}_{C}\otimes\rho}$

A morphism $(M,\rho_{M})\rightarrow(N,\rho_{N})$ of right $C$-comodules is a
morphism $f:M\rightarrow N$ in $\mathsf{C}$ such that the following diagram
commutes:

${M}$${N}$${M\otimes C}$${N\otimes
C.}$$\scriptstyle{f}$$\scriptstyle{\rho_{M}}$$\scriptstyle{\rho_{N}}$$\scriptstyle{f\otimes\mathsf{id}_{C}}$

This morphism will be referred as a _(right) $C$-colinear homomorphism_. A
$(C,D)$-bicolinear homomorphism is a morphism that is both a left $C$-colinear
map and a right $D$-colinear homomorphism. The notation
$\mathsf{CoMod}_{C}(\mathsf{C})$ will denote the category of right
$C$-comodules in $\mathsf{C}$. Similarly, we use
${}_{C}\mathsf{CoMod}(\mathsf{C})$ to denote the category of left
$C$-comodules in $\mathsf{C}$, and ${}_{C}\mathsf{CoMod}_{D}(\mathsf{C})$ for
the category of $(C,D)$-bicomodules. Notice that we have isomorphisms of
categories
${}_{C}\mathsf{CoMod}_{\mathbb{I}}(\mathsf{C})\cong{}_{C}\mathsf{CoMod}(\mathsf{C})$
and
${}_{\mathbb{I}}\mathsf{CoMod}_{C}(\mathsf{C})\cong\mathsf{CoMod}_{C}(\mathsf{C})$.

######  Definition 1.8.

We say a coalgebra $(C,\Delta,\varepsilon)$ is _flat_ in a symmetric monoidal
category $(\mathsf{C},\otimes,\mathbb{I})$ if the induced functor
$C\otimes-:\mathsf{C}\rightarrow\mathsf{C}$ preserves equalizers when they
exist.

######  Remark 1.9.

Given a symmetric monoidal category $(\mathsf{C},\otimes,\mathbb{I})$, and
coalgebras $C$ and $D$ in $\mathsf{C}$, note that we obtain an isomorphism of
categories

${}_{C}\mathsf{CoMod}_{D}(\mathsf{C})\cong\mathsf{CoMod}_{C^{\mathsf{op}}\otimes
D}(\mathsf{C}).$

In particular, given a $(C,D)$-bicomodule $(M,\lambda,\rho)$, we obtain a
right $(C^{\mathsf{op}}\otimes D)$-comodule via

${M}$${M\otimes D}$${C\otimes M\otimes D}$${M\otimes C\otimes
D.}$$\scriptstyle{\rho}$$\scriptstyle{\lambda\otimes\mathsf{id}_{D}}$$\scriptstyle{\tau\otimes\mathsf{id}_{D}}$

######  Definition 1.10.

Let $(\mathsf{C},\otimes,\mathbb{I})$ be a symmetric monoidal category, let
$C,D,$ and $E$ be coalgebras in $\mathsf{C}$, and let
$(M,\lambda_{M},\rho_{M})$ be a $(D,C)$-bicomodule and
$(N,\lambda_{N},\rho_{N})$ be a $(C,E)$-bicomodule. Define the _cotensor
product_ $M\square_{C}N$ to be the following equalizer in $\mathsf{C}$:

${M\square_{C}N}$${M\otimes N}$${M\otimes C\otimes
N.}$$\scriptstyle{\rho_{M}\otimes\mathsf{id}_{N}}$$\scriptstyle{\mathsf{id}_{M}\otimes\lambda_{N}}$

If $D$ and $E$ are flat coalgebras, then $M\square_{C}N$ is a
$(D,E)$-bicomodule via

${M\square_{C}N}$${M\otimes N}$${M\otimes C\otimes
N.}$${D\otimes(M\square_{C}N)\otimes E}$${D\otimes M\otimes N\otimes
E}$${D\otimes M\otimes C\otimes N\otimes
E.}$$\scriptstyle{\exists!}$$\scriptstyle{\lambda_{X}\otimes\rho_{Y}}$$\scriptstyle{\rho_{M}\otimes\mathsf{id}_{N}}$$\scriptstyle{\mathsf{id}_{M}\otimes\lambda_{N}}$$\scriptstyle{\lambda_{M}\otimes\mathsf{id}_{C}\otimes\rho_{N}}$$\scriptstyle{\mathsf{id}_{D}\otimes\rho_{M}\otimes\mathsf{id}_{N}\otimes\mathsf{id}_{E}}$$\scriptstyle{\mathsf{id}_{D}\otimes\mathsf{id}_{M}\otimes\lambda_{N}\otimes\mathsf{id}_{E}}$

######  Definition 1.11.

Given a symmetric monoidal category $(\mathsf{C},\otimes,\mathbb{I})$, we say
two coalgebras $C$ and $D$ in $\mathsf{C}$ are Morita-Takeuchi equivalent if
the categories ${}_{C}\mathsf{CoMod}(\mathsf{C})$ and
${}_{D}\mathsf{CoMod}(\mathsf{C})$ are equivalent.

######  Definition 1.12.

Given a closed symmetric monoidal category
$(\mathsf{C},\otimes,\mathbb{I},[-,-])$, and a coalgebra $C$, we can provide
an enrichment of $\mathsf{CoMod}_{C}(\mathsf{C})$ as follows. Given right
$C$-comodules $M$ and $N$, define $\mathsf{Hom}_{C}(M,N)$ as the equalizer in
$\mathsf{C}$

${\mathsf{Hom}_{C}(M,N)}$${{[M,N]}}$${{[M,N\otimes C]},}$

where the first parallel map is induced by the coaction $N\rightarrow N\otimes
C$ while the second is defined by forgetful-cofree adjointness. We denote
$\mathsf{End}_{C}(M)=\mathsf{Hom}_{C}(M,M)$. We can similarly define
${}_{C}\mathsf{Hom}(-,-)$ and ${}_{C}\mathsf{End}(-)$ for left $C$-comodules.

######  Definition 1.13.

Let $\mathsf{A}$ be an abelian symmetric monoidal category with enough
injective objects. Suppose $C$ is a flat coalgebra in $\mathsf{A}$. Then
${}_{C}\mathsf{CoMod}(\mathsf{A})$ also has enough injective objects. If $M$
is a right $C$-comodule in $\mathsf{A}$, then the functor
$M\square_{C}-:{}_{C}\mathsf{CoMod}(\mathsf{A})\rightarrow\mathsf{A}$ is left
exact between abelian categories with enough injectives and thus we can right
derive it. We denote by $\mathsf{CoTor}^{i}_{C}(M,-)$ the $i^{th}$ right
derived functor. Similarly, if $N$ is a left $C$-comodule, we can right derive
the functor
$-\square_{C}N:\mathsf{CoMod}_{C}(\mathsf{A})\rightarrow\mathsf{A}$ and obtain
functors $\mathsf{CoTor}^{i}_{C}(-,N)$. Per usual, we can check that
$\mathsf{CoTor}^{i}_{C}(M,N)$ is unambiguous.

##  2\. The universal cotrace

In this section, we introduce the traces appearing in Theorem 1.1 and Theorem
1.3. Because it will be crucial in order to dualize it, we carefully recall
the various definitions and approaches of the Hattori-Stallings trace as well.

###  2.1. Hattori-Stallings trace

We recall [Hat65, Sta65]. Let $R$ be a ring, not necessarily commutative, and
let $A$ be an abelian group. Denote the ring of square matrices of size $n$
with coefficients in $R$ by $\mathcal{M}_{n}(R)$. A trace function on $R$ with
values in $A$ is a collection of set maps $T_{n}:\mathcal{M}_{n}(R)\rightarrow
A$ for each $n\geq 1$, such that $T_{n}(M+N)=T_{n}(M)+T_{n}(N)$ and
$T_{n}(MN)=T_{n}(NM)$. Each map $T_{n}:\mathcal{M}_{n}(R)\rightarrow A$ is
entirely determined by the map $T_{1}:R=\mathcal{M}_{1}(R)\rightarrow A$,
[Sta65, 1.4,1.5] as $T_{n}(M)=\sum_{i=1}^{n}T_{1}(m_{ii})$, where
$M=[m_{ij}]$.

Therefore we can rephrase a trace on $R$ with values in $A$ to be a
$\mathbb{Z}$-linear homomorphism $T:R\rightarrow A$ such that $T(rs)=T(sr)$,
for all $r,s\in R$, i.e. the following diagram commutes:

${R\otimes_{\mathbb{Z}}R}$${R}$${A,}$$\scriptstyle{m}$$\scriptstyle{m\circ\tau}$$\scriptstyle{T}$

where $m$ denotes the multiplication on $R$.

If we let $[R,R]=\\{rs-sr|r,s\in R\\}$ be the subgroup of commutators, then we
obtain a universal trace $\mathrm{tr}:R\rightarrow R/[R,R]$ by the quotient
homomorphism, and every trace $T:R\rightarrow A$ is uniquely determined by a
$\mathbb{Z}$-linear homomorphism $R/[R,R]\rightarrow A$:

${R}$${A.}$${R/{[R,R]}}$$\scriptstyle{T}$$\scriptstyle{\mathrm{tr}}$$\scriptstyle{\exists!}$

Given $M$ a finitely generated free right $R$-module, then an $R$-linear
endomorphism $f:M\rightarrow M$ is represented by a square matrix on $R$, and
thus any trace $T:R\rightarrow M$ defines a trace
$T:\mathsf{End}_{R}(M)\rightarrow M$ as
$\mathsf{End}_{R}(M)\cong\mathcal{M}_{n}(R)$, where $n$ is the rank of $M$.
More generally, if $M$ is a finitely generated projective right $R$-module,
then there exists another projective module $N$ such that $M\oplus N$ is a
free $R$-module. Therefore if we extend an $R$-linear endomorphism
$f:M\rightarrow M$ to the endomorphism $f+0:M\oplus N\rightarrow M\oplus N$,
we again obtain a square matrix, and thus any trace $T:R\rightarrow A$ defines
a trace $T:\mathsf{End}_{R}(M)\rightarrow A$.

######  Definition 2.1.

Let $R$ be a ring, and $M$ be a finitely generated projective right
$R$-module. The Hattori-Stallings trace on $M$ is the trace induced by the
universal trace $\mathrm{tr}:R\rightarrow R/[R,R]$ on the endomorphisms of
$M$:

$\mathrm{tr}:\mathsf{End}_{R}(M)\longrightarrow R/[R,R].$

The Hattori-Stallings rank of $M$ is then the trace of the identity
endomorphism on $M$. The abelian group $R/[R,R]$ is isomorphic to
$\mathrm{HH}_{0}(R)=R\otimes_{R\otimes R^{\mathsf{op}}}R$, the zeroth
Hochschild homology of $R$.

The definition above depends on the choice of generators of $M$. However, the
Hattori-Stallings can be defined coordinate-free as follows. Given a finitely
generated projective right $R$-module $M$, we have an isomorphism

$M\otimes_{R}\mathsf{Hom}_{R}(M,R)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathsf{Hom}_{R}(M,M)=\mathsf{End}_{R}(M),$

and thus the Hattori-Stallings trace is the composition

${\mathsf{End}_{R}(M)}$${M\otimes_{R}\mathsf{Hom}_{R}(M,R)}$${R}$${\mathrm{HH}_{0}(R).}$$\scriptstyle{\cong}$$\scriptstyle{\mathrm{ev}}$$\scriptstyle{\mathrm{tr}}$

Let $\mathsf{P}(R)$ be the set of isomorphism classes of finitely generated
projective right $R$-modules. It is a commutative monoid with respect to
direct sum. Then the Hattori-Stallings rank defines an additive map

$\displaystyle\mathrm{rk}:\mathsf{P}(R)$
$\displaystyle\longrightarrow\mathrm{HH}_{0}(R)$ $\displaystyle[M]$
$\displaystyle\longrightarrow\mathrm{tr}(\mathsf{id}_{M}).$

Let $K_{0}(R)$ be the group completion of $\mathsf{P}(R)$. Then the additive
map factors through a $\mathbb{Z}$-linear homomorphism

$\mathrm{rk}:K_{0}(R)\longrightarrow\mathrm{HH}_{0}(R).$

This map is the zeroth level of the Dennis trace
$K_{*}(R)\rightarrow\mathrm{HH}_{*}(R)$.

######  Remark 2.2.

A key observation of [Pon10, 4.2.2] is the following. We can generalize a
trace $T:R\rightarrow A$ to a trace on any $(R,R)$-bimodule $P$ as a
$\mathbb{Z}$-linear homomorphism $T:P\rightarrow A$ such that $T(rx)=T(xr)$
for all $r\in R$, $x\in P$. A universal trace is then given by
$P\rightarrow\mathrm{HH}_{0}(R,P):=R\otimes_{R\otimes R^{\mathsf{op}}}P$. If
$M$ is a right $R$-module, then $\mathsf{Hom}_{R}(M,R)\otimes_{\mathbb{Z}}M$
is an $(R,R)$-bimodule. The adjoint of the identity morphism
$\mathbb{Z}\otimes_{\mathbb{Z}}M\cong M\rightarrow M$ provides a
$\mathbb{Z}$-linear homomorphism $\mathbb{Z}\rightarrow\mathsf{End}_{R}(M)$.
The Hattori-Stallings trace, $\mathrm{tr}(f)$, on an endomorphism
$f:M\rightarrow M$, where $M$ is finitely generated projective right
$R$-module, is thus the image of $1$ under the composition

${\mathbb{Z}}$${M\otimes_{R}M^{\star}}$${M\otimes_{R}M^{\star}}$${\mathrm{HH}_{0}(R,M^{\star}\otimes_{\mathbb{Z}}M)}$${\mathrm{HH}_{0}(R),}$$\scriptstyle{f\otimes\mathsf{id}}$$\scriptstyle{\cong}$$\scriptstyle{\mathrm{ev}}$

where $M^{\star}=\mathsf{Hom}_{R}(M,R)$. This is a bicategorical trace, see
Example 6.3. We shall use this approach to obtain a cotrace in comodules.

######  Definition 2.3 ([Pon10, 2.3.4]).

Let $R$ be a ring, $M$ a finitely generated projective right $R$-module, and
$P$ an $(R,R)$-bimodule. The Hattori-Stallings trace of a right $R$-linear
homomorphism $f:M\rightarrow M\otimes_{R}P$, referred to as an endomorphism on
$M$ twisted by $P$, is the image of $f$ under the composition

${\mathsf{Hom}_{R}(M,M\otimes_{R}P)}$${(M\otimes_{R}P)\otimes_{R}\mathsf{Hom}_{R}(M,R)}$${P}$${\mathrm{HH}_{0}(P,R).}$$\scriptstyle{\cong}$$\scriptstyle{\mathrm{ev}}$$\scriptstyle{\mathrm{tr}}$

Just as in Remark 2.2, we can define the twisted trace of $f$ in a similar
fashion by [Pon10, 4.2.2]. It is the image of $1$ under the composition

${\Bbbk}$${M\otimes M^{\star}}$${M\otimes P\otimes
M^{\star}\cong\mathrm{HH}_{0}(R,P\otimes M^{\star}\otimes
M)}$${\mathrm{HH}_{0}(R,P).}$$\scriptstyle{f\otimes\mathsf{id}}$$\scriptstyle{\mathrm{ev}}$

###  2.2. A universal cotrace

We can dualize the above approach as follows. Throughout the rest of the
section, let $C$ be a $\Bbbk$-coalgebra, not necessarily cocommutative. Let
$Q$ be a $(C,C)$-bicomodule, with left coaction $\lambda:Q\rightarrow C\otimes
Q$ and right coaction $\rho:Q\rightarrow Q\otimes C$. Let $V$ be a
$\Bbbk$-module. A $\Bbbk$-linear map $T:V\rightarrow Q$ is said to be a
cotrace if the following diagram commutes:

${V}$${Q}$${C\otimes
Q.}$$\scriptstyle{T}$$\scriptstyle{\lambda}$$\scriptstyle{\tau\circ\rho}$

Define the cocommutator submodule ${\langle\langle}Q{\rangle\rangle}_{C}$ to
be the kernel in $\Bbbk$-modules:

${{\langle\langle}Q{\rangle\rangle}_{C}}$${Q}$${C\otimes
Q.}$$\scriptstyle{\mathrm{cotr}}$$\scriptstyle{\lambda}$$\scriptstyle{\tau\circ\rho}$

We obtain a universal cotrace
$\mathrm{cotr}:{\langle\langle}Q{\rangle\rangle}_{C}\hookrightarrow Q$ in the
following sense. Every cotrace $T:V\rightarrow Q$ is uniquely determined by a
$\Bbbk$-linear homomorphism
$V\rightarrow{\langle\langle}Q{\rangle\rangle}_{C}$:

${{\langle\langle}Q{\rangle\rangle}_{C}}$${V}$${Q}$$\scriptstyle{\mathrm{cotr}}$$\scriptstyle{\exists!}$$\scriptstyle{T}$

The universal cotrace already appeared in [Qui88]. Notice that
${\langle\langle}Q{\rangle\rangle}_{C}$ is isomorphic to the zeroth
coHochschild homology $\mathrm{coHH}_{0}(Q,C):=Q\square_{C\otimes
C^{\mathsf{op}}}C$. We also denote
$\mathrm{coHH}_{0}(C):=\mathrm{coHH}_{0}(C,C)\cong{\langle\langle}C{\rangle\rangle}_{C}$.
We recall the definition of $\mathrm{coHH}_{q}(Q,C)$ more generally in
Definition 4.3.

######  Example 2.4.

Given a $\Bbbk$-coalgebra $C$ and a $\Bbbk$-algebra $L$ with a $\Bbbk$-linear
trace $T:L\rightarrow V$, and cotrace $T^{\prime}:W\rightarrow C$, we then
obtain a new trace

$T^{T^{\prime}}:\mathsf{Hom}_{\Bbbk}(C,L)\rightarrow\mathsf{Hom}_{k}(W,V),$

where $f:C\rightarrow L$ is sent to

${W}$${C}$${L}$${V.}$$\scriptstyle{T^{\prime}}$$\scriptstyle{f}$$\scriptstyle{T}$

Here $\mathsf{Hom}_{\Bbbk}(C,L)$ is given a $\Bbbk$-algebra structure via
convolution [Swe69]. The idea can also be extended to chain complexes. In
particular, one can consider the zeroth coHochschild homology of the bar
resolution $B$ of a $\Bbbk$-algebra $R$, which Quillen identified with the
cyclic complex of $R$ up to a dimension shift [Qui88]. Considering the
universal cotrace induced on $B$, Quillen provides a trace $T^{\mathrm{cotr}}$
on the differential graded algebra $\mathsf{Hom}_{\Bbbk}(B,L)$ just as above.
As $B_{1}=R$, a linear homomorphism $\rho:R\rightarrow L$ defines a 1-cochain
over $R$ considered as the algebraic analogue of a connection. So its
curvature $\omega$ is defined as $\delta\rho+\rho^{2}$ where $\delta$ is the
differential of $\mathsf{Hom}_{\Bbbk}(B,L)$. Just as in Chern-Weil theory, the
trace $T^{\mathrm{cotr}}$ on $\mathsf{Hom}_{\Bbbk}(B,L)$ defines classes
$T^{\mathrm{cotr}}(\omega^{n})$ which correspond to cyclic cocycles. This idea
is used to construct Connes’s odd cyclic classes, their even analogues, which
turn out to be Chern-Simons forms, and the Jaffe-Lesniewski-Osterwalder
version of the Chern character of Connes’s $\theta$-summable Fredholm modules.

###  2.3. The coendomorphism coalgebra

In order to show how the universal cotrace provides a dual analogue of the
Hattori-Stallings trace, we need to know the analogue of finitely generated
and projective modules for comodules. We shall make use of the notion of
“quasi-finite” comodules, as introduced by Takeuchi in [Tak77b]. More modern
reviews, in more general settings, can be found in [BW03] and [Al-02].

######  Definition 2.5.

We say a left $C$-comodule $M$ is finitely cogenerated if there is an
injective $C$-colinear homomorphism $M\hookrightarrow C^{\oplus n}$ for some
$n\geq 0$.

Let ${}_{C}\mathsf{CoMod}$ be the category of left $C$-comodules. If $M$ is a
left $C$-comodule and $V$ is a $\Bbbk$-module, then $M\otimes V$ is a left
$C$-comodule. This defines a functor:

$\displaystyle M\otimes-:\mathsf{Mod}_{\Bbbk}$
$\displaystyle\longrightarrow{}_{C}\mathsf{CoMod}$ $\displaystyle V$
$\displaystyle\longmapsto M\otimes V.$

######  Definition 2.6.

We say a left $C$-comodule $M$ is _quasi-finite_ if the functor
$M\otimes-:\mathsf{Mod}_{\Bbbk}\longrightarrow{}_{C}\mathsf{CoMod}$ is a right
adjoint. In this case, we denote its left adjoint by
$h_{C}(M,-):{}_{C}\mathsf{CoMod}\rightarrow\mathsf{Mod}_{\Bbbk}$, and refer to
it as the cohom functor. In other words, given any left $C$-comodule $N$, the
cohom $h_{C}(M,N)$ is the universal $\Bbbk$-module providing a natural
$\Bbbk$-linear isomorphism

$\mathsf{Mod}_{\Bbbk}\Big{(}h_{C}(M,N),V\Big{)}\cong{}_{C}\mathsf{CoMod}\Big{(}N,M\otimes
V\Big{)},$

for any $\Bbbk$-module $V$. In particular, for $V=\Bbbk$, we obtain the
isomorphism

$h_{C}(M,N)^{*}\cong{}_{C}\mathsf{Hom}(N,M).$

######  Example 2.7.

An example of a quasi-finite left $C$-comodule $M$ is given by cofree
comodules $M=C\otimes V$, where $V$ is a finitely generated $\Bbbk$-module.
Since we have

$\displaystyle\mathsf{Mod}_{\Bbbk}\Big{(}h_{C}(C\otimes V,N),W\Big{)}$
$\displaystyle\cong{}_{C}\mathsf{CoMod}\Big{(}N,C\otimes V\otimes W\Big{)}$
$\displaystyle\cong\mathsf{Mod}_{\Bbbk}\Big{(}N,V\otimes W\Big{)}$
$\displaystyle\cong\mathsf{Mod}_{\Bbbk}\Big{(}V^{*}\otimes N,W\Big{)},$

for any $\Bbbk$-module $W$, we obtain $h_{C}(C\otimes V,N)\cong V^{*}\otimes
N$.

Any finitely cogenerated comodule is quasi-finite. The unit of the adjunction
provides a $\Bbbk$-linear homomorphism called the coevaluation:

$\displaystyle\mathrm{coev}:N\longrightarrow M\otimes h_{C}(M,N).$

If $D$ is another coalgebra, and $N$ is a $(D,C)$-bicomodule, then
$h_{C}(M,N)$ is a right $D$-comodule, and the map $\mathrm{coev}$ is a
morphism of $(C,D)$-bicomodules, see [Tak77b, 1.7, 1.19]. Therefore, for any
quasi-finite left $C$-comodule $M$, we obtain that $h_{C}(M,C)$ is a right
$C$-comodule, and we have a $(C,C)$-bicomodule homomorphism
$\mathrm{coev}:C\rightarrow M\otimes h_{C}(M,C)$.

######  Lemma 2.8.

If $M$ is a left $C$-comodule, and $N$ is a right $C$-comodule, then $M\otimes
N$ is a $(C,C)$-bicomodule and we obtain an isomorphism of $\Bbbk$-modules

$\mathrm{coHH}_{0}(M\otimes N,C)\cong N\square_{C}M.$

###### Proof.

This will be a particular case of Theorem 4.5. ∎

Let $T$ be a left $C$-comodule, $M$ a quasi-finite left $C$-comodule, and $N$
a $(C,C)$-bicomodule. Denote by $\partial:h_{C}(M,N\square_{C}T)\rightarrow
h_{C}(M,N)\square_{C}T$ the adjoint to the left $C$-colinear homomorphism

$\mathrm{coev}\square\mathsf{id}:N\square_{C}T\longrightarrow\Big{(}M\otimes
h_{C}(M,N)\Big{)}\square_{C}T\cong
M\otimes\Big{(}h_{C}(M,N)\square_{C}T\Big{)}.$

It is noted that, in [Tak77b, 1.14], the $\Bbbk$-linear homomorphism
$\partial$ above is an isomorphism whenever $M$ is an injective quasi-finite
left $C$-comodule. If we choose $N=C$ and $T=M$, we obtain an isomorphism of
$\Bbbk$-modules:

${h_{C}(M,M)\cong
h_{C}(M,C\square_{C}M)}$${h_{C}(M,C)\square_{C}M.}$$\scriptstyle{\partial}$$\scriptstyle{\cong}$

######  Definition 2.9.

Let $M$ be a quasi-finite left $C$-comodule. The coendomorphism coalgebra on
$M$ denoted $e_{C}(M)$ is the $\Bbbk$-module $h_{C}(M,M)$ endowed with the
comultiplication:

$h_{C}(M,M)\longrightarrow h_{C}(M,M)\otimes h_{C}(M,M),$

which is the adjoint of the $C$-colinear homomorphism:

${M}$${M\otimes h_{C}(M,M)}$${M\otimes h_{C}(M,M)\otimes
h_{C}(M,M).}$$\scriptstyle{\mathrm{coev}}$$\scriptstyle{\mathrm{coev}\otimes\mathsf{id}}$

The counit $h_{C}(M,M)\rightarrow\Bbbk$ is the adjoint of the identity of the
$C$-colinear homomorphism
$M\stackrel{{\scriptstyle\cong}}{{\rightarrow}}M\otimes\Bbbk$.

######  Example 2.10 ([BW03, 12.20]).

Let $M=C^{\oplus n}=C\otimes\Bbbk^{\oplus n}$; it is a finitely cogenerated
cofree comodule and is thus quasi-finite and injective. Then the
coendomorphism coalgebra on $C^{\oplus n}$ is the $C$-matrix coalgebra
$\mathcal{M}_{n}(C):=C\otimes\mathcal{M}_{n}(\Bbbk)$, for which the coalgebra
structure on $\mathcal{M}_{n}(\Bbbk)$ is defined as follows. Given any
finitely generated $\Bbbk$-module $V$, as it is dualizable, we obtain that
$V^{*}\otimes V$ is a $\Bbbk$-coalgebra via evaluation $V^{*}\otimes
V\rightarrow\Bbbk$ (providing the counit) and coevaluation $\Bbbk\rightarrow
V^{*}\otimes V$ (providing the comultiplication by applying it on
$V^{*}\otimes V\otimes\Bbbk$). If we let $V=\Bbbk^{\oplus n}$, this provides a
coalgebra structure on $\mathcal{M}_{n}(\Bbbk)$. The isomorphism of coalgebras
$\mathcal{M}_{n}(C)\cong e_{C}(C^{\oplus n})$ follows from:

${C\otimes\mathcal{M}_{n}(\Bbbk)\cong
C\otimes\mathsf{Hom}_{\Bbbk}(\Bbbk^{\oplus n},\Bbbk^{\oplus
n})}$${C\otimes\Bbbk^{\oplus n}\otimes\Bbbk^{\oplus n}\cong C^{\oplus
n}\square_{C}C^{\oplus n}}$${e_{C}(C^{\oplus
n})}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\partial}$

Just as in the case for modules, the categories of left $C$-comodules and left
$\mathcal{M}_{n}(C)$-comodules are equivalent, providing an instance of
Morita-Takeuchi equivalence.

###  2.4. A Hattori-Stallings cotrace

We define a Hattori-Stallings cotrace using a dual approach of Remark 2.2. We
shall also see that it is induced by the universal cotrace.

######  Definition 2.11.

Let $M$ be a finitely cogenerated injective left $C$-comodule. Let
$f:M\rightarrow M$ be a $C$-colinear endomorphism. We can define a Hattori-
Stallings cotrace of $f$ as a $\Bbbk$-linear homomorphism
$\mathrm{coHH}_{0}(C)\rightarrow\Bbbk$ defined as follows:

${\mathrm{coHH}_{0}(C)}$${\mathrm{coHH}_{0}(M\otimes h_{C}(M,C),C)\cong
h_{C}(M,C)\square_{C}M}$${h_{C}(M,C)\square_{C}M}$${e_{C}(M)}$${\Bbbk.}$$\scriptstyle{{\langle\langle}\mathrm{coev}{\rangle\rangle}}$$\scriptstyle{\mathsf{id}\square
f}$$\scriptstyle{\cong}$$\scriptstyle{\partial}$

This defines a $\Bbbk$-linear homomorphism
${}_{C}\mathsf{End}(M)\longrightarrow\mathsf{Hom}_{\Bbbk}\left(\mathrm{coHH}_{0}(C),\Bbbk\right)$
and by considering its adjoint, we obtain the Hattori-Stallings cotrace as a
$\Bbbk$-linear homomorphism:

$\mathrm{cotr}:{}_{C}\mathsf{End}(M)\otimes\mathrm{coHH}_{0}(C)\longrightarrow\Bbbk.$

######  Example 2.12.

If $C=\Bbbk$, then $\mathrm{coHH}_{0}(\Bbbk)=\Bbbk$ and $\Bbbk$-comodules are
precisely $\Bbbk$-modules. A finitely cogenerated injective left
$\Bbbk$-comodule $M$ is therefore precisely defined as a finitely generated
$\Bbbk$-module. In this case
$e_{\Bbbk}(M)=h_{\Bbbk}(M,M)=\mathsf{Hom}_{\Bbbk}(M,M)$. Given a
$\Bbbk$-linear endomorphism $f$, the cotrace of $f$ is the $\Bbbk$-linear
homomorphism $\Bbbk\rightarrow\Bbbk$ sending $1$ to $\mathrm{tr}(f)$, the
usual Hattori-Stallings trace of $f$.

######  Example 2.13.

Suppose $M=C^{\oplus n}$ is a finitely cogenerated cofree left $C$-comodule.
Denote the comultiplication $C\rightarrow C\otimes C$ by $c\mapsto
c_{(1)}\otimes c_{(2)}$, using Sweedler notation. Then a $C$-colinear
endomorphism $f:C^{\oplus n}\rightarrow C^{\oplus n}$ is determined by a
$\Bbbk$-linear homomorphism $C^{\oplus n}\rightarrow\Bbbk^{\oplus n}$, and
thus is determined by a $\Bbbk$-linear homomorphism

$\displaystyle C$
$\displaystyle\longrightarrow\mathsf{Hom}_{\Bbbk}(\Bbbk^{\oplus
n},\Bbbk^{\oplus n})\cong\mathcal{M}_{n}(\Bbbk)$ $\displaystyle c$
$\displaystyle\longmapsto f_{c}.$

Then the Hattori-Stallings cotrace is the following $\Bbbk$-linear
homomorphism:

$\displaystyle\mathrm{cotr}:{}_{C}\mathsf{End}(C^{\oplus
n})\otimes\mathrm{coHH}_{0}(C)$ $\displaystyle\longrightarrow\Bbbk$
$\displaystyle f\otimes c$
$\displaystyle\longmapsto\varepsilon(c_{(1)})\mathrm{tr}\left(f_{c_{(2)}}\right).$

######  Remark 2.14.

The Hattori-Stallings cotrace is also induced by the universal cotrace, by
carefully dualizing the approach of Definition 2.1. Recall that for any
$\Bbbk$-module $W$ we have a natural injective $\Bbbk$-linear homomorphism
$W\hookrightarrow W^{**}$, given by evaluation. Given any cotrace
$T:V\rightarrow C$, we can lift it to a cotrace on
$V\rightarrow\mathcal{M}_{n}(C)=C\otimes\mathcal{M}_{n}(\Bbbk)$ as follows

$\displaystyle V$ $\displaystyle\longrightarrow\mathcal{M}_{n}(C)$
$\displaystyle v$ $\displaystyle\longmapsto T(v)\otimes I_{n},$

where $I_{n}$ is the identity matrix of size $n$. We can then postcompose with
the injection $\mathcal{M}_{n}(C)\hookrightarrow\mathcal{M}_{n}(C)^{**}$.
Recall that $\mathcal{M}_{n}(C)^{**}\cong e_{C}(C^{\oplus
n})^{**}\cong{}_{C}\mathsf{End}(C^{\oplus n})^{*}$. Considering the adjoint of
the resulting $\Bbbk$-linear homomorphism
$V\rightarrow\mathcal{M}_{n}(C)^{**}$ above leads to a pairing:

${}_{C}\mathsf{End}(C^{\oplus n})\otimes V\longrightarrow\Bbbk.$

Considering the universal cotrace
$T=\mathrm{cotr}:\mathrm{coHH}_{0}(C)\hookrightarrow C$, we recover the
Hattori-Stallings cotrace on the finitely cogenerated cofree left $C$-comodule
$C^{\oplus n}$.

More generally, given any finitely cogenerated injective left $C$-comodule
$M$, by the $C$-colinear homomorphism $M\hookrightarrow C^{\oplus n}$, there
exists a left $C$-comodule $N$ such that $M\oplus N\cong C^{\oplus n}$ as left
$C$-comodules. Therefore, given any $C$-colinear endomorphism on $M$, we can
extend by zero to obtain an endomorphism on $C^{\oplus n}$. This defines a
$\Bbbk$-linear homomorphism

${}_{C}\mathsf{End}(M)\rightarrow{}_{C}\mathsf{End}(C^{\oplus n}).$

Therefore, if we postcompose the mapping $V\rightarrow\mathcal{M}_{n}(C)^{**}$
defined above with the induced map
$\mathcal{M}_{n}(C)^{**}\cong{}_{C}\mathsf{End}(C^{\oplus
n})^{*}\rightarrow{}_{C}\mathsf{End}(M)^{*}$, we obtain a $\Bbbk$-linear
homomorphism

$V\longrightarrow{}_{C}\mathsf{End}(M)^{*},$

whose adjoint is ${}_{C}\mathsf{End}(M)\otimes V\rightarrow\Bbbk$. If we
choose $T$ as the universal cotrace
$T=\mathrm{cotr}:\mathrm{coHH}_{0}(C)\hookrightarrow C$, we recover the
Hattori-Stallings cotrace on $M$ as in Definition 2.11. We prefer the approach
of Definition 2.11 to the one just described as it is more directly dual from
the usual case of modules, while above we have to consider the extra injective
homomorphism $\mathcal{M}_{n}(C)\hookrightarrow\mathcal{M}_{n}(C)^{**}$, which
feels less natural at first. Moreover, our approach is coordinate-free.

######  Remark 2.15.

Although in Definition 2.1, we have the evaluation
$M\otimes_{R}\mathsf{Hom}_{R}(M,R)\rightarrow R$ for a right $R$-module $M$,
we do not seem to have a homomorphism $C\rightarrow h_{C}(M,C)\square_{C}M$
for a left $C$-comodule $M$, but only a morphism $C\rightarrow M\otimes
h_{C}(M,C)$. This is why Lemma 2.8 is crucial for Definition 2.11. This is an
instance of a shadow property, and we will generalize in the following
sections (see Theorem 4.5).

###  2.5. A $K$-theory for coalgebras

Just as in the case of rings, considering the cotrace of the identity yields a
dual notion of a Hattori-Stallings rank. Dually to $K_{0}(R)$, we need to
consider a $K$-theory of coalgebras.

######  Definition 2.16.

Let $\mathsf{I}(C)$ be the isomorphism class of finitely cogenerated injective
left $C$-comodules. It is a commutative monoid with respect to direct sum.
Denote its group completion by $K_{0}(C)$.

######  Definition 2.17.

For any $c\in\mathrm{coHH}_{0}(C)$, we can define a Hattori-Stallings corank
as an additive homomorphism

$\displaystyle\mathrm{cork}(-,c):\mathsf{I}(C)$
$\displaystyle\longrightarrow\Bbbk$ $\displaystyle[M]$
$\displaystyle\longmapsto\mathrm{cotr}(\mathsf{id}_{M}\otimes c).$

This lifts to a $\mathbb{Z}$-linear homomorphism
$\mathrm{cork}(-,c):K_{0}(C)\longrightarrow\Bbbk$. As the homomorphism
$\mathrm{cork}(-,[M])$ is $\mathbb{Z}$-linear, since
$\mathrm{cotr}(\mathsf{id}_{M}\otimes-)$ is $\Bbbk$-linear, the corank is a
$\mathbb{Z}$-linear homomorphism

$K_{0}(C)\otimes_{\mathbb{Z}}\mathrm{coHH}_{0}(C)\longrightarrow\Bbbk.$

######  Definition 2.18.

Define the co-$K$-theory of $C$ as the $\Bbbk$-module
$K^{\vee}_{0}(C):=\mathsf{Hom}_{\mathbb{Z}}(K_{0}(C),\Bbbk)$. The adjoint of
the corank defines a $\Bbbk$-linear homomorphism

$\mathrm{coHH}_{0}(C)\longrightarrow K^{\vee}_{0}(C).$

We first present how the corank relates to the rank in some particular cases.

######  Example 2.19.

If $C=\Bbbk$, then $\Bbbk$-comodules are simply $\Bbbk$-modules, and
$\mathsf{I}(\Bbbk)=\mathsf{P}(\Bbbk)$ (as every $\Bbbk$-module is both
injective and projective). Therefore $K_{0}(C)=K_{0}(\Bbbk)\cong\mathbb{Z}$,
$\mathrm{coHH}_{0}(\Bbbk)=\Bbbk$, and the corank is simply the identity on
$\Bbbk$.

######  Example 2.20.

Suppose $C$ is finitely generated as a $\Bbbk$-module. Since we obtain an
equivalence of symmetric monoidal categories on finitely generated
$\Bbbk$-modules

$\mathsf{Mod}_{\Bbbk}^{fg}\cong(\mathsf{Mod}_{\Bbbk}^{fg})^{\mathsf{op}},$

then we can show that $\mathsf{I}(C)\cong\mathsf{P}(C^{*})$ as commutative
monoids, and thus $K_{0}(C)\cong K_{0}(C^{*})$. Note also that we have an
isomorphism

$\mathrm{coHH}_{0}(C)\cong\mathrm{HH}_{0}(C^{*})^{*}.$

Therefore, the Hattori-Stallings rank on $C^{*}$ will induce the corank as
follows. Consider the free $\Bbbk$-linear homomorphism induced on the rank

$\Bbbk\otimes_{\mathbb{Z}}K_{0}(C^{*})\longrightarrow\mathrm{HH}_{0}(C^{*}).$

Apply the $\Bbbk$-linear dual on both side to obtain

$\mathrm{coHH}_{0}(C)\longrightarrow\mathsf{Hom}_{\Bbbk}(\Bbbk\otimes_{\mathbb{Z}}K_{0}(C^{*}),\Bbbk)\cong\mathsf{Hom}_{\mathbb{Z}}(K_{0}(C^{*}),\Bbbk).$

Thus, taking the adjoint, we re-obtain the corank,
$K_{0}(C^{*})\otimes\mathrm{coHH}_{0}(C)\rightarrow\Bbbk$. This approach was
already noted in [BP23, 4.4] for $C$ any $\mathbb{E}_{1}$-coalgebra spectrum,
with some finiteness assumption, over an $\mathbb{E}_{\infty}$-ring spectrum.
When $C$ is a $\Bbbk$-coalgebra with no finiteness assumptions, our approach
provides a new trace method, more general than the Hattori-Stallings rank.

######  Remark 2.21 (Eilenberg Swindle).

Just as in the case for rings, we need to restrict to finitely cogenerated
comodules, as otherwise $K_{0}(C)=0$. Indeed, let $\Bbbk^{\oplus\infty}$ be
the $\Bbbk$-vector space with countably infinite basis. Let
$C^{\oplus\infty}=C\otimes\Bbbk^{\oplus\infty}$ be an infinitely cogenerated
cofree left $C$-comodule. Then we obtain isomorphisms of left $C$-comodules

$\displaystyle C^{\oplus\infty}$ $\displaystyle=C\otimes\Bbbk^{\oplus\infty}$
$\displaystyle\cong C\otimes\big{(}\Bbbk^{\oplus n}\oplus\Bbbk^{\oplus
n}\oplus\cdots\big{)}$ $\displaystyle\cong(C\otimes\Bbbk^{\oplus
n})\oplus(C\otimes\Bbbk^{\oplus n})\oplus\cdots$ $\displaystyle\cong C^{\oplus
n}\oplus C^{\oplus n}\oplus\cdots.$

Therefore if $M\oplus N\cong C^{\oplus n}$, then we obtain

$\displaystyle M\oplus C^{\oplus\infty}$ $\displaystyle\cong M\oplus(N\oplus
M)\oplus(N\oplus M)\oplus\cdots$ $\displaystyle\cong(M\oplus N)\oplus(M\oplus
N)\oplus\cdots$ $\displaystyle\cong C^{\oplus\infty}.$

Hence, in $K_{0}(C)$ we would get $[M]=0$.

######  Proposition 2.22.

If $C$ is a cocommutative and irreducible $\Bbbk$-coalgebra, then
$K_{0}(C)\cong\mathbb{Z}$ and $K_{0}^{\vee}(C)\cong\Bbbk$.

###### Proof.

By [Tak77a, A.2.1, A.2.2], injective $C$-comodules are all cofree when $C$ is
cocommutative and irreducible. Then $\mathsf{I}(C)\cong\mathbb{N}$, and the
result follows. ∎

######  Proposition 2.23.

If $C$ is a cocommutative $\Bbbk$-coalgebra, then $K_{0}(C)$ is a commutative
ring and $K_{0}^{\vee}(C)$ is a cocommutative $\Bbbk$-coalgebra if $K_{0}(C)$
is finitely generated as an abelian group.

###### Proof.

If $C$ is cocommutative, then the cotensor product forms a symmetric monoidal
structure on $C$-comodules. The cotensor product of injective $C$-comodules is
injective [Doi81, Proposition 1]. The cotensor product of finitely cogenerated
$C$-comodules is finitely cogenerated. Therefore $K_{0}(C)$ is a commutative
ring via cotensor product of $C$-comodules. ∎

######  Example 2.24.

By construction, our $K$-theory is Morita-Takeuchi invariant. In particular,
for any $n\geq 1$ and any coalgebra $C$ we obtain
$K_{0}(\mathcal{M}_{n}(C))\cong K_{0}(C)$.

###  2.6. Another trace for coalgebras

Instead of dualizing the trace, we can also lift the trace of a twisted
endomorphism as in Definition 2.3 over comodules. Let $M$ be a _right_
$C$-comodule that is finitely generated (and automatically projective) as a
$\Bbbk$-module, and let $M^{*}$ be its usual linear dual. A key observation is
that in this case we can give a left $C$-coaction on $M^{*}$ via

$\displaystyle\lambda:M^{*}$
$\displaystyle\longrightarrow\mathsf{Hom}_{\Bbbk}(M,C)\cong C\otimes M^{*}$
$\displaystyle\left(M\stackrel{{\scriptstyle\alpha}}{{\rightarrow}}\Bbbk\right)$
$\displaystyle\longmapsto\left(M\stackrel{{\scriptstyle\rho}}{{\rightarrow}}M\otimes
C\stackrel{{\scriptstyle\alpha\otimes\mathsf{id}}}{{\rightarrow}}C\right),$

where $\rho:M\rightarrow M\otimes C$ is the right $C$-coaction of $M$. By
[BW03, 10.11], we have an isomorphism of $\Bbbk$-modules
$M\square_{C}M^{*}\cong\mathsf{Hom}_{C}(M,M)$. Notice that the following
diagram commutes

${M^{*}\otimes M}$${\mathsf{Hom}_{\Bbbk}(M,C)\otimes M}$${M^{*}\otimes
M\otimes
C}$${C,}$$\scriptstyle{\lambda\otimes\mathsf{id}}$$\scriptstyle{\mathsf{id}\otimes\rho}$

where the unlabeled maps are defined by evaluating. This defines the
$C$-bicolinear evaluation homomorphism $\varepsilon:M^{*}\otimes M\rightarrow
C$.

######  Definition 2.25.

The Hattori-Stallings $C$-trace of a right $C$-colinear homomorphism
$f:M\rightarrow M$, denoted by $\mathrm{tr}^{C}(f)$, is the Hattori-Stallings
trace of $\rho\circ f:M\rightarrow M\otimes C$. In other words
$\mathrm{tr}^{C}(f)=\mathrm{tr}(\rho\circ f)$. More precisely, it can be
defined as the image of $1$ under the composition

${\Bbbk}$${\mathsf{End}_{C}(M)\cong
M\square_{C}M^{*}}$${M\square_{C}M^{*}\cong\mathrm{coHH}_{0}(M^{*}\otimes
M,C)}$${\mathrm{coHH}_{0}(C).}$$\scriptstyle{f\square\mathsf{id}}$$\scriptstyle{{\langle\langle}\varepsilon{\rangle\rangle}}$

The second isomorphism follows from Lemma 2.8. This is simply restricting the
trace of $\rho\circ f$, as in Definition 2.3. We restrict $M\otimes
M^{*}\rightarrow\mathrm{HH}_{0}(C,\Bbbk)=C$ to $M\square_{C}M^{*}$, and the
fact that $f$ is $C$-colinear guarantees we land in $\mathrm{coHH}_{0}(C)$.
The $C$-trace defines a $\Bbbk$-linear homomorphism

$\mathrm{tr}^{C}:\mathsf{End}_{C}(M)\longrightarrow\mathrm{coHH}_{0}(C).$

######  Example 2.26.

Let $(e_{1},\ldots,e_{n})$ be a basis of $M$, and
$(e_{1}^{*},\ldots,e_{n}^{*})$ be the dual basis of $M^{*}$. Denote the
coaction $\rho:M\rightarrow M\otimes C$ by $\rho(m)=m_{(0)}\otimes m_{(1)}$.
Then the $C$-trace of a $C$-colinear endomorphism $f$ on $M$ is explicitly
given by

$\mathrm{tr}^{C}(f)=\sum_{i=1}^{n}e_{i}^{*}(f(e_{i})_{(0)}){e_{i}}_{(1)}\in\mathrm{coHH}_{0}(C).$

######  Definition 2.27.

Denote by $\mathsf{P}^{C}(\Bbbk)$ the set of isomorphism classes of right
$C$-comodules that are finitely generated (and projective) as $\Bbbk$-modules.
It is a commutative monoid with respect to direct sum. Denote its group
completion by $K_{0}^{C}(\Bbbk)$ .

######  Definition 2.28.

The Hattori-Stallings $C$-rank is an additive homomorphism

$\displaystyle\mathrm{rk}^{C}:\mathsf{P}^{C}(\Bbbk)$
$\displaystyle\longrightarrow\mathrm{coHH}_{0}(C)$ $\displaystyle[(M,\rho)]$
$\displaystyle\longmapsto\mathrm{tr}^{C}(\mathsf{id}_{M})=\mathrm{tr}(\rho).$

It lifts to a $\mathbb{Z}$-linear homomorphism,
$\mathrm{rk}^{C}:K^{C}_{0}(\Bbbk)\rightarrow\mathrm{coHH}_{0}(C)$.

######  Remark 2.29.

While $K_{0}(C)$ captures the coalgebraic structure on $C$, $K^{C}_{0}(\Bbbk)$
is more closely related to relative $K$-theory, $K_{0}(\Bbbk;C)$, of
dualizable $\Bbbk$-modules $M$ together with twisted endomorphisms
$M\rightarrow M\otimes C$ as in [DM94]. Indeed, $K_{0}^{C}(\Bbbk)$ makes only
the additional assumption that the twisted endomorphism defines a right
$C$-coaction.

######  Remark 2.30.

Notice that if $C$ and $D$ are Morita-Takeuchi equivalent, then $K^{C}(\Bbbk)$
and $K^{D}(\Bbbk)$ might not be isomorphic. This phenomenon can already be
observed in algebras. Suppose $R$ and $S$ are $\Bbbk$-algebras such that left
$R$-modules and left $S$-modules are categorically equivalent; it does not
imply that left $R$-modules that are finitely generated as $\Bbbk$-modules are
equivalent to $S$-modules that are finitely generated as $\Bbbk$-modules. A
simple example arises from $R=\Bbbk[x]$ and $S=\mathcal{M}_{2\times
2}(\Bbbk[x])$, and considering $\Bbbk$ as a $\Bbbk[x]$-module by assuming $x$
acts trivially.

##  3\. Bicategories of bicomodules

In this section, we introduce the bicategories of bicomodules in Definition
3.5 and Definition 3.13. For convenience, we recall the definition of a
bicategory, further details of which may be found in [Pon10, PS13, CP19].

######  Definition 3.1.

A bicategory $\mathcal{B}$ consists of following.

  * •

A collection of objects, $ob(\mathcal{B})$, called _0-cells_. We write
$C\in\mathcal{B}$ instead of $C\in ob(\mathcal{B})$.

  * •

Categories $\mathcal{B}(C,D)$ for each pair of objects $C,D\in\mathcal{B}$.
The objects in these categories are referred to as _1-cells_ and the morphisms
as _2-cells_. The composition is referred to as _vertical composition_.

  * •

Unit functors $U_{C}\in\mathcal{B}(C,C)$ for all $C\in\mathcal{B}$.

  * •

Horizontal composition functors for $C,D,E\in\mathcal{B}$

$\odot:\mathcal{B}(C,D)\times\mathcal{B}(D,E)\to\mathcal{B}(C,E),$

which are not required to be strictly associative or unital.

  * •

Natural isomorphisms for $M\in\mathcal{B}(C,D)$, $N\in\mathcal{B}(D,E)$,
$P\in\mathcal{B}(E,F)$, and $Q\in\mathcal{B}(F,G)$ given
$C,D,E,F,G\in\mathcal{B}$:

$a:(M\odot N)\odot P\xlongrightarrow{\cong}M\odot(N\odot P),$ $\ell:U_{C}\odot
M\xlongrightarrow{\cong}M,\quad\quad\quad r:M\odot
U_{D}\xlongrightarrow{\cong}M,$

which satisfy the triangle identity

${(M\odot U_{D})\odot N}$${M\odot(U_{D}\odot N)}$${M\odot
N,}$$\scriptstyle{r\odot\mathsf{id}_{N}}$$\scriptstyle{a}$$\scriptstyle{\mathsf{id}_{M}\odot\ell}$

and the pentagon identity

${(M\odot N)\odot(P\odot Q)}$${((M\odot N)\odot P)\odot
Q}$${M\odot(N\odot(P\odot Q))}$${(M\odot(N\odot P))\odot Q}$${M\odot((N\odot
P)\odot
Q).}$$\scriptstyle{a}$$\scriptstyle{a}$$\scriptstyle{a\odot\mathsf{id}_{Q}}$$\scriptstyle{a}$$\scriptstyle{\mathsf{id}_{M}\odot
a}$

A bicategory with one object is precisely a monoidal category. The bicategory
of categories in which $0$-cells are categories, $1$-cells are functors, and
$2$-cells are natural transformations is a classical example. We recall other
examples below that will be motivating in our context.

######  Example 3.2.

We can define a bicategory whose 0-cells are rings and whose 1- and 2-cells
come from the category of $(R,S)$-bimodules for rings $R$ and $S$. In this
case the horizontal composition is given by the tensor product of bimodules.
The derived version of this bicategory has 0-cells which are rings and 1- and
2-cells from the derived category of $(R,S)$-bimodules, where horizontal
composition is given by the derived tensor product $\otimes^{\mathbb{L}}$.
Recall that if $M$ is a right $S$-module, and $N$ is a left $S$-module, then
$M\otimes_{S}^{\mathbb{L}}N$ is equivalent to the two-sided bar construction,
$\mathsf{Bar}(M,S,N)$.

######  Example 3.3.

When the 0-cells are given by ring spectra and the 1- and 2-cells come from
the homotopy category of $(R,S)$-bimodules for ring spectra $R$ and $S$, then
the horizontal composition is given by the derived smash product of spectra.
Note that as in the previous example, $\mathsf{Bar}(M,S,N)\simeq
M\wedge^{\mathbb{L}}_{S}N$, see [EKMM97, IV.7.5], and so we may also consider
this horizontal composition as the two-sided bar construction.

In this paper, we will introduce two bicategories formed by bicomodules
instead of bimodules. Their horizontal compositions will be given by the
(derived) cotensor product.

######  Proposition 3.4.

Let $(\mathsf{C},\otimes,\mathbb{I})$ be a symmetric monoidal category, and
let $C$ be a flat coalgebra in $\mathsf{C}$. Then
$({}_{C}\mathsf{CoMod}_{C}(\mathsf{C}),\square_{C},C)$ is a monoidal category.
Further, it is symmetric monoidal if $C$ is cocommutative.

######  Definition 3.5.

Define $\mathsf{CoMod}$ to be the bicategory whose $0$-cells are coalgebras in
$\mathsf{Mod}_{\Bbbk}$, and whose $1$-cells, $2$-cells, and vertical
compositions are given by the category
${}_{C}\mathsf{CoMod}_{D}(\mathsf{Mod}_{\Bbbk})$. The unit $U_{C}$ is the
$(C,C)$-bicomodule $C$, and horizontal composition is given by the cotensor
product:

$\displaystyle\odot:\mathcal{B}(C,D)\times\mathcal{B}(D,E)$
$\displaystyle\to\mathcal{B}(C,E)$ $\displaystyle(M,N)$ $\displaystyle\mapsto
M\odot N:=M\square_{D}N.$

For $(C,D)$-bicomodule $M$, $(D,E)$-bicomodule $N$, and $(E,F)$-bicomodule
$P$, the natural isomorphisms

$a:(M\odot N)\odot P\xlongrightarrow{\cong}M\odot(N\odot
P),\quad\quad\ell:U_{C}\odot M\xlongrightarrow{\cong}M,\quad\quad r:M\odot
U_{D}\xlongrightarrow{\cong}M,$

follow as in [Doi81] from the natural isomorphisms

$(M\square_{D}N)\square_{E}P\cong M\square_{D}(N\square_{E}P),\quad\quad
C\square_{C}M\cong M,\quad\quad M\square_{D}D\cong M.$

More specifically, these maps are given by

$\displaystyle a:(M\square_{D}N)\square_{E}P$
$\displaystyle\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}M\square_{D}(N\square_{E}P)$
$\displaystyle\ell:C\square_{C}M$
$\displaystyle\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}M$
$\displaystyle r:M\square_{D}D$
$\displaystyle\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}M$
$\displaystyle(m\otimes n)\otimes p$ $\displaystyle\longmapsto
m\otimes(n\otimes p)$ $\displaystyle c\otimes m$
$\displaystyle\longmapsto\varepsilon_{C}(c)m$ $\displaystyle m\otimes d$
$\displaystyle\longmapsto\varepsilon_{D}(d)m.$

We now consider the derived case. We first need a homotopy theory of
bicomodules. Let $\mathsf{M}={\mathsf{Ch}_{\Bbbk}^{\geq 0}}$ be the category
of non-negative chain complexes over $\Bbbk$. There is a model structure on
${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$ in which weak equivalences are quasi-
isomorphisms, cofibrations are monomorphisms, and fibrations are positive
levelwise epimorphisms, see [Hov99]. It is a combinatorial symmetric monoidal
model category with its usual tensor product, and a simplicial model category
via the Dold-Kan correspondence. In addition, every object is cofibrant and
fibrant.

######  Proposition 3.6.

Let $C$ and $D$ be dg-coalgebras over $\Bbbk$. There exists a simplicial
combinatorial model structure on
${}_{C}\mathsf{CoMod}_{D}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$ in which

  * •

the weak equivalences $\mathsf{W}$ are the morphisms of $(C,D)$-bicomodules
that are quasi-isomorphisms in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$;

  * •

the cofibrations are the morphisms of $(C,D)$-bicomodules that are
monomorphisms in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$.

In particular, every object is cofibrant.

###### Proof.

The model structure is left-induced in the sense of [HKRS17] via the
forgetful-cofree adjunction

${{}_{C}\mathsf{CoMod}_{D}\left({\mathsf{Ch}_{\Bbbk}^{\geq
0}}\right)}$${{\mathsf{Ch}_{\Bbbk}^{\geq
0}}.}$$\scriptstyle{U}$$\scriptstyle{\perp}$$\scriptstyle{C\otimes-\otimes D}$

Indeed, this follows from [Pér20, 2.12] and [HKRS17, 6.3.7] by Remark 1.9. ∎

The above result is true more generally for unbounded chain complexes over any
ring. However, it is not known if this forms an algebraic model for homotopy
coherent comodules in this generality.

######  Definition 3.7.

A (connective) _dg-coalgebra over $\Bbbk$_ is a coalgebra $C$ in
${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$. We say it is _simply connected_ if
$C_{0}\cong\Bbbk$ and $C_{1}=0$.

When $C$ and $D$ are simply connected, we show the model structure from the
previous proposition defines the correct homotopy type, see Theorem A.4.
Fibrant objects in ${}_{C}\mathsf{CoMod}_{D}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$
are retracts of Postnikov towers, see more details in Appendix A.

In order to define horizontal composition for the bicategory in the derived
setting, we now need to derive the cotensor product. However, it is more
subtle than the usual tensor product of modules because the cotensor product
is neither a left nor a right adjoint in general. Nevertheless, one can right
derive the cotensor product using methods of [Pér20, Pér21]. We leave the
details in Appendix A, but essentially, we show that the cotensor product of
fibrant bicomodules is again fibrant (Proposition A.10) and that the cotensor
preserves weak equivalences on fibrant objects (Proposition A.17). Moreover,
just as the derived tensor product can be interpreted as a two-sided bar
construction, the derived cotensor product can be interpreted as a two-sided
cobar construction, as we shall now explain.

######  Definition 3.8.

Let $C$ and $D$ be simply connected dg-coalgebras over $\Bbbk$, with
comultiplication and counit of $C$ given by $\Delta:C\rightarrow C\otimes C$
and $\varepsilon:C\rightarrow\Bbbk$ respectively. Let $M$ be a
$(D,C)$-bicomodule in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$, and let
$\rho:M\rightarrow M\otimes C$ denote its right coaction over $C$. The _two-
sided cosimplicial cobar construction_ $\Omega^{\bullet}(M,C,C)$ of $M$ is the
cosimplicial object in ${}_{D}\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq
0}})$:

${M\otimes C}$${M\otimes C\otimes C}$${\cdots,}$

defined as follows.

  * •

For all $n\geq 0$, $\Omega^{n}(M,C,C)=M\otimes C^{\otimes n+1}$.

  * •

The zeroth coface map $d^{0}:\Omega^{n}(M,C,C)\rightarrow\Omega^{n+1}(M,C,C)$
is given by $d^{0}=\rho\otimes\mathsf{id}_{C^{n+1}}$.

  * •

For $1\leq i\leq n+1$, the $i^{th}$ coface map
$d^{i}:\Omega^{n}(M,C,C)\rightarrow\Omega^{n+1}(M,C,C)$ is given by

$d^{i}=\mathsf{id}_{M}\otimes\mathsf{id}_{C^{\otimes
i-1}}\otimes\Delta\otimes\mathsf{id}_{C^{\otimes n+1-i}}.$

  * •

For all $0\leq j\leq n$, the $j^{th}$ codegeneracy map
$s^{j}:\Omega^{n+1}(M,C,C)\rightarrow\Omega^{n}(M,C,C)$ is given by

$s^{j}=\mathsf{id}_{M}\otimes\mathsf{id}_{C^{\otimes
j}}\otimes\varepsilon\otimes\mathsf{id}_{C^{\otimes n+1-j}}.$

Since ${}_{D}\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$ is a
simplicial model category, homotopy limits over cosimplicial diagrams are
computed as in [Hir03, 18.1.8]. We denote the homotopy limit of the
cosimplicial diagram $\Omega^{\bullet}(M,C,C)$ in
${}_{D}\mathsf{CoMod}_{C}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$ by $\Omega(M,C,C)$,
and we say it is the _two-sided cobar resolution of $M$._ By [Pér20, 3.13], we
have $M\simeq\Omega(M,C,C)$ as a $(D,C)$-bicomodule if $M$ is fibrant as a
left $D$-comodule. Notice that each object in the cosimplicial diagram
$\Omega^{\bullet}(M,C,C)$ is a fibrant right $C$-comodule by Lemma A.8. Thus
$\Omega(M,C,C)$ is a fibrant $(D,C)$-bicomodule by [Hir03, 18.5.2] if $M$ is a
fibrant left $D$-comodule.

######  Definition 3.9.

Let $C$, $D$ and $E$ be simply connected dg-coalgebras over $\Bbbk$. Let $M$
be a $(D,C)$-bicomodule and $N$ be a $(C,E)$-bicomodule. We define the _two-
sided cosimplicial cobar construction of $M$ and $N$_ to be the cosimplicial
object $\Omega^{\bullet}(M,C,N)$ in
${}_{D}\mathsf{CoMod}_{E}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$ given by
$\Omega^{\bullet}(M,C,C)\square_{C}N$. We write $\Omega(M,C,N)$ for the
homotopy limit of $\Omega^{\bullet}(M,C,N)$ in
${}_{D}\mathsf{CoMod}_{E}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$, computed as in
[Hir03, 18.1.8]. As noted in [Pér20], it is equivalent to compute the homotopy
limit in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$.

######  Proposition 3.10.

Let $C$, $D$ and $E$ be simply connected dg-coalgebras over $\Bbbk$. Let $M$
be a $(D,C)$-bicomodule and $N$ be a fibrant $(C,E)$-bicomodule. Then we
obtain an equivalence $\Omega(M,C,C)\square_{C}N\simeq\Omega(M,C,N)$ as
$(D,E)$-bicomodules.

###### Proof.

This argument is similar to that of [Pér20, 4.33] (the cocommutative
requirement there was not needed). ∎

######  Remark 3.11.

By the Dold-Kan correspondence, since $C$ is a simply connected dg-coalgebra,
the two-sided cosimplicial cobar resolution $\Omega(M,C,N)$ from Definition
3.8 is quasi-isomorphic to $\underline{\Omega}(M,C,N)$, the _conormalized
cobar resolution of $M$ and $N$ over $C$_, which we now define. We first
establish the following notation conventions.

  * •

Given $V$ a graded $\Bbbk$-module, we define

$T(V)=\bigoplus_{n\geq 0}V^{\otimes n}.$

Elements in the summands are denoted $v_{1}|\cdots|v_{n}$, where $v_{i}\in V$.

  * •

Let $s^{-1}$ denote the desuspension functor on graded $\Bbbk$-modules where,
for $V=\bigoplus_{i\in\mathbb{Z}}V_{i}$, we define $(s^{-1}V)_{i}=V_{i+1}$.
Given a homogeneous element $v$ in $V$, we write $s^{-1}v$ for the
corresponding element in $s^{-1}V$.

  * •

We denote the kernel of the counit $\varepsilon:C\rightarrow\Bbbk$ by
$\underline{C}$, often referred to as the coideal of $C$.

The conormalized cobar resolution of $C$ is the chain complex
$\underline{\Omega}C:=(T(s^{-1}\underline{C}),d_{\Omega})$ where, if $d$
denotes the differential on $C$, then

$\displaystyle d_{\Omega}(s^{-1}c_{1}|\cdots|s^{-1}c_{n})$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\pm
s^{-1}c_{1}|\cdots|s^{-1}(dc_{j})|\cdots|s^{-1}c_{n}$
$\displaystyle+\sum_{j=1}^{n}\pm
s^{-1}c_{1}|\cdots|s^{-1}{c_{j}}_{(1)}|s^{-1}{c_{j}}_{(2)}|\cdots|s^{-1}c_{n},$

where $\Delta(c_{j})={c_{j}}_{(1)}\otimes{c_{j}}_{(2)}$ denotes the (reduced)
comultiplication of $\underline{C}$ on the element $c_{j}$ using the Sweedler
notation. To determine the signs above, one needs to apply the Koszul rule.
More generally, we define the conormalized cobar resolution of $M$ and $N$
over $C$ to be the chain complex $\underline{\Omega}(M,C,N):=(M\otimes
T(s^{-1}\underline{C})\otimes N,\delta),$ where up to Koszul sign, the
differential $\delta$ is defined as

$\displaystyle\delta(m\otimes s^{-1}c_{1}|\cdots|s^{-1}c_{n}\otimes n)$
$\displaystyle=$ $\displaystyle dm\otimes
s^{-1}c_{1}|\cdots|s^{-1}c_{n}\otimes n$ $\displaystyle\pm m_{(0)}\otimes
s^{-1}m_{(1)}|s^{-1}c_{1}|\cdots|s^{-1}c_{n}\otimes n$ $\displaystyle\pm
m\otimes d_{\Omega}\Big{(}s^{-1}c_{1}|\cdots|s^{-1}c_{n}\Big{)}\otimes n$
$\displaystyle\pm m\otimes s^{-1}c_{1}|\cdots|s^{-1}c_{n}\otimes dn$
$\displaystyle\pm m\otimes s^{-1}c_{1}|\cdots|s^{-1}c_{n}|s^{-1}n_{(1)}\otimes
n_{(0)},$

where $d$ denotes either the differential on $M$ or $N$, and $m\mapsto
m_{(0)}\otimes m_{(1)}$ denotes the coaction of $M\rightarrow M\otimes C$
applied to an element $m$, and $n\mapsto n_{(1)}\otimes n_{(0)}$ denotes the
coaction of $N\rightarrow C\otimes N$ applied to an element $n$.

By the Dold-Kan correspondence, there is an isomorphism of categories between
cosimplicial objects and non-positive cochain complexes of bicomodules given
by the conormalization functor

$N^{*}:\left({}_{D}\mathsf{CoMod}_{E}({\mathsf{Ch}_{\Bbbk}^{\geq
0}})\right)^{\Delta}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathsf{coCh}^{\leq
0}\left({}_{D}\mathsf{CoMod}_{E}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})\right).$

Therefore, we denote $N^{*}(\Omega^{\bullet}(M,C,N))$ by
$\underline{\Omega}^{\bullet}(M,C,N)$. It is a double complex, and one can
show that its total complex is precisely $\underline{\Omega}(M,C,N)$. A word
of warning however: a double complex in general has two possible
totalizations, one given using coproducts and one using products (see [Wei94,
1.2.6]). The product-total complex of $\underline{\Omega}^{\bullet}(M,C,N)$ is
quasi-isomorphic to the homotopy limit of $\Omega^{\bullet}(M,C,N)$, i.e.,
$\Omega(M,C,N)$ (see [Bun12, 4.23] for instance). On the other hand, the
coproduct-total complex of $\underline{\Omega}^{\bullet}(M,C,N)$ is
$\underline{\Omega}(M,C,N)$. Of course, if the double complex is bounded,
these are equal. For instance, when $C$ is simply-connected, then
$\underline{\Omega}^{-q}(M,C,N)_{p}=0$ for $0\leq p\leq 2q-1$, and is
therefore bounded. Thus when $C$ is simply-connected, we obtain a quasi-
isomorphism

$\Omega(M,C,N)\simeq\underline{\Omega}(M,C,N).$

But in general, if $C$ is not simply-connected, no such claim can be made.

######  Remark 3.12.

Let $C$ be a simply-connected coalgebra in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$.
Let $M$ and $N$ be left and right $C$-comodules respectively. As noted in
[Rav86, A1.2.12], for all $i\geq 0$ we have an isomorphism

$\mathsf{CoTor}_{C}^{i}(M,N)\cong
H^{i}\left(\underline{\Omega}^{\bullet}(M,C,N)\right).$

Therefore, on the homotopy category of comodules, we have a derived cotensor
product, which we denote by $\widehat{\square}$. In fact,
$M\widehat{\square}_{C}N$ is quasi-isomorphic to the two-sided cobar
resolution $\Omega(M,C,N)$, see Corollary A.18. The derived cotensor preserves
any homotopy coherent coactions, see [Pér20].

Having established the necessary definitions for each of the components, we
now define the appropriate bicategory for the derived bicomodule setting.

######  Definition 3.13.

Define $\mathcal{D}\mathsf{CoMod}$ to be the bicategory whose $0$-cells are
simply connected dg-coalgebras over $\Bbbk$, and whose 1-cells, 2-cells, and
vertical compositions are given by the homotopy category of
${}_{C}\mathsf{CoMod}_{D}({\mathsf{Ch}_{\Bbbk}^{\geq 0}})$. The unit $U_{C}$
is the fibrant $(C,C)$-bicomodule $C$, and horizontal composition is given by
the derived cotensor product of bicomodules. Given a fibrant
$(C,D)$-bicomodule $M$ and a fibrant $(D,E)$-bicomodule $N$, their horizontal
composition $M\odot N$ is the fibrant $(C,E)$-bicomodule

$M\square_{D}N\simeq M\widehat{\square}_{D}N\simeq\Omega(M,D,N).$

If $P$ is a fibrant $(E,F)$-bicomodule, then we define the natural
isomorphisms:

$a:(M\square_{D}N)\square_{E}P\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}M\square_{D}(N\square_{E}P),$
$\ell:C\square_{C}M\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}M,\quad
r:M\square_{D}D\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}M,$

as follows. The isomorphism $a$ follows from the natural isomorphism
$(M\otimes N)\otimes P\cong M\otimes(N\otimes P)$, and therefore automatically
respects the pentagon identity. The natural isomorphisms $\ell$ and $r$ are
induced by the counits $C\rightarrow\Bbbk$ and $D\rightarrow\Bbbk$
respectively, and the triangle identities follow for the cotensor products
since they hold for the tensor product.

Since we shall need it in next section, we provide an explicit definition of
$a$, $\ell$, and $r$ from above, where we instead use the cobar construction
as a model for the derived cotensor product. The associative equivalence

$a:\Omega(\Omega(M,D,N),E,P)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\Omega(M,D,\Omega(N,E,P))$

is induced by an isomorphism of cosimplicial objects
$\Omega^{\bullet}(\Omega^{\bullet}(M,D,N),E,P)\cong\Omega^{\bullet}(M,D,\Omega^{\bullet}(N,E,P))$,
using Corollary A.21 from Appendix A. Indeed, the $(i,j)$-spot corresponds to
the $(j,i)$-spot up to isomorphism

$(M\otimes D^{\otimes j}\otimes N)\otimes E^{\otimes i}\otimes
P\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}M\otimes D^{\otimes
j}\otimes(N\otimes E^{\otimes i}\otimes P)$

from the usual associative isomorphism. The equivalences
$\ell:\Omega(C,C,M)\stackrel{{\scriptstyle\simeq}}{{\rightarrow}}M$ and
$r:\Omega(M,D,D)\stackrel{{\scriptstyle\simeq}}{{\rightarrow}}M$ are defined
as in Definition 3.8, and are induced by the counits. For instance, the map
$C^{\otimes n+1}\otimes M\rightarrow M$ is induced by repeatedly applying the
counit on $C$.

##  4\. CoHochschild homology as a shadow

Here we show our main results, that coHochschild homology provides a shadow
structure on the previously defined bicategories of bicomodules.

######  Definition 4.1 ([Pon10, PS13]).

A _shadow functor_ for a bicategory $\mathcal{B}$ consists of functors

${\langle\langle}-{\rangle\rangle}_{C}:\mathcal{B}(C,C)\to\textbf{T}$

for every $C\in\mathcal{B}$ and some fixed category T equipped with a natural
isomorphism for $M\in\mathcal{B}(C,D)$, $N\in\mathcal{B}(D,C)$:

$\displaystyle\theta:{\langle\langle}M\odot
N{\rangle\rangle}_{C}\xlongrightarrow{\cong}{\langle\langle}N\odot
M{\rangle\rangle}_{D}.$

For $P\in\mathcal{B}(C,C)$, these functors must satisfy the following
commutative diagrams

${{\langle\langle}(M\odot N)\odot
P{\rangle\rangle}_{C}}$${{\langle\langle}P\odot(M\odot
N){\rangle\rangle}_{C}}$${{\langle\langle}(P\odot M)\odot
N{\rangle\rangle}_{C}}$${{\langle\langle}M\odot(N\odot
P){\rangle\rangle}_{C}}$${{\langle\langle}(N\odot P)\odot
M{\rangle\rangle}_{D}}$${{\langle\langle}N\odot(P\odot
M){\rangle\rangle}_{D},}$$\scriptstyle{\theta}$$\scriptstyle{{\langle\langle}a{\rangle\rangle}}$$\scriptstyle{{\langle\langle}a{\rangle\rangle}}$$\scriptstyle{\theta}$$\scriptstyle{{\langle\langle}a{\rangle\rangle}}$$\scriptstyle{\theta}$
${{\langle\langle}P\odot
U_{C}{\rangle\rangle}_{C}}$${{\langle\langle}U_{C}\odot
P{\rangle\rangle}_{C}}$${{\langle\langle}P\odot
U_{C}{\rangle\rangle}_{C}}$${{\langle\langle}P{\rangle\rangle}_{C}.}$$\scriptstyle{\theta}$$\scriptstyle{{\langle\langle}r{\rangle\rangle}}$$\scriptstyle{{\langle\langle}\ell{\rangle\rangle}}$$\scriptstyle{{\langle\langle}r{\rangle\rangle}}$

In this case, we say $\mathcal{B}$ is a _shadowed bicategory_ , and we write
$(\mathcal{B},{\langle\langle}-{\rangle\rangle})$ for the bicategory and its
shadow.

######  Example 4.2.

We can now consider shadows for the bicategories that we introduced in
Examples 3.2 and 3.3. For a ring $R$, the $0^{th}$ Hochschild homology,
$\mathrm{HH}_{0}(R,-)$, is a shadow on the bicategory with 1- and 2-cells from
the category of $(R,R)$-bimodules to the category of abelian groups,
$\mathsf{Ab}$:

$\displaystyle{\langle\langle}-{\rangle\rangle}_{R}:_{R}\\!\mathsf{Mod}_{R}$
$\displaystyle\to\mathsf{Ab}$ $\displaystyle M$ $\displaystyle\mapsto
R\otimes_{R\otimes R^{\mathsf{op}}}M\cong\mathrm{HH}_{0}(R,M).$

More generally, a Dennis-Waldhausen Morita argument shows that Hochschild
homology, $\mathrm{HH}(R,-)$, is a shadow in the derived setting [Wal79].
Further, [BM12] shows that topological Hochschild homology,
$\mathrm{THH}(R,-)$, is a shadow to the homotopy category of spectra:

$\displaystyle{\langle\langle}-{\rangle\rangle}_{R}:\textup{Ho}(_{R}\mathsf{Mod}_{R})$
$\displaystyle\to\textup{Ho}(\textup{Sp})$ $\displaystyle M$
$\displaystyle\mapsto\mathrm{THH}(R,M).$

Since (topological) Hochschild homology provides a bicategorical shadow for
the setting of modules, we want to consider the analogue of this construction
for the context of comodules. Work of Doi defines coHochschild homology,
denoted $\mathrm{coHH}$, as an invariant of coalgebras analogous to Hochschild
homology.

######  Definition 4.3 ([Doi81]).

For a commutative ring $R$, a coassociative, counital $R$-coalgebra $C$, and a
$(C,C)$-bicomodule $M$, build the cochain complex $\mathcal{H}(M,C)$:

$\cdots\xlongleftarrow{}M\otimes_{R}C\otimes_{R}C\xlongleftarrow{}M\otimes_{R}C\xlongleftarrow{}M\xlongleftarrow{}0,$

as follows. Let $\mathcal{H}^{r}(M,C)=M\otimes C^{\otimes r}$ for $r\geq 0$
with coboundary map
$\delta^{r}:\mathcal{H}^{r}(M,C)\rightarrow\mathcal{H}^{r+1}(M,C)$ defined by

$\delta^{r}=\sum_{i=0}^{r+1}(-1)^{i}d_{i},$

for $d_{i}$ given by

$\displaystyle d_{i}$
$\displaystyle=\begin{cases}\rho\otimes\mathsf{id}_{C}^{\otimes r}&i=0\\\
\mathsf{id}_{M}\otimes\mathsf{id}_{C}^{\otimes
i-1}\otimes\Delta\otimes\mathsf{id}_{C}^{\otimes(r-i)}&1\leq i\leq r\\\
\tilde{t}\circ(\lambda\otimes\mathsf{id}_{C}^{\otimes r})&i=r+1,\end{cases}$

where $\rho:M\rightarrow M\otimes_{R}C$ denotes the right coaction,
$\lambda:M\rightarrow C\otimes_{R}M$ denotes the left coaction, and
$\tilde{t}$ is the map that twists the first factor to the last. Then the
$q^{th}$-coHochschild homology of $C$ with coefficients in $M$ is given by the
cohomology of the cochain complex

$\mathrm{coHH}_{q}(M,C):=H^{q}(\mathcal{H}(M,C)).$

######  Remark 4.4.

We can see that $\mathcal{H}(M,C)=\underline{\Omega}(M,C^{e},C)$ as in Remark
3.11, where $C^{e}=C\otimes_{R}C^{\mathsf{op}}$, and in particular we get that
$\mathrm{coHH}_{q}(M,C)\cong\mathsf{CoTor}^{q}_{C^{e}}(M,C)$, see Definition
1.13 and Remark 3.12. In particular $\mathrm{coHH}_{0}(M,C)\cong
M\square_{C^{e}}C$.

######  Theorem 4.5.

The $0^{th}$ coHochschild homology, $\mathrm{coHH}_{0}$, is a shadow on the
bicategory $\mathsf{CoMod}$. That is, it gives a family of functors

$\displaystyle\mathrm{coHH}_{0}(-,C):{}_{C}\mathsf{CoMod}_{C}$
$\displaystyle\to\mathsf{Mod}_{\Bbbk}$ $\displaystyle M$
$\displaystyle\mapsto\mathrm{coHH}_{0}(M,C)$

that satisfy the required shadow properties.

###### Proof.

Let $C$ and $D$ be coalgebras over $\Bbbk$. Given $M$ a $(C,D)$-bicomodule and
$N$ a $(D,C)$-bicomodule, we need to show that we have an isomorphism

$\theta:\mathrm{coHH}_{0}(M\square_{D}N,C)\longrightarrow\mathrm{coHH}_{0}(N\square_{C}M,D).$

Using Sweedler notation, we denote the coactions on $M$ as follows:

$\displaystyle M$ $\displaystyle\longrightarrow C\otimes M,$ $\displaystyle M$
$\displaystyle\longrightarrow M\otimes D$ $\displaystyle m$
$\displaystyle\longmapsto m^{C}_{(1)}\otimes m^{C}_{(0)}$ $\displaystyle m$
$\displaystyle\longmapsto m^{D}_{(0)}\otimes m^{D}_{(1)},$

and the coactions on $N$ as follows:

$\displaystyle N$ $\displaystyle\longrightarrow D\otimes N,$ $\displaystyle N$
$\displaystyle\longrightarrow N\otimes C$ $\displaystyle n$
$\displaystyle\longmapsto n^{D}_{(1)}\otimes n^{D}_{(0)}$ $\displaystyle n$
$\displaystyle\longmapsto n^{C}_{(0)}\otimes n^{C}_{(1)}.$

Recall that $\mathrm{coHH}_{0}(M\square_{D}N,C)$ is defined as the kernel of

$\delta^{0}:M\square_{D}N\rightarrow(M\square_{D}N)\otimes C.$

The desired isomorphism will be induced by

$\displaystyle\tau:M\otimes N$ $\displaystyle\longrightarrow N\otimes M$
$\displaystyle m\otimes n$ $\displaystyle\longmapsto n\otimes m.$

We need to verify that if we restrict $\tau$ to
$\mathrm{coHH}_{0}(M\square_{D}N,C)$, we indeed corestrict to
$\mathrm{coHH}_{0}(N\square_{C}M,D)$. In other words, we need to verify that
if $m\otimes n\in\mathrm{coHH}_{0}(M\square_{D}N,C)$, then $n\otimes
m\in\mathrm{coHH}_{0}(N\square_{C}M,D)$. Notice that because $m\otimes
n\in\mathrm{coHH}_{0}(M\square_{D}N,C)$, it follows that

  1. (1)

$m^{D}_{(0)}\otimes m^{D}_{(1)}\otimes n=m\otimes n^{D}_{(1)}\otimes
n^{D}_{(0)}$, since $m\otimes n\in M\square_{D}N$;

  2. (2)

$m\otimes n^{C}_{(0)}\otimes n^{C}_{(1)}=m^{C}_{(0)}\otimes n\otimes
m^{C}_{(1)}$, since $m\otimes n\in\ker(\delta^{0})$.

To see that $n\otimes m\in\mathrm{coHH}_{0}(N\square_{C}M,D)$, notice that
$n\otimes m\in N\square_{C}M$ follows from (2) above, while $n\otimes
m\in\ker(\delta^{0})$ follows from (1) above. Therefore we have obtained the
desired homomorphism, $\theta$. An analogous argument defines its inverse, and
thus $\theta$ is an isomorphism.

Next we must show that for a $(C,C)$-bicomodule $P$, the following diagram is
commutative:

${{\mathrm{coHH}_{0}((M\square_{D}N)\square_{C}P,C)}}$${{\mathrm{coHH}_{0}(P\square_{C}(M\square_{D}N),C)}}$${{\mathrm{coHH}_{0}((P\square_{C}M)\square_{D}N,C)}}$${{\mathrm{coHH}_{0}(M\square_{D}(N\square_{C}P),C)}}$${{\mathrm{coHH}_{0}((N\square_{C}P)\square_{C}M,D)}}$${{\mathrm{coHH}_{0}(N\square_{C}(P\square_{C}M),D)}}$$\scriptstyle{{\langle\langle}a{\rangle\rangle}}$$\scriptstyle{\theta}$$\scriptstyle{\theta}$$\scriptstyle{{\langle\langle}a{\rangle\rangle}}$$\scriptstyle{{\langle\langle}a{\rangle\rangle}}$$\scriptstyle{\theta}$

We check its commutativity directly by applying the definition of $\theta$
given above and $a$ from Definition 3.5:

${(m\otimes n)\otimes p}$${p\otimes(m\otimes n)}$${(p\otimes m)\otimes
n.}$${m\otimes(n\otimes p)}$${(n\otimes p)\otimes m}$${n\otimes(p\otimes m)}$

Similarly, we need to check the commutativity of the diagram below:

${{\mathrm{coHH}_{0}(P\square_{C}C,C)}}$${{\mathrm{coHH}_{0}(C\square_{C}P,C)}}$${{\mathrm{coHH}_{0}(P\square_{C}C,C)}}$${{\mathrm{coHH}_{0}(P,C)}}$$\scriptstyle{{\langle\langle}\ell{\rangle\rangle}}$$\scriptstyle{\theta}$$\scriptstyle{{\langle\langle}r{\rangle\rangle}}$$\scriptstyle{\theta}$$\scriptstyle{{\langle\langle}r{\rangle\rangle}}$

This follows again by applying the definitions of $\theta$, $r$, and $\ell$
(see Definition 3.5):

${p\otimes c}$${c\otimes p}$${p\otimes c}$${\varepsilon(c)p}$

This proves that the $0^{th}$ coHochschild homology is a shadow in this
bicategorical setting. ∎

Having established that $\mathrm{coHH}_{0}$ is a bicategorical shadow on
$\mathsf{CoMod}$, we now consider the derived setting. Recall that
$\underline{\Omega}$ of Remark 3.11 is not invariant under quasi-isomorphisms
in general. Therefore particular care is required in order to show that
$\mathrm{coHH}$ is a shadow in the derived setting. To do so, we must instead
consider coalgebras in chain complexes that are simply connected. Extending
the definitions of [HS21] and [BGH+18], the first author introduced in [Kla22,
2.8] the notion of coHochschild homology with coefficients for any model
category with a symmetric monoidal structure.

######  Definition 4.6.

Let $(\mathsf{M},\otimes,\mathbb{I})$ be a symmetric monoidal category with a
model structure, and let $C\in\mathsf{M}$ be a coalgebra with coassociative
comultiplication $\Delta:C\to C\otimes C$ and counit
$\varepsilon:C\to\mathbb{I}$. Further, let $M$ be a $(C,C)$-bicomodule with
left and right coactions $\lambda:M\to C\otimes M$ and $\rho:M\to M\otimes C$
respectively. Define $\mathrm{coHH}^{\mathsf{M}}(M,C)^{\bullet}$ to be the
cosimplicial object with $r$-simplices
$\mathrm{coHH}^{\mathsf{M}}(M,C)^{r}=M\otimes C^{\otimes r}$, with coface maps

$\displaystyle d_{i}$
$\displaystyle=\begin{cases}\rho\otimes\mathsf{id}_{C}^{\otimes r}&i=0\\\
\mathsf{id}_{M}\otimes\mathsf{id}_{C}^{\otimes
i-1}\otimes\Delta\otimes\mathsf{id}_{C}^{\otimes(r-i)}&1\leq i\leq r\\\
\tilde{t}\circ(\lambda\otimes\mathsf{id}_{C}^{\otimes r})&i=r+1,\end{cases}$

where $\tilde{t}$ is the map that twists the first factor to the last, and
with codegeneracy maps $s_{i}:M\otimes C^{\otimes(r+1)}\to M\otimes C^{\otimes
r}$, for $0\leq i\leq r$,

$s_{i}=\mathsf{id}_{M}\otimes\mathsf{id}_{C}^{\otimes
i}\otimes\varepsilon\otimes\mathsf{id}_{C}^{\otimes r-i}.$

This gives a cosimplicial object of the form

${\cdots}$${{M\otimes C\otimes C}}$${{M\otimes C}}$${M.}$

The coHochschild homology in $\mathsf{M}$ of the coalgebra $C$ with
coefficients in $M$ is then defined by

$\mathrm{coHH}^{\mathsf{M}}(M,C)=\mathsf{holim}_{\Delta}\big{(}\mathrm{coHH}^{\mathsf{M}}(M,C)^{\bullet}\big{)}.$

We shall give a higher categorical version of coHochschild homology in
Appendix B. Notably, we show in Proposition B.2 that the two definitions
agree.

######  Remark 4.7.

In the following, we shall always assume
$\mathsf{M}={\mathsf{Ch}_{\Bbbk}^{\geq 0}}$, and thus will omit it from the
notation. We show in Proposition B.4 that
$\mathrm{coHH}(M,C)\simeq\Omega(M,C^{e},C)$ for $C$ simply connected and $M$ a
fibrant $(C,C)$-bicomodule.

######  Theorem 4.8.

CoHochschild homology defines a shadow on the bicategory of derived
bicomodules over simply connected coalgebras $\mathcal{D}\mathsf{CoMod}$.

###### Proof.

Let $C$ and $D$ be simply connected coalgebras in ${\mathsf{Ch}_{\Bbbk}^{\geq
0}}$. Given $M$ a fibrant $(C,D)$-bicomodule, and $N$ a fibrant
$(D,C)$-bicomodule, we need to show that we have a quasi-isomorphism:

$\theta:\mathrm{coHH}(M\widehat{\square}_{D}N,C)\longrightarrow\mathrm{coHH}(N\widehat{\square}_{C}M,D).$

As $M$ and $N$ are fibrant, the derived cotensor $M\widehat{\square}_{D}N$ is
modeled by $M\square_{D}N$ or $\Omega(M,D,N)$. We choose the cobar
construction as a model: it automatically gives us the desired quasi-
isomorphism at the cost of some combinatorics. To provide this equivalence, we
apply Corollary A.22 from Appendix A. In particular, we claim that $\theta$ is
induced by an isomorphism of bicosimplicial objects:

$\theta:\mathrm{coHH}(\Omega^{\bullet}(M,D,N),C)^{\bullet}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathrm{coHH}(\Omega^{\bullet}(N,C,M),D)^{\bullet}.$

This is the dual to the Dennis-Waldhausen Morita argument. Indeed, we define
$\theta$ by the usual shuffling isomorphism in the bicosimplicial map below,
in which the rows correspond to the two-sided cobar construction, while the
columns correspond to the coHochschild complex. For ease of understanding, the
diagrams below have been color-coded to illustrate this isomorphism via
shuffling.

${{\color[rgb]{0.83921568627451,0.6,0.36078431372549}\definecolor[named]{pgfstrokecolor}{rgb}{0.83921568627451,0.6,0.36078431372549}M\otimes
N}}$${{\color[rgb]{0.176470588235294,0.784313725490196,0.247058823529412}\definecolor[named]{pgfstrokecolor}{rgb}{0.176470588235294,0.784313725490196,0.247058823529412}M\otimes
N\otimes
C}}$${{\color[rgb]{0.36078431372549,0.36078431372549,0.83921568627451}\definecolor[named]{pgfstrokecolor}{rgb}{0.36078431372549,0.36078431372549,0.83921568627451}M\otimes
N\otimes C^{\otimes
2}}}$${\cdots}$${{\color[rgb]{0.83921568627451,0.36078431372549,0.36078431372549}\definecolor[named]{pgfstrokecolor}{rgb}{0.83921568627451,0.36078431372549,0.36078431372549}M\otimes
D\otimes
N}}$${{\color[rgb]{0.188235294117647,0.752941176470588,0.752941176470588}\definecolor[named]{pgfstrokecolor}{rgb}{0.188235294117647,0.752941176470588,0.752941176470588}M\otimes
D\otimes N\otimes
C}}$${{\color[rgb]{1,0.462745098039216,0.0196078431372549}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.462745098039216,0.0196078431372549}M\otimes
D\otimes N\otimes C^{\otimes
2}}}$${\cdots}$${{\color[rgb]{0.83921568627451,0.36078431372549,0.83921568627451}\definecolor[named]{pgfstrokecolor}{rgb}{0.83921568627451,0.36078431372549,0.83921568627451}M\otimes
D^{\otimes 2}\otimes
N}}$${{\color[rgb]{0.858823529411765,0.0235294117647059,0.286274509803922}\definecolor[named]{pgfstrokecolor}{rgb}{0.858823529411765,0.0235294117647059,0.286274509803922}M\otimes
D^{\otimes 2}\otimes N\otimes
C}}$${{\color[rgb]{0.96078431372549,0.8,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.96078431372549,0.8,0}M\otimes
D^{\otimes 2}\otimes N\otimes C^{\otimes
2}}}$${\cdots}$${\vdots}$${\vdots}$${\vdots}$${{\color[rgb]{0.83921568627451,0.6,0.36078431372549}\definecolor[named]{pgfstrokecolor}{rgb}{0.83921568627451,0.6,0.36078431372549}N\otimes
M}}$${{\color[rgb]{0.83921568627451,0.36078431372549,0.36078431372549}\definecolor[named]{pgfstrokecolor}{rgb}{0.83921568627451,0.36078431372549,0.36078431372549}N\otimes
M\otimes
D}}$${{\color[rgb]{0.83921568627451,0.36078431372549,0.83921568627451}\definecolor[named]{pgfstrokecolor}{rgb}{0.83921568627451,0.36078431372549,0.83921568627451}N\otimes
M\otimes D^{\otimes
2}}}$${\cdots}$${{\color[rgb]{0.176470588235294,0.784313725490196,0.247058823529412}\definecolor[named]{pgfstrokecolor}{rgb}{0.176470588235294,0.784313725490196,0.247058823529412}N\otimes
C\otimes
M}}$${{\color[rgb]{0.188235294117647,0.752941176470588,0.752941176470588}\definecolor[named]{pgfstrokecolor}{rgb}{0.188235294117647,0.752941176470588,0.752941176470588}N\otimes
C\otimes M\otimes
D}}$${{\color[rgb]{0.858823529411765,0.0235294117647059,0.286274509803922}\definecolor[named]{pgfstrokecolor}{rgb}{0.858823529411765,0.0235294117647059,0.286274509803922}N\otimes
C\otimes M\otimes D^{\otimes
2}}}$${\cdots}$${{\color[rgb]{0.36078431372549,0.36078431372549,0.83921568627451}\definecolor[named]{pgfstrokecolor}{rgb}{0.36078431372549,0.36078431372549,0.83921568627451}N\otimes
C^{\otimes 2}\otimes
M}}$${{\color[rgb]{1,0.462745098039216,0.0196078431372549}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.462745098039216,0.0196078431372549}N\otimes
C^{\otimes 2}\otimes M\otimes
D}}$${{\color[rgb]{0.96078431372549,0.8,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.96078431372549,0.8,0}N\otimes
C^{\otimes 2}\otimes M\otimes D^{\otimes
2}}}$${\cdots}$${\vdots}$${\vdots}$${\vdots}$$\scriptstyle{\theta}$

In particular, the $(i,j)$-spot in
$\mathrm{coHH}(\Omega^{\bullet}(M,D,N),C)^{\bullet}$ is isomorphic to the
$(j,i)$-spot in $\mathrm{coHH}(\Omega^{\bullet}(N,C,M),D)^{\bullet}$:

$\theta:M\otimes D^{\otimes i}\otimes N\otimes C^{\otimes
j}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}N\otimes C^{\otimes
j}\otimes M\otimes D^{\otimes i},$

which incorporates a Koszul sign. Moreover, the map $\theta$ is compatible
with the cofaces and codegeneracies. By [Hir03, 18.5.3], we obtain the
equivalence

$\theta:\mathrm{coHH}(\Omega(M,D,N),C)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\mathrm{coHH}(\Omega(N,C,M),D),$

using the model of homotopy limit as in [Hir03, 18.1.8], providing the desired
quasi-isomorphism $\theta$.

Given a fibrant $(C,C)$-bicomodule $P$, we next must show that the following
diagram commutes:

${{\mathrm{coHH}(\Omega(\Omega(M,D,N),C,P),C)}}$${{\mathrm{coHH}(\Omega(P,C,\Omega(M,D,N)),C)}}$${{\mathrm{coHH}(\Omega(\Omega(P,C,M),D,N),C)}}$${{\mathrm{coHH}(\Omega(M,D,\Omega(N,C,P)),C)}}$${{\mathrm{coHH}(\Omega(\Omega(N,C,P),C,M),D)}}$${{\mathrm{coHH}(\Omega(N,C,\Omega(P,C,M)),D).}}$$\scriptstyle{{\langle\langle}a{\rangle\rangle}}$$\scriptstyle{\theta}$$\scriptstyle{\theta}$$\scriptstyle{{\langle\langle}a{\rangle\rangle}}$$\scriptstyle{{\langle\langle}a{\rangle\rangle}}$$\scriptstyle{\theta}$

This follows a similar argument as above. We need to consider an isomorphism
of “tri-cosimplicial” isomorphisms in which we keep track of the swapping of
the grading:

${\big{(}(M\otimes D^{\otimes k}\otimes N)\otimes C^{\otimes j}\otimes
P\big{)}\otimes C^{\otimes i}}$${\big{(}P\otimes C^{\otimes i}\otimes(M\otimes
D^{\otimes k}\otimes N)\big{)}\otimes C^{\otimes j}}$${\big{(}(P\otimes
C^{\otimes i}\otimes M)\otimes D^{\otimes k}\otimes N\big{)}\otimes C^{\otimes
j}.}$${\big{(}M\otimes D^{\otimes k}\otimes(N\otimes C^{\otimes j}\otimes
P)\big{)}\otimes C^{\otimes i}}$${\big{(}(N\otimes C^{\otimes j}\otimes
P)\otimes C^{\otimes i}\otimes M\big{)}\otimes D^{\otimes
k}}$${\big{(}N\otimes C^{\otimes j}\otimes(P\otimes C^{\otimes i}\otimes
M)\big{)}\otimes D^{\otimes
k}}$$\scriptstyle{\theta}$$\scriptstyle{{\langle\langle}a{\rangle\rangle}}$$\scriptstyle{{\langle\langle}a{\rangle\rangle}}$$\scriptstyle{\theta}$$\scriptstyle{{\langle\langle}a{\rangle\rangle}}$$\scriptstyle{\theta}$

We then need to check if the following diagram commutes:

${{\mathrm{coHH}(\Omega(P,C,C),C)}}$${{\mathrm{coHH}(\Omega(C,C,P),C)}}$${\mathrm{coHH}({\Omega(P,C,C),C)}}$${{\mathrm{coHH}(P,C)}.}$$\scriptstyle{{\langle\langle}\ell{\rangle\rangle}}$$\scriptstyle{\theta}$$\scriptstyle{{\langle\langle}r{\rangle\rangle}}$$\scriptstyle{\theta}$$\scriptstyle{{\langle\langle}r{\rangle\rangle}}$

We will prove the left triangle is commutative, but the argument for the right
triangle will follow analogously. We apply the definition of $\theta$, $\ell$,
and $r$ on the bicosimplicial objects:

${(P\otimes C^{\otimes\bullet}\otimes C)\otimes C^{\otimes\ast}}$${(C\otimes
C^{\otimes\ast}\otimes P)\otimes C^{\otimes\bullet}}$${P\otimes
C^{\otimes\ast}}$${P\otimes
C^{\otimes\bullet}.}$$\scriptstyle{{\langle\langle}r{\rangle\rangle}}$$\scriptstyle{\theta}$$\scriptstyle{{\langle\langle}\ell{\rangle\rangle}}$$\scriptstyle{\cong}$

Here we kept track of the different gradings by $\bullet$ and $\ast$.
Therefore, once we apply the homotopy limit, we obtain the desired result. ∎

##  5\. Duality

To apply the trace of a shadow in the setting of a bicategory, we must have a
notion of a dualizable object as endomorphisms on dualizable objects
generalize square matrices.

######  Definition 5.1.

Let $\mathcal{B}$ be a bicategory. A $1$-cell $M\in\mathcal{B}(C,D)$ is _right
dualizable_ if there is another $1$-cell $M^{\star}$ in $\mathcal{B}(D,C)$,
with $2$-cells $\eta:U_{C}\rightarrow M\odot M^{\star}$ and
$\varepsilon:M^{\star}\odot M\rightarrow U_{D}$, called _coevaluation_ and
_evaluation_ respectively, such that the compositions in $\mathcal{B}(C,D)$
and $\mathcal{B}(D,C)$ respectively are the identity $2$-cells:

${M\cong{U_{C}\odot M}}$${{(M\odot M^{\star})\odot M\cong
M\odot(M^{\star}\odot M)}}$${{M\odot U_{D}}\cong
M,}$$\scriptstyle{\eta\odot\mathsf{id}}$$\scriptstyle{\mathsf{id}\odot\varepsilon}$
${M^{\star}\cong{M^{\star}\odot U_{C}}}$${{M^{\star}\odot(M\odot
M^{\star})\cong(M^{\star}\odot M)\odot M^{\star}}}$${{U_{D}\odot
M^{\star}}\cong
M^{\star}.}$$\scriptstyle{\mathsf{id}\odot\eta}$$\scriptstyle{\varepsilon\odot\mathsf{id}}$

We call $M^{\star}$ the _right dual_ of $M$ and say that $(M,M^{\star})$ form
a _dual pair_ in $\mathcal{B}$.

For instance, in the bicategory of bimodules, an $(R,S)$-bimodule $M$ is right
dualizable if and only if it is finitely generated and projective as a right
$S$-module, and its dual is given by its linear dual $\mathsf{Hom}_{S}(M,S)$,
see [PS13, 6.1]. We shall obtain a very similar result for bicomodules with
the subtlety that our bicategory is not “closed” (see [Pon10, 4.1.4]), and
thus we cannot recognize dualizable objects (as in [Pon10, 4.3.3]) since we
are not provided with an internal hom. Our bicategories will almost be “co-
closed” thanks to the introduction of a cohom functor (Definition 2.6).

######  Example 5.2.

Consider the bicategory $\mathsf{CoMod}$ of Definition 3.5. A comodule $M$ in
${}_{C}\mathsf{CoMod}_{\Bbbk}$ is right dualizable if it is quasi-finite
(Definition 2.6) and injective as a left $C$-comodule. The dual of $M$ is then
the right $C$-comodule $M^{\star}:=h_{C}(M,C)$, together with the coevaluation
$\eta:C\rightarrow M\otimes h_{C}(M,C)$ and evaluation
$\varepsilon:h_{C}(M,C)\square_{C}M\rightarrow\Bbbk$ as in Definition 2.9. The
desired triangle identities follow from the adjunction of the cohom functor.

######  Example 5.3.

In Example 5.2, recall an example of a quasi-finite left $C$-comodule $M$ is
given by $M=C\otimes V$, where $V$ is a dualizable $\Bbbk$-module (i.e.,
finitely generated) by Example 2.7. We obtain $M^{\star}=h_{C}(C\otimes
V,C)\cong V^{*}\otimes C$. We can then explicitly define the coevaluation and
evaluation. Let us denote the comultiplication and counit of $C$ by
$\Delta_{C}:C\rightarrow C\otimes C$ and $\varepsilon_{C}:C\rightarrow\Bbbk$
respectively. Similarly, denote the coevaluation and evaluation of the
dualizable $\Bbbk$-module $V$ by $\eta_{V}:\Bbbk\rightarrow V\otimes V^{*}$
and $\varepsilon_{V}:V^{*}\otimes V\rightarrow\Bbbk$ respectively. Then the
coevaluation $\eta:C\rightarrow M\square_{\Bbbk}M^{\star}\cong(C\otimes
V)\otimes(V^{*}\otimes C)$ is the composite

${C}$${C\otimes C\cong C\otimes\Bbbk\otimes C}$${C\otimes(V\otimes
V^{*})\otimes
C.}$$\scriptstyle{\Delta_{C}}$$\scriptstyle{\mathsf{id}\otimes\eta_{V}\otimes\mathsf{id}}$

Explicitly, if we write $(e_{1},\ldots,e_{n})$ for a basis of $V$, and
$(e^{*}_{1},\ldots,e_{n}^{*})$ for the dual basis of $V^{*}$, and write
$\Delta_{C}(c)=c_{(1)}\otimes c_{(2)}$, we obtain the formula

$c_{(1)}\otimes\left(\sum_{i=1}^{n}e_{i}\otimes e^{*}_{i}\right)\otimes
c_{(2)}.$

The evaluation $\varepsilon:M^{*}\square_{C}M\cong V^{*}\otimes C\otimes
V\rightarrow\Bbbk$ is then the composite

${V^{*}\otimes C\otimes V}$${V^{*}\otimes\Bbbk\otimes V\cong V^{*}\otimes
V}$${\Bbbk.}$$\scriptstyle{\mathsf{id}\otimes\varepsilon_{C}\otimes\mathsf{id}}$$\scriptstyle{\varepsilon_{V}}$

Explicitly, we obtain the formula

$\varepsilon(f\otimes c\otimes v)=\varepsilon_{C}(c)f(v).$

######  Example 5.4.

We now generalize Example 5.2. A comodule $M$ in ${}_{C}\mathsf{CoMod}_{D}$ is
right dualizable whenever it is quasi-finite and injective as a left
$C$-comodule. Define $M^{\star}=h_{C}(M,C)$, which is a $(D,C)$-bicomodule.
Indeed, by [Tak77b, 1.8], for any $(C,D)$-bicomodule $M$ such that $M$ is
quasi-finite as a left $C$-comodule, and any left $C$-comodule $N$, we have
that $h_{C}(M,N)$ is a left $D$-comodule with coaction $h_{C}(M,N)\rightarrow
D\otimes h_{C}(M,N)$ given by the adjoint to the left $C$-colinear map

${N}$${M\otimes h_{C}(M,N)}$${M\otimes D\otimes
h_{C}(M,N).}$$\scriptstyle{\eta}$$\scriptstyle{\rho\otimes\mathsf{id}}$

Moreover, by [Tak77b, 1.7], the $C$-bicolinear map $\eta:C\rightarrow M\otimes
h_{C}(M,C)$ factors through the $(C,C)$-bicomodule $M\square_{D}h_{C}(M,C)$,
and the induced map $\eta:C\rightarrow M\square_{D}h_{C}(M,C)$ remains
$C$-bicolinear by [Tak77b, 1.9]. Therefore this defines the desired
$C$-bicolinear coevaluation $\eta:C\rightarrow M\square_{D}M^{\star}$. In
fact, by [Tak77b, 1.10], the cohom functor
$h_{C}:{}_{C}\mathsf{CoMod}\rightarrow\mathsf{Mod}_{\Bbbk}$ can be promoted to
a functor $h_{C}:{}_{C}\mathsf{CoMod}\rightarrow{}_{D}\mathsf{CoMod}$ whenever
the quasi-finite left $C$-comodule $M$ is endowed with a right $D$-coaction,
and it is the left adjoint of the functor

$\displaystyle{}_{D}\mathsf{CoMod}$
$\displaystyle\longrightarrow{}_{C}\mathsf{CoMod}$ $\displaystyle L$
$\displaystyle\longrightarrow M\square_{D}L,$

i.e., we obtain an equivalence

${}_{D}\mathsf{CoMod}\Big{(}h_{C}(M,N),L\Big{)}\cong{}_{C}\mathsf{CoMod}\Big{(}N,M\square_{D}L\Big{)}$

for any left $C$-comodule $N$ and left $D$-comodule $L$. In particular, if we
choose $M=N$ and $L=D$, the adjoint of the identity map on $M$ provides a left
$D$-colinear map $h_{C}(M,M)\rightarrow D$, which is in fact $D$-bicolinear by
[Tak77b, 1.11]. Moreover, by [Tak77b, 1.15], the map

${h_{C}(M,M)\cong
h_{C}(M,C\square_{C}M)}$${h_{C}(M,C)\square_{C}M,}$$\scriptstyle{\partial}$

is $D$-bicolinear. When $M$ is injective as a left $C$-comodule, the map
$\partial$ is an isomorphism. Therefore, the desired evaluation
$\varepsilon:M^{\star}\square_{C}M\rightarrow D$ is given by composing the
previous $D$-bicolinear maps

${h_{C}(M,C)\square_{C}M}$${h_{C}(M,C\square_{C}M)\cong
h_{C}(M,M)}$${D.}$$\scriptstyle{\partial^{-1}}$$\scriptstyle{\cong}$

Just as in Example 5.2, the triangle identities follow from the adjunction of
the cohom functor.

######  Example 5.5.

A particular case of Example 5.4 is when $C=\Bbbk$. A bicomodule $M$ in
${}_{\Bbbk}\mathsf{CoMod}_{D}$ is right dualizable when $M$ is dualizable as a
$\Bbbk$-module (i.e., finitely generated), and its right dual is $M^{*}$, the
usual linear dual just as above in Definition 2.25.

We can further generalize the examples above to the differential graded case.
Arguments for quasi-finite comodules are entirely categorical (see [Al-02]).
However, the notion has not been documented before in this context, and
therefore we introduce them here.

######  Definition 5.6.

Let $C$ be a simply connected dg-coalgebra over $\Bbbk$. A left
dg-$C$-comodule $M$ is _quasi-finite_ if the functor

$\displaystyle{\mathsf{Ch}_{\Bbbk}^{\geq 0}}$
$\displaystyle\longrightarrow{}_{C}\mathsf{CoMod}$ $\displaystyle V$
$\displaystyle\longmapsto M\otimes V$

admits a left adjoint, denoted
$h_{C}(M,-):{}_{C}\mathsf{CoMod}\rightarrow{\mathsf{Ch}_{\Bbbk}^{\geq 0}}$,
called the _cohom functor_. In other words, we obtain an equivalence

${\mathsf{Ch}_{\Bbbk}^{\geq
0}}\Big{(}h_{C}(M,N),V\Big{)}\cong{}_{C}\mathsf{CoMod}\Big{(}N,M\otimes
V\Big{)},$

for any left dg-$C$-comodule $N$ and $\Bbbk$-chain complex $V$.

######  Example 5.7.

If $V$ is a perfect $\Bbbk$-chain complex, then just as in Example 2.7, we
have that $C\otimes V$ is quasi-finite, and $h_{C}(C\otimes V,N)\cong
V^{*}\otimes N$. If $C=\Bbbk$, then a left dg $C$-comodule is just a chain
complex, and quasi-finite comodules are precisely the perfect chain complexes.

If $M$ is also endowed with a right $D$-coaction, then $h_{C}(M,N)$ is a right
$D$-comodule and we obtain the adjunction

${{}_{C}\mathsf{CoMod}}$${{}_{D}\mathsf{CoMod}.}$$\scriptstyle{h_{C}(M,-)}$$\scriptstyle{M\square_{D}-}$$\scriptstyle{\perp}$

Therefore, just as in the discrete case, the right dual of a
$(C,D)$-bicomodule $M$ should be the $(D,C)$-bicomodule $h_{C}(M,C)$. Notice,
even if $M$ is a fibrant $(C,D)$-bicomodule, there is no reason to expect
$h_{C}(M,C)$ is fibrant as a $(D,C)$-bicomodule. However, this is not needed.
Indeed, by Proposition A.17, if $M$ is a fibrant $(C,D)$-bicomodule, then

$\displaystyle M\square_{D}h_{C}(M,C)$ $\displaystyle\simeq
M\square_{D}\Omega(D,D,h_{C}(M,C))$
$\displaystyle\simeq\Omega(M,D,h_{C}(M,C))$ $\displaystyle\simeq
M\widehat{\square}_{D}h_{C}(M,C).$

Similarly, we obtain $h_{C}(M,C)\square_{C}M\simeq
h_{C}(M,C)\widehat{\square}_{C}M$.

Just as in the discrete case, the $C$-bicolinear map $\eta:C\rightarrow
M\square_{D}h_{C}(M,C)$ from the unit of the cohom adjunction induces the
desired coevaluation on the dual pair $(M,h_{C}(M,C))$.

We now describe how to obtain the evaluation. Recall that a $\Bbbk$-chain
complex $V$ is perfect if and only if the natural map
$V\otimes\mathsf{Hom}_{\Bbbk}(V,\Bbbk)\rightarrow\mathsf{Hom}_{\Bbbk}(V,V)$ is
an isomorphism. Similarly, as in Example 5.2, for any quasi-finite left
$C$-comodule $M$, we have a natural $D$-bicolinear map
$\partial:h_{C}(M,M)\rightarrow h_{C}(M,C)\square_{C}M$. We saw in the
discrete case that if $M$ is injective and quasi-finite, then $\partial$ is an
isomorphism.

######  Definition 5.8.

A fibrant left $C$-comodule $M$ is said to be _coperfect_ if it is quasi-
finite and the induced map $\partial:h_{C}(M,M)\rightarrow
h_{C}(M,C)\square_{C}M$ is an isomorphism.

Given a coperfect $(C,D)$-bicomodule $M$, define the evaluation as

${\varepsilon:h_{C}(M,C)\square_{C}M}$${h_{C}(M,M)}$${D.}$$\scriptstyle{\partial^{-1}}$$\scriptstyle{\cong}$

Combining our arguments above, we obtain the desired triangle identities by
adjunction of the cohom functor, and thus the following result.

######  Proposition 5.9.

Let $C$ and $D$ be simply connected dg-coalgebras. A (fibrant)
$(C,D)$-bicomodule $M$ is right dualizable if it is coperfect as a left
$C$-comodule. Its right dual is given by $h_{C}(M,C)$.

######  Example 5.10.

Suppose $D=\Bbbk$, and $M=C\otimes V$, where $V$ is a perfect chain complex.
Then $M$ is quasi-finite as a left $C$-comodule and $M^{\star}=h_{C}(M,C)\cong
V^{*}\otimes C$. Moreover, $M$ is coperfect because

$V^{*}\otimes M\cong h_{C}(C\otimes V,M)\rightarrow h_{C}(C\otimes
V,C)\square_{C}M\cong V^{*}\otimes M$

is an isomorphism.

######  Example 5.11.

If $C=\Bbbk$, then if $M$ is a right $D$-comodule such that $M$ is a perfect
chain complex, then $M$ is dualizable and
$M^{*}=\mathsf{Hom}_{\Bbbk}(M,\Bbbk)$.

##  6\. Traces

Every shadowed bicategory defines a notion of traces on its dualizable
objects. This extends the notion of traces on symmetric monoidal categories,
as reviewed in [PS14]. We first recall the general definition of a trace.

######  Definition 6.1.

[PS13] Let $(\mathcal{B},{\langle\langle}-{\rangle\rangle})$ be a shadowed
bicategory. Let $M\in\mathcal{B}(C,D)$ be a right dualizable $1$-cell. The
_trace_ of a $2$-cell $f:M\rightarrow M$, denoted
$\mathrm{tr}^{\mathcal{B}}(f)$, is the composite

${{{\langle\langle}U_{C}{\rangle\rangle}}_{C}}$${{\langle\langle}M\odot
M^{\star}{\rangle\rangle}_{C}}$${{\langle\langle}M\odot
M^{\star}{\rangle\rangle}_{C}}$${{\langle\langle}M^{\star}\odot
M{\rangle\rangle}_{D}}$${{\langle\langle}U_{D}{\rangle\rangle}_{D}.}$$\scriptstyle{{\langle\langle}\eta{\rangle\rangle}}$$\scriptstyle{{\langle\langle}f\odot\mathsf{id}{\rangle\rangle}}$$\scriptstyle{\theta}$$\scriptstyle{\cong}$$\scriptstyle{{\langle\langle}\varepsilon{\rangle\rangle}}$

More generally, given $1$-cells $P\in\mathcal{B}(D,D)$ and
$Q\in\mathcal{B}(C,C)$, the trace of a $2$-cell $f:Q\odot M\rightarrow M\odot
P$ is the composite

${{{\langle\langle}Q{\rangle\rangle}}_{C}}$${{\langle\langle}Q\odot M\odot
M^{\star}{\rangle\rangle}_{C}}$${{\langle\langle}M\odot P\odot
M^{\star}{\rangle\rangle}_{C}}$${{\langle\langle}M^{\star}\odot
M{\rangle\rangle}_{D}}$${{\langle\langle}P{\rangle\rangle}_{D}.}$$\scriptstyle{{\langle\langle}\mathsf{id}\odot\eta{\rangle\rangle}}$$\scriptstyle{{\langle\langle}f\odot\mathsf{id}{\rangle\rangle}}$$\scriptstyle{\theta}$$\scriptstyle{\cong}$$\scriptstyle{{\langle\langle}\varepsilon\odot\mathsf{id}{\rangle\rangle}}$

The _Euler characteristic_ of $M$, denoted $\chi(M),$ is the trace of its
identity $2$-cell $\mathrm{tr}^{\mathcal{B}}(\mathsf{id}_{M})$.

Bicategorical traces enjoy many useful properties that are recorded in [PS13,
Section 7]. Notably, we obtain the following cyclicity property.

######  Proposition 6.2 ([PS13, 7.3]).

If $M$ and $N$ are right dualizable $1$-cells in a shadowed bicategory
$\mathcal{B}$ and if $f:M\rightarrow N$ and $g:N\rightarrow M$ are $2$-cells
in $\mathcal{B}$, then $\mathrm{tr}^{\mathcal{B}}(f\circ
g)=\mathrm{tr}^{\mathcal{B}}(g\circ f)$.

######  Example 6.3 ([Pon10, 4.2.2]).

Let $R$ be a ring and let $M$ be a finitely generated right $R$-module. Let
$f$ be an $R$-linear endomorphism on $M$. The Hattori-Stalling trace can be
regarded as a bicategorical trace
$\mathsf{tr}(f):\mathbb{Z}\rightarrow\mathrm{HH}_{0}(R)$, i.e.,
$\mathrm{tr}(f)\in\mathrm{HH}_{0}(R)$.

######  Example 6.4.

Let $C$ be a $\Bbbk$-coalgebra. Given a dualizable left $C$-comodule $M$, the
trace of a $C$-colinear endomorphism $f:M\rightarrow M$ is then the Hattori-
Stallings cotrace $\mathrm{coHH}_{0}(C)\rightarrow\Bbbk$, as in Definition
2.11.

######  Remark 6.5.

In upcoming work of Justin Barhite, Hochschild cohomology is shown to be a so-
called coshadow structure on the bicategory of bimodules over a
$\Bbbk$-algebra $A$. He obtains what is called a cotrace
$\mathrm{HH}^{n}(A)\rightarrow\mathsf{End}_{A}(M)$ for any $A$-linear
endomorphism on a finitely generated projective right $A$-module $M$. If $C$
is a $\Bbbk$-coalgebra that is finitely generated as $\Bbbk$-module, then the
bicategory of bimodules over $C^{*}$ is intimately related to the bicategory
of bicomodules over $C$ since
$M\square_{C}N\cong{}_{C^{*}}\mathsf{Hom}_{C^{*}}(C^{*},M\otimes N)$ by [BW03,
10.10]. From our perspective however, our cotrace comes from a shadow and not
a coshadow, and we do not assume that our bicategory is closed. We see that we
recover his cotrace $\mathrm{HH}^{n}(A)\rightarrow\mathsf{End}_{A}(M)$ when
$n=0$ and $A=C^{*}$. Nonetheless, it does not seem that the language of
coshadows is the appropriate vocabulary for the Hattori-Stallings cotrace, nor
the $C$-trace.

######  Example 6.6.

Let $C$ be a $\Bbbk$-coalgebra. Given a right $C$-comodule $M$ that is
finitely generated as a $\Bbbk$-module, recall that
$M\square_{C}M^{*}\cong\mathsf{Hom}_{C}(M,M)$, and
$\mathrm{coHH}_{0}(M\square_{C}M^{*},\Bbbk)\cong M\square_{C}M^{*}$. The trace
of a $C$-colinear endomorphism $f:M\rightarrow M$ is a $\Bbbk$-linear map
$\Bbbk\rightarrow\mathrm{coHH}_{0}(C)$ given by

${\Bbbk}$${\mathsf{Hom}_{C}(M,M)}$${\mathsf{Hom}_{C}(M,M)}$${\mathrm{coHH}_{0}(M^{*}\otimes
M,C)}$${\mathrm{coHH}_{0}(C).}$$\scriptstyle{\eta}$$\scriptstyle{f_{*}}$$\scriptstyle{\theta}$$\scriptstyle{\cong}$$\scriptstyle{\mathrm{coHH}_{0}(\varepsilon)}$

This is the Hattori-Stallings $C$-trace for $f$ as in Definition 2.25.

######  Example 6.7.

We can generalize the Hattori-Stallings cotrace for chain complexes. Let $C$
be a simply connected dg-coalgebra over $\Bbbk$, let $M$ be a coperfect left
$C$-comodule, and let $f$ be an $C$-colinear endomorphism on $M$. Then we
obtain the cotrace

${\mathrm{coHH}(C)}$${\mathrm{coHH}(M\otimes
M^{*},C)}$${\mathrm{coHH}(M\otimes
M^{*},C)}$${M^{*}\widehat{\square}_{C}M}$${\Bbbk.}$$\scriptstyle{\mathrm{coHH}(\eta)}$$\scriptstyle{\mathrm{coHH}(f\otimes\mathsf{id})}$$\scriptstyle{\theta}$$\scriptstyle{\cong}$$\scriptstyle{\varepsilon}$

Since the homomorphism is in ${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$, and $\Bbbk$ is
concentrated in degree zero, then the above trace is entirely determined by
$\mathrm{coHH}_{0}(C)$. Unfortunately, as $C$ is simply connected,
$\mathrm{coHH}_{0}(C)=\Bbbk$, and this cotrace is simply the alternating sum
of the usual trace of $f$ on each degree. Similarly, we can also generalize
the $C$-trace for chain complexes. Letting $V$ denote a perfect chain complex
with a right $C$-comodule structure, given a $C$-colinear endomorphism
$f:V\rightarrow V$, its $C$-trace will define a chain homomorphism
$\Bbbk\rightarrow\mathrm{coHH}(C)$. It is non-trivial only in degree zero, but
as $C$ is simply connected, we get that the $C$-trace is the alternating sum
of the trace of $f$ on each degree.

######  Example 6.8.

Let $X$ be a simply connected CW-complex, and let $Y$ be a finite CW-complex.
Let $C_{*}(-;\Bbbk)=C_{*}(-)$ denote the singular chain complex over $\Bbbk$
associated to a space. Recall that $C_{*}(X)$ is a dg-coalgebra over $\Bbbk$
using Alexander-Whitney formula and the diagonal $X\rightarrow X\times X$.
Then $C_{*}(X\times Y)\cong C_{*}(X)\otimes C_{*}(Y)$ is a coperfect
$C_{*}(X)$-comodule. Let $f:Y\rightarrow Y$ be an endomorphism. This defines
an endomorphism on $X\times Y$ that is the identity on $X$, and thus an
endomorphism $\widehat{f}$ on the comodule $C_{*}(X\times Y)$. Therefore the
cotrace of $\widehat{f}$ is a map $\mathrm{coHH}(C_{*}(X))\rightarrow\Bbbk$ in
which its image is the usual Lefschetz number of $f$.

######  Example 6.9.

Let $X$ be a simply connected CW-complex, and let $Y$ be a finite CW-complex
with a continuous map $\rho:Y\rightarrow X$. Then $C_{*}(Y)$ is a comodule
over $C_{*}(X)$. Let $f$ be an endomorphism over $Y$. Then the
$C_{*}(X)$-trace induced on coHochschild homology
$\Bbbk\rightarrow\mathrm{coHH}(C_{*}(X))$ is the usual alternating sum of the
traces of $C_{*}(Y)\stackrel{{\scriptstyle
f}}{{\rightarrow}}C_{*}(Y)\stackrel{{\scriptstyle\rho}}{{\rightarrow}}C_{*}(X)$.

##  7\. Morita-Takeuchi invariance

In bicategories, Morita equivalence is the natural notion of an equivalence in
a bicategory. It extends the usual notion of equivalence of categories and
Morita equivalence between rings.

######  Definition 7.1 ([CP19, 4.1, 4.2]).

Let $\mathcal{B}$ be a bicategory, let $M\in\mathcal{B}(C,D)$ be a right
dualizable $1$-cell, and denote its right dual by $M^{\star}$. We say the dual
pair $(M,M^{\star})$ is a _Morita equivalence_ in $\mathcal{B}$ if the
coevaluation $\eta:U_{C}\rightarrow M\odot M^{\star}$ and evaluation
$\varepsilon:M^{\star}\odot M\rightarrow D$ maps are isomorphisms in
$\mathcal{B}(C,C)$ and $\mathcal{B}(D,D)$ respectively. If such a pair exists,
we say $C$ and $D$ are _Morita equivalent_.

######  Proposition 7.2 ([CP19, 4.5]).

Let $(\mathcal{B},{\langle\langle}-{\rangle\rangle})$ be a shadowed
bicategory. Let $M\in\mathcal{B}(C,D)$ be a right dualizable $1$-cell and
denote its right dual by $M^{\star}$. If $(M,M^{\star})$ is a Morita
equivalence, then the Euler characteristic $\chi(M)$ is an isomorphism, with
inverse $\chi(M^{\star})$. In particular, if $C$ and $D$ are Morita
equivalent, then

${\langle\langle}U_{C}{\rangle\rangle}_{C}\cong{\langle\langle}U_{D}{\rangle\rangle}_{D}.$

######  Example 7.3.

Consider the bicategory $\mathsf{CoMod}$ of bicomodules over $\Bbbk$ as in
Definition 3.5. Then in [Tak77b, 2.3], a Morita equivalence is referred to as
a _set of equivalence data_. It is then shown than a Morita equivalence in
this bicategory recovers the notion of Morita-Takeuchi equivalence in
$\Bbbk$-modules (Definition 1.11). Therefore, if $C$ and $D$ are Morita-
Takeuchi equivalent, then $\mathrm{coHH}_{0}(C)\cong\mathrm{coHH}_{0}(D)$ by
Proposition 7.2. This recovers the results of [FS98] at level zero. By
[Tak77b, 3.5], if $M$ is a left $C$-comodule that is a quasi-finite injective
cogenerator, and $D:=e_{C}(M)$, the coalgebra of coendomorphisms, then $C$ and
$D$ are Morita-Takeuchi equivalent, and $M$ and $M^{\star}$ form a Morita
equivalence in the bicategory $\mathsf{CoMod}$.

######  Example 7.4.

Consider the bicategory of derived bicomodules $\mathcal{D}\mathsf{CoMod}$ as
in Definition 3.13. We say two simply connected coalgebras $C$ and $D$ in
${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$ are homotopically Morita-Takeuchi equivalent
if they are Morita equivalent in the bicategory $\mathcal{D}\mathsf{CoMod}$.
In this situation

$\mathrm{coHH}(C)\cong\mathrm{coHH}(D).$

This extends the results of [HS21], where we suspect that the simply-connected
condition was forgotten to be mentioned.

######  Proposition 7.5.

Suppose $(M,M^{\star})$ form a dual pair in $\mathcal{D}\mathsf{CoMod}$. Then
we obtain a Quillen adjunction

${M^{\star}\square_{C}-:{}_{C}\mathsf{CoMod}({\mathsf{Ch}_{\Bbbk}^{\geq
0}})}$${{}_{D}\mathsf{CoMod}({\mathsf{Ch}_{\Bbbk}^{\geq
0}}):M\square_{D}-.}$$\scriptstyle{\perp}$

Moreover, it is a Quillen equivalence if and only if $(M,M^{\star})$ is a
homotopical Morita-Takeuchi equivalence.

###### Proof.

Since $M$ and $M^{\star}$ are dual to each other, the maps $\eta:C\rightarrow
M\square_{D}M^{\star}$ and $\varepsilon:M^{\star}\square_{C}M\rightarrow D$
induce the adjunction. For instance, given a left $C$-comodule $P$, a left
$D$-comodule $Q$, and a $D$-colinear map
$\alpha:M^{\star}\square_{C}P\rightarrow Q$, we obtain a $C$-colinear map
$P\rightarrow M\square_{D}Q$ via the composite

${P\cong C\square_{C}P}$${(M\square_{D}M^{\star})\square_{C}P\cong
M\square_{D}(M^{\star}\square_{C}P)}$${M\square_{D}Q.}$$\scriptstyle{\eta\square\mathsf{id}}$$\scriptstyle{\mathsf{id}\square\alpha}$

The axioms on $\eta$ and $\varepsilon$ guarantee that the procedure gives a
correspondence:

${}_{D}\mathsf{CoMod}({\mathsf{Ch}_{\Bbbk}^{\geq
0}})\Big{(}M^{\star}\square_{C}P,Q\Big{)}\cong{}_{C}\mathsf{CoMod}({\mathsf{Ch}_{\Bbbk}^{\geq
0}})\Big{(}P,M\square_{D}Q\Big{)}.$

Since the cotensor product is left exact, then the functor
$M^{\star}\square_{D}-$ preserves monomorphisms, i.e., cofibrations. Since
$M^{\star}$ is a fibrant right $C$-comodule, then $M^{\star}\square_{C}-$
preserves quasi-isomorphisms, i.e., weak equivalences (by Proposition A.17).
Therefore we obtain that it is a Quillen adjunction.

Suppose $(M,M^{\star})$ is a homotopical Morita-Takeuchi equivalence and
$P\rightarrow P^{\prime}$ is a $C$-colinear map such that the induced map

$M^{\star}\square_{C}P\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}M^{\star}\square_{C}P^{\prime}$

is a quasi-isomorphism. Then if we apply $M\square_{D}-$, since $M$ is a
fibrant $D$-comodule, we obtain the quasi-isomorphism by Proposition A.17:

${\underbrace{(M\square_{D}M^{\star})\square_{C}P}_{\simeq P}\cong
M\square_{D}(M^{\star}\square_{C}P)}$${M\square_{D}(M^{\star}\square_{C}P^{\prime})\cong\underbrace{(M\square_{D}M^{\star})\square_{C}P^{\prime}}_{\simeq
P^{\prime}}.}$$\scriptstyle{\simeq}$

Therefore $P\rightarrow P^{\prime}$ is a quasi-isomorphism, and thus
$M^{\star}\square_{C}-$ reflects quasi-isomorphism. Moreover, for any fibrant
left $D$-comodule $Q$, we have a quasi-isomorphism

$Q\cong D\square_{D}Q\simeq M^{\star}\square_{C}M\square_{D}Q\longrightarrow
Q.$

Thus by [Hov99, 1.3.16], we obtain that the adjunction is a Quillen
equivalence.

Conversely, if we supposed the adjunction to be a Quillen equivalence, then if
we apply the adjoints on $C$ and $D$, we recover that the coevaluation and
evaluation $C\rightarrow M\square_{D}M^{\star}$ and
$M^{\star}\square_{C}M\rightarrow D$ are quasi-isomorphisms by again applying
[Hov99, 1.3.16]. ∎

######  Example 7.6.

Given a map of simply connected coalgebras $f:C\rightarrow D$, we recover the
expected result that if $f$ is a quasi-isomorphism, then $C$ and $D$ are
homotopically Morita-Takeuchi equivalent. In more details, recall that $(C,C)$
form a dual pair of bicomodules over $C$, and $h_{C}(C,C)\cong C$. Moreover,
$C$ can be regarded as a left $D$-comodule via $f$:

${C}$${C\otimes C}$${D\otimes C.}$$\scriptstyle{f\otimes\mathsf{id}}$

In fact, any left $C$-comodule $P$ can be regarded as a left $D$-comodule this
way, we shall denote it by $f^{*}(C)$. Notice that $f^{*}(C)\cong
C\square_{C}P$ as a left $D$-comodule. We obtain a Quillen adjunction:

${f^{*}:{}_{C}\mathsf{CoMod}({\mathsf{Ch}_{\Bbbk}^{\geq
0}})}$${{}_{D}\mathsf{CoMod}({\mathsf{Ch}_{\Bbbk}^{\geq
0}}):C\square_{D}-,}$$\scriptstyle{\perp}$

which is a Quillen equivalence if and only if $f$ is a quasi-isomorphism. This
recovers the usual change of coalgebras, see [Pér20, 4.29].

##  Appendix A Homotopy theory of connective bicomodules

The goal of the appendix is to show that the homotopy theory of bicomodules is
endowed with a derived cotensor product which provides a homotopy coherent
monoidal structure.

Let $\mathcal{C}^{\otimes}$ be a symmetric monoidal $\infty$-category, as in
[Lur17, 2.0.0.7]. Subsequently, we only refer to its underlying
$\infty$-category, $\mathcal{C}$, as in [Lur17, 2.1.2.20]. By an
_${\mathbb{A}_{\infty}}$ -algebra in $\mathcal{C}$_, we mean an associative
algebra in the sense of [Lur17, 4.1.1.6]. Given
${\mathbb{A}_{\infty}}$-algebras $A$ and $B$ in $\mathcal{C}$, denote the
$\infty$-category of $(A,B)$-bimodule objects in $\mathcal{C}$ by
${}_{A}\mathsf{Mod}_{B}(\mathcal{C})$, as in [Lur17, Section 4.3]. From
[Pér22a, 2.1], define an _${\mathbb{A}_{\infty}}$ -coalgebra_ in $\mathcal{C}$
to be an ${\mathbb{A}_{\infty}}$-algebra in the opposite category
$\mathcal{C}$. Similarly as in [Lur17, 4.1.1.7], given an
${\mathbb{A}_{\infty}}$-coalgebra $C$ in $\mathcal{C}$, we can define
$C^{\mathsf{op}}$, the _opposite ${\mathbb{A}_{\infty}}$-coalgebra_ of $C$.

######  Definition A.1.

Let $\mathcal{C}$ be a symmetric monoidal $\infty$-category. Let $C$ and $D$
be ${\mathbb{A}_{\infty}}$-coalgebras in $\mathcal{C}$. A _$(C,D)$
-bicomodule_ object in $\mathcal{C}$ is a $(C,D)$-bimodule object in
$\mathcal{C}^{\mathsf{op}}$. The $\infty$-category of $(C,D)$-bicomodules in
$\mathcal{C}$ is defined as

${}_{C}\mathsf{CoMod}_{D}(\mathcal{C}):=({}_{C}\mathsf{Mod}_{D}(\mathcal{C}^{\mathsf{op}}))^{\mathsf{op}}.$

We define the $\infty$-categories of left $C$-comodules
${}_{C}\mathsf{CoMod}(\mathcal{C})$ and right $D$-comodules
$\mathsf{CoMod}_{D}(\mathcal{C})$ similarly.

######  Remark A.2.

Just as in Remark 1.9, given ${\mathbb{A}_{\infty}}$-coalgebras $C$ and $D$ in
a symmetric monoidal $\infty$-category $\mathcal{C}$, we obtain an equivalence
of $\infty$-categories

${}_{C}\mathsf{CoMod}_{D}(\mathcal{C})\simeq\mathsf{CoMod}_{C^{\mathsf{op}}\otimes
D}(\mathcal{C}).$

This follows from [Lur17, 4.6.3.11] applied to the opposite category. Notice
that the statement requires the monoidal product of $\mathcal{C}$ to commute
with totalizations. However, as noted above [Lur17, 4.6.3.3], this requirement
is not essential and remains true for any symmetric monoidal $\infty$-category
$\mathcal{C}$.

######  Remark A.3.

Let $\mathsf{C}$ be a symmetric monoidal category. Let $\mathsf{C}^{\otimes}$
be its operator category as in [Lur17, 2.0.0.1]. Then its nerve
$\mathcal{N}(\mathsf{C}^{\otimes})$ is a symmetric monoidal $\infty$-category
whose underlying $\infty$-category is $\mathcal{N}(\mathsf{C})$, see [Lur17,
2.1.2.21]. Let $C$ be a coalgebra in $\mathsf{C}$. It can be regarded as an
$\mathbb{A}_{\infty}$-coalgebra in $\mathcal{N}(\mathsf{C})$, see [Pér22a,
2.3]. If $D$ is also a coalgebra in $\mathsf{C}$, then we obtain an
equivalence of $\infty$-categories

$\mathcal{N}\left({}_{C}\mathsf{CoMod}_{D}(\mathsf{C})\right)\simeq{}_{C}\mathsf{CoMod}_{D}\left(\mathcal{N}(\mathsf{C})\right).$

Let $\mathsf{M}$ be a symmetric monoidal model category as in [Hov99, 4.2.6],
with a class of weak equivalence denoted $\mathsf{W}$. We assume every object
is cofibrant. Its Dwyer-Kan localization is the $\infty$-category denoted
$\mathcal{N}(\mathsf{M})[\mathsf{W}^{-1}]$ following [Lur17, 1.3.4.15] whose
homotopy category is the homotopy category of $\mathsf{M}$. It is endowed with
a symmetric monoidal structure via the derived tensor product, see [Lur17,
4.1.7.6]. Suppose now that the model structure on $\mathsf{M}$ is
combinatorial. Given algebras $A$ and $B$ in $\mathsf{M}$, the category of
bimodules ${}_{A}\mathsf{Mod}_{B}(\mathsf{M})$ is endowed with a model
structure whose class of weak equivalences, denoted $\mathsf{W}^{\prime}$, are
the morphisms of bimodules which are weak equivalences regarded as in
$\mathsf{M}$. By [Lur17, 4.3.3.17], we obtain an equivalence of
$\infty$-categories

$\mathcal{N}\Big{(}{}_{A}\mathsf{Mod}_{B}(\mathsf{M})\Big{)}[\mathsf{W}^{\prime-1}]\simeq{}_{A}\mathsf{Mod}_{B}\left(\mathcal{N}(\mathsf{M})[\mathsf{W}^{-1}]\right).$

The same statement for bicomodules is challenging if we insist on considering
a combinatorial monoidal model category (and not “co-combinatorial”).
Following [Pér20], we show in Theorem A.4 when the equivalence above does hold
for bicomodules. We denote the Dwyer-Kan localization of
${\mathsf{Ch}_{\Bbbk}^{\geq 0}}$ by $\mathcal{D}^{\geq 0}(\Bbbk)$; it is
equivalent to the symmetric monoidal $\infty$-category of connective
$H\Bbbk$-modules in spectra.

######  Theorem A.4.

Let $C$ and $D$ be simply connected dg-coalgebras over $\Bbbk$. Then the
natural functor

$\mathcal{N}\left({}_{C}\mathsf{CoMod}_{D}({\mathsf{Ch}_{\Bbbk}^{\geq
0}})\right)\left[\mathsf{W}^{-1}\right]\longrightarrow{}_{C}\mathsf{CoMod}_{D}(\mathcal{D}^{\geq
0}(\Bbbk))$

is an equivalence of $\infty$-categories.

###### Proof.

The result was proved in [Pér20, 1.1] for right comodules. We can deduce the
result for bicomodules using Remarks 1.9 and A.2. In more details, if $C$ and
$D$ are simply connected, then so is $C\otimes D$:

$(C\otimes D)_{0}=C_{0}\otimes D_{0}\cong\Bbbk\otimes\Bbbk\cong\Bbbk,\hskip
15.0pt(C\otimes D)_{1}=(C_{1}\otimes D_{0})\oplus(C_{0}\otimes D_{1})=0.$

Therefore we can apply [Pér20, 1.1] to the model category of right
$(C^{\mathsf{op}}\otimes D)$-comodules. In particular, we obtain the following
diagram of $\infty$-categories:

${\mathcal{N}\left({}_{C}\mathsf{CoMod}_{D}({\mathsf{Ch}_{\Bbbk}^{\geq
0}})\right)\left[\mathsf{W}^{-1}\right]}$${\mathcal{N}\left(\mathsf{CoMod}_{C^{\mathsf{op}}\otimes
D}({\mathsf{Ch}_{\Bbbk}^{\geq
0}})\right)\left[\mathsf{W^{\prime}}^{-1}\right]}$${{}_{C}\mathsf{CoMod}_{D}(\mathcal{D}^{\geq
0}(\Bbbk))}$${\mathsf{CoMod}_{C^{\mathsf{op}}\otimes D}(\mathcal{D}^{\geq
0}(\Bbbk)).}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

Here, we let $\mathsf{W}^{\prime}$ denote the class of right