# Shadows are Bicategorical Traces

Kathryn Hess and Nima Rasekh  École Polytechnique Fédérale de Lausanne, SV BMI
UPHESS, Station 8, CH-1015 Lausanne, Switzerland<EMAIL_ADDRESS>Max
Planck Institute for Mathematics, Bonn, Germany<EMAIL_ADDRESS>
(Date: April 2023)

###### Abstract.

The theory of shadows is an axiomatic, bicategorical framework that
generalizes topological Hochschild homology (THH) and satisfies analogous
important properties, such as Morita invariance. Using Berman’s extension of
$\mathrm{THH}$ to bicategories, we prove that there is an equivalence between
functors out of Hochschild homology of a bicategory and shadows on that
bicategory. As part of that proof we give a computational description of
pseudo-functors out of the truncated simplex category and a variety thereof,
which can be of independent interest.

Building on this result we present three applications:

  1. (1)

We provide a new, conceptual proof that shadows, and as a result the Euler
characteristic and trace introduced by Campbell and Ponto, are Morita
invariant.

  2. (2)

We strengthen this result by using an explicit computation of the Hochschild
homology of the free adjunction bicategory to show that the construction of
the Euler characteristic is homotopically unique.

  3. (3)

We generalize the construction of $\mathscr{V}$-enriched Hochschild homology,
where $\mathscr{V}$ is a presentably symmetric monoidal $\infty$-category, to
bimodules, and prove it gives us a shadow.

###### Contents

  1. 1 Introduction
    1. 1.1 From THH to shadows
    2. 1.2 The rise of homotopy coherence
    3. 1.3 From shadows back to THH
    4. 1.4 Where do we go from here?
    5. 1.5 Organization
    6. 1.6 Notation
    7. 1.7 Background
    8. 1.8 Acknowledgments
  2. 2 Background
    1. 2.1 Shadows of bicategories
    2. 2.2 $\mathrm{THH}$ of enriched $\infty$-categories
  3. 3 Shadows vs. Traces
    1. 3.1 Hochschild homology of bicategories
    2. 3.2 The equivalence between shadows and traces
  4. 4 Morita Invariance of Shadows via Hochschild Homology
  5. 5 Hochschild Homology of $2$-Categories and Homotopy Coherence
  6. 6 Towards shadows on $(\infty,2)$-categories
  7. A Truncated Simplicial Pseudo-Diagrams

##  1\. Introduction

###  1.1. From THH to shadows

Hochschild homology of associative rings, first studied by Hochschild [Hoc45]
and Cartan-Eilenberg [CE56], generalizes the notion of Kähler differentials
from the commutative to the merely associative setting. Hochschild homology is
an important tool for non-commutative geometry [CM08], which satisfies
interesting properties such as Morita invariance [Lod98] and which provides a
useful approximation to algebraic $K$-theory via the Dennis trace [Den76,
Wal79]. The homotopical analogue of classical Hochschild homology, topological
Hochschild homology ($\mathrm{THH}$) of ring spectra, admits an analogous
Dennis trace map from algebraic $K$-theory [Bök85, EKMM97].

Since the definition of algebraic $K$-theory has been extended to many types
of structured categories [Wal85], the important connection between
$\mathrm{THH}$ and algebraic $K$-theory of ring spectra inspired the
definition of similar extensions of (topological) Hochschild homology: to
exact categories [McC94], to dg-categories [Kel99], and to spectral categories
[BM12], as a special case of topological Hochschild homology of bimodules over
spectral categories. It can be be challenging to establish properties, such as
Morita invariance, of certain of these versions of THH.

An axiomatic framework that is able to incorporate simultaneously all of these
different settings is therefore worth developing. Ponto provided one potential
such framework, when she introduced the notion of a shadow of bicategories, as
tool to study fixed point phenomena [Pon10]. In [CP19] she and Campbell proved
that topological Hochschild homology of bimodules over spectral categories
indeed provides an example of a shadow. They showed moreover that Morita
invariance of $\mathrm{THH}$ of ring spectra is simply a special case of a
more general, abstract Morita invariance of shadows. This observation,
combined with the possibility of performing concrete computations with shadows
via string diagrams [PS13], provides strong incentive to determine whether the
extensions of $\mathrm{THH}$ to other types of structured categories are also
shadows, and, if so, how homotopy coherent they are. The issue of homotopy
coherence is the focus of our attention in this article.

###  1.2. The rise of homotopy coherence

Over the past two decades, a rich theory of $\infty$-categories has been
developed [Joy08a, Joy08b, Lur09], in which, instead of focusing only on
homotopy categories, one studies objects of interest in a homotopy-coherent
fashion. The focus on $\infty$-categories has had a significant influence on
the study of algebraic $K$-theory and $\mathrm{THH}$. In this framework,
algebraic $K$-theory can be described as a functor associating spaces to
stable $\infty$-categories of a certain type, which satisfies certain
universal properties, giving rise to a new definition of the Dennis trace to
$\mathrm{THH}$ [BGT13, BGT14, BGMN21]. The $\infty$-categorical perspective
provides new insight into $\mathrm{THH}$ of ring spectra and its cyclotomic
structure [NS18, NS19].

As sketched above, the development of various notions of $\mathrm{THH}$
eventually inspired the notion of a shadow on a bicategory, which captures the
essential structure of those diverse constructions. Given that many of these
construction have now been generalized to the $\infty$-categorical framework,
it is natural to seek an analogous definition of an $\infty$-categorical
shadow. Since shadows were originally defined on bicategories, the
$\infty$-categorical analogue should be defined on $(\infty,2)$-categories,
which are the higher categorical analogue of bicategories [Ber20].

The existence of shadows on $(\infty,2)$-categories would have interesting
practical consequences as well. For example, there is a natural definition of
coHochschild homology ($\mathrm{co}\mathrm{THH}$) of coalgebra spectra [HS21,
BGH+18, BGS21], dual to that of $\mathrm{THH}$. Given this duality, there
should be an axiomatic, shadow-type approach to $\mathrm{co}\mathrm{THH}$.
However, since essentially the only examples of coalgebras in point-set models
of spectra are suspension spectra [PS19], a strict bicategorical approach to
studying $\mathrm{co}\mathrm{THH}$ would be of limited interest. An
$(\infty,2)$-categorical approach is necessary, and elaborating and analyzing
an $(\infty,2)$-categorical version of shadows is an important first step in
this direction.

###  1.3. From shadows back to THH

To determine how best to generalize shadows from bicategories to
$(\infty,2)$-categories, it is helpful to examine more carefully the
relationship between shadows and $\mathrm{THH}$. In particular, given that
shadows are defined in the abstract setting of bicategories, how is it that
they satisfy Morita invariance, which is often thought of as a concrete
property of rings and $\mathrm{THH}$?

The goal of this work is to tackle this question, exploiting the construction
of $\mathrm{THH}$ of enriched $\infty$-categories introduced by Berman
[Ber22], in the case of enrichments in the $\infty$-category of categories.
Berman formulated an efficient definition of Hochschild homology of enriched
$\infty$-categories in terms of the colimit of a certain cyclic bar
construction taking values in the enriching $\infty$-category, avoiding the
sophisticated machinery of enriched $\infty$-categories due to Gepner and
Haugseng [GH15]. Applying Berman’s construction to the $\infty$-category of
categories, we prove following result relating Hochschild Homology of
bicategories and shadows.

###### Theorem 3.19.

For any bicategory $\mathscr{B}$ and category $\mathscr{D}$, there is a
natural equivalence of categories

$\mathrm{Fun}\big{(}\mathrm{biHH}(\mathscr{B}),\mathscr{D}\big{)}\xrightarrow{\
\simeq\ }\mathscr{S}\mathrm{ha}(\mathscr{B},\mathscr{D}).$

Here $\mathrm{biHH}(\mathscr{B})$, which is a category, is Hochschild homology
of a bicategory $\mathscr{B}$ (2.10), $\mathrm{Fun}(-,-)$ denotes the functor
category, and $\mathscr{S}\mathrm{ha}$ is the category of shadows on
$\mathscr{B}$ taking values in $\mathscr{D}$ (3.17).

This result has many interesting implications of a philosophical, as well as
mathematical, nature.

Why shadows resemble THH: Our main theorem enables us to explain why shadows
satisfy many properties that also hold for THH, since it implies that the
axioms of a shadow are precisely the conditions required to construct a cocone
out of the diagram that defines $\mathrm{THH}$. Working in an
$\infty$-categorical framework, rather than in the context of model
categories, facilitated making this connections, since (co)limits of diagrams
of $\infty$-categories make natural sense [Lur09].

Tricategorical Shadows: A key step in the proof of 3.19 is to characterize
precisely pseudo-categorical $2$-truncated simplicial cocones (cf. Appendix A)
and then to relate this characterization to shadows. This opens up the
possibility of defining tricategorical shadows by a similar method, via a
characterization of $3$-truncated simplicial cocones.

Morita invariance: Our reformulation of shadow functors out of $\mathrm{THH}$
of bicategories allows us to give a new proof of the Morita invariance of
shadows, by lifting both the “Euler characteristic” construction and the
“trace construction” of Campbell and Ponto [CP19] to a functor (4.9/4.16),
providing insight into the essence of this fundamental property. The proofs of
these statement rely on having a basic understanding of the structure of
Hochschild homology of the free adjunction category and a variety thereof.
Beyond those implications, this new proof thus strongly hints at further
generalizations of Morita invariance to other (enriched) settings.

Homotopy canonical Euler characteristic and trace: Reformulating Morita
invariance using certain structures of the Hochschild homology of the free
adjunction enables us to make a much deeper analysis of the Euler
characteristic and trace. Concretely, by explicitly computing the whole
Hochschild homology of the free adjunction $2$-category (5.23) and a
generalization thereof (5.26), we deduce that the definition of the Euler
characteristic is canonical (5.25), whereas the definition of trace depends on
a choice of automorphism (5.28). Our result provides another example of how a
computation related to the free adjunction bicategory leads to formal results,
similar to recent work of Ayala and Francis [AF21]111Using an alternative
approach to $\mathrm{THH}$ of enriched categories via factorization homology
[AFR17].

Property vs. computation: A key part of the previous items was to translate a
desirable property regarding $\mathrm{THH}$, such as Morita invariance, into
computing $\mathrm{THH}$ of the representing $2$-category, in this case the
free adjunction $2$-category. This should be seen as one example of a general
method we introduce that translates properties of $\mathrm{THH}$ into
$\mathrm{THH}$ computations and is expected to have many further applications.

Shadows of $(\infty,2)$-categories and $\mathrm{THH}$ of $\infty$-bimodules:
Knowing that shadows on a bicategory $\mathscr{B}$ are equivalent to functors
out of the category $\mathrm{THH}(\mathscr{B})$ enables us to formulate a
reasonable $(\infty,2)$-categorical generalization that coincides
appropriately with the classical definition upon restriction to the homotopy
bicategory. As a first application we define $\mathrm{THH}$ of enriched
$\infty$-categorical bimodules in the sense of Haugseng [Hau16] and prove it
lifts to a functor
$\widehat{\mathrm{THH}}:\mathrm{THH}(\mathscr{M}\mathrm{od}_{\mathscr{V}})\to\mathrm{Ho}\mathscr{V}$
(6.11).

###  1.4. Where do we go from here?

The results proven here suggest several interesting and natural next steps.

  1. (1)

$\mathrm{co}\mathrm{THH}$ via shadows: As discussed above, we cannot expect to
study $\mathrm{co}\mathrm{THH}$ at a point-set level. The results here create
an opportunity to define $\mathrm{co}\mathrm{THH}$ of a coalgebra spectrum as
$\mathrm{THH}$ of its category of comodules.

  2. (2)

Functoriality of $\mathrm{THH}$: One current challenge when studying
$\mathrm{THH}$ of enriched $\infty$-categories is that the definition given in
[Ber22] is not known to be functorial. Having a working functorial
construction would, for example, permit us to generalize the Morita invariance
of shadows beyond bicategories.

  3. (3)

$\mathrm{THH}$ is a trace: A key result in [CP19] is that $\mathrm{THH}$ for
bimodules over spectral categories itself is a shadow. In 6.11 we have
generalized this result to a functor of $\infty$-categories valued in
$\mathrm{Ho}(\mathscr{V})$, however, the expectation is that it should lift to
an $\infty$-categorical functor

$\mathrm{THH}(\mathscr{M}\mathrm{od}_{V})\to\mathscr{V},$

which should, under the right circumstances, even be $\mathscr{V}$-enriched,
as has been discussed in 6.2, if some theoretical challenges regarding
enriched $(\infty,2)$-categories can be resolved (6.14).

###  1.5. Organization

The paper is organized as follows. We begin by reviewing in the first section
relevant content about shadows and $\mathrm{THH}$ of enriched
$\infty$-categories. In the second section, we compute $\mathrm{THH}$ of
bicategories and use the result to prove that traces of bicategories are
equivalent to shadows. Based on this characterization, we provide our new
proof for Morita invariance of shadows on bicategories in the third section.
Finally, in the last section, we deduce from the equivalence between shadows
and $\mathrm{THH}$ that it is reasonable to define a homotopy-coherent shadow
to be a trace on an $(\infty,2)$-category.

###  1.6. Notation

We use several types of categories throughout this article and thus need to
distinguish between them carefully.

A category is a $(1,1)$-category. We denote by $\mathscr{C}\mathrm{at}$ the
large $(2,1)$-category of small categories, i.e., $\mathscr{C}\mathrm{at}$ has

  * •

small categories as objects,

  * •

functors as morphisms, and

  * •

natural isomorphisms as $2$-morphisms.

In particular, the existence of a commuting diagram of functors

${\mathscr{C}_{1}}$${\mathscr{C}_{2}}$${\mathscr{C}_{3}}$${\mathscr{C}_{4}}$$\scriptstyle{F_{1}}$$\scriptstyle{F_{2}}$$\scriptstyle{F_{3}}$$\scriptstyle{F_{4}}$

$\scriptstyle\cong$

means that there is a natural isomorphism

$\alpha:F_{3}\circ F_{1}\xrightarrow{\ \cong\ }F_{4}\circ F_{2}.$

For any two categories $\mathscr{C},\mathscr{D}$, we let
$\mathrm{Fun}(\mathscr{C},\mathscr{D})$ denote the functor category, whose
objects are functors from $\mathscr{C}$ to $\mathscr{D}$, and whose morphisms
are natural transformations.

In this article a bicategory $\mathscr{B}$ is a $(2,2)$-category, composed of

  * •

a class of objects $X,Y,Z,...$,

  * •

a category of morphisms $\mathscr{B}(X,Y)$ for every pair of objects $X,Y$,
where the unit object in $\mathscr{B}(X,X)$ is denoted $U_{X}$, and

  * •

natural isomorphisms $a$, $r$ and $l$ that witness associativity, right
unitality, and left unitality, respectively and that satisfy certain axioms
that the reader can find, for example, in [Bén67, Lei98].

Note that $\mathscr{C}\mathrm{at}$ is an example of a bicategory.

In what follows, the terminology $\infty$-category is used as a synonym for
quasi-category, one important model of $(\infty,1)$-categories, popularized by
Lurie [Lur09]. We denote the large $\infty$-category of small
$\infty$-categories by $\mathscr{C}\mathrm{at}_{\infty}$. The nerve functor
$N$ from the category $\mathscr{C}\mathrm{at}$ of small categories to that of
simplicial sets enables us to see $\mathscr{C}\mathrm{at}$ as a subcategory of
$\mathscr{C}\mathrm{at}_{\infty}$. We routinely suppress the functor $N$ to
simplify notation. The inclusion of $\mathscr{C}\mathrm{at}$ into
$\mathscr{C}\mathrm{at}_{\infty}$ admits a left adjoint

$\mathrm{Ho}:\mathscr{C}\mathrm{at}_{\infty}\to\mathscr{C}\mathrm{at}$

that takes each $\infty$-category to its homotopy category.

If $\mathscr{C}$ is an $\infty$-category, we let $\mathcal{P}(\mathscr{C})$
denote the $\infty$-category of space-valued presheaves on $\mathscr{C}$.

Though we also use $(\infty,2)$-categories in this article, we assume no prior
knowledge of them. All the necessary theory of $(\infty,2)$-categories is
reviewed in Subsection 2.2.

We work as well with various notions of the “category of spectra”. In the
first part of Subsection 2.1, we consider a monoidal model category of
spectra, denoted $\mathcal{S}p$, e.g., symmetric spectra or orthogonal spectra
[MMSS01]. We denote the monoidal product on $\mathcal{S}p$ by $\wedge$. We let
$\mathrm{Sp}$ denote the $\infty$-category of spectra and
$\mathrm{Ho}(\mathrm{Sp})$ its homotopy category, which is equivalent to the
homotopy category of the model category $\mathcal{S}p$.

###  1.7. Background

We assume only basic familiarity with category theory and a healthy curiosity
about $\mathrm{THH}$. We review in Section 2 the relevant definitions of
$\mathrm{THH}$ of spectral categories, shadows, enriched $\infty$-categories,
and the $\infty$-categorical definition of $\mathrm{THH}$ of enriched
$\infty$-categories.

###  1.8. Acknowledgments

We would like to thank John Berman for helpful conversations and for
clarifying several points in his paper [Ber22]. We also like to thank Rune
Haugseng for helpful discussions regarding his paper [Hau16]. Finally, we
would like to thank Thomas Nikolaus for helpful discussions and suggestions.

##  2\. Background

In this section we recall relevant definitions in two (seemingly distinct)
areas: shadows of bicategories and $\mathrm{THH}$ of enriched
$\infty$-categories. The first subsection is written at the point-set level,
in the language of bicategories and model categories. The material presented
here can be found in more detail in [CP19]. The second subsection, based on
[Ber22], reviews the relevant concepts about enriched $\infty$-categories and
their topological Hochschild homology, which is not defined on the point-set
level.

###  2.1. Shadows of bicategories

We review here the notion of a shadow, which can be viewed as an
axiomatization of $\mathrm{THH}$. Shadows first appeared in [Pon10], though
[CP19] is our main reference. Topological Hochschild homology ($\mathrm{THH}$)
was originally defined for ring spectra [Bök85], but was then generalized to
spectral categories, i.e., categories enriched over a monoidal model category
of spectra $\mathcal{S}p$ [BM12].

######  Definition 2.1.

Let $\mathscr{C}$ and $\mathscr{D}$ be spectral categories.

  1. (1)

A $\mathscr{C}$-module is a spectral functor

$\mathscr{C}\to\mathcal{S}p.$

  2. (2)

A $(\mathscr{C},\mathscr{D})$-bimodule is a spectral functor

$\mathscr{C}^{op}\wedge\mathscr{D}\to\mathcal{S}p,$

where $\mathscr{C}^{op}\wedge\mathscr{D}$ is the spectral category with as
objects ordered pairs $(c,d)$, where $c\in\mathscr{C}$ and $d\in\mathscr{D}$,
and with mapping spectrum

$\mathrm{Map}_{\mathscr{C}^{op}\wedge\mathscr{D}}\big{(}(c_{1},d_{1}),(c_{2},d_{2})\big{)}=\mathrm{Map}_{\mathscr{C}}(c_{1},c_{2})\wedge\mathrm{Map}_{\mathscr{D}}(d_{1},d_{2}).$

######  Definition 2.2 ([CP19, Definition 2.5]).

Let $\mathscr{C}$ be a be a pointwise-cofibrant spectral category (i.e.,
$\mathscr{C}(c,d)$ is a cofibrant spectrum for all objects $c,d$) and
$\mathscr{D}$ a $(\mathscr{C},\mathscr{C})$-bimodule that is pointwise-
cofibrant as a spectral category. The topological Hochschild homology of
$\mathscr{C}$ with coefficients in $\mathscr{D}$, denoted
$\mathrm{THH}(\mathscr{C};\mathscr{D})$, is the geometric realization of the
cyclic bar construction $N^{cy}_{\bullet}(\mathscr{C},\mathscr{D})$, i.e., the
simplicial spectrum that at level $n$ is

$N^{cy}_{n}(\mathscr{C},\mathscr{D})=\bigvee_{c_{0},...,c_{n}}\mathscr{C}(c_{0},c_{1})\wedge\mathscr{C}(c_{1},c_{2})\wedge...\wedge\mathscr{D}(c_{n},c_{0}),$

with faces given by composition in $\mathscr{C}$ or the action of
$\mathscr{C}$ on $\mathscr{D}$ (together with a cyclic permutation for the
last face) and degeneracies given by the unit, i.e.,

$\mathrm{THH}(\mathscr{C};\mathscr{D})=|N^{cy}_{\bullet}(\mathscr{C},\mathscr{D})|.$

Ponto’s key insight was that this definition could be axiomatized. Concretely,
let $\mathrm{Ho}(\mathscr{M}\mathrm{od})$ be the bicategory with pointwise-
cofibrant small spectral categories as objects and with the morphism category
from $\mathscr{C}$ to $\mathscr{C}^{\prime}$ equal to the homotopy category of
$(\mathscr{C},\mathscr{C}^{\prime})$-bimodules,
$\mathrm{Ho}(\mathscr{M}\mathrm{od}_{(\mathscr{C},\mathscr{C}^{\prime})})$.
For every pointwise-cofibrant spectral category $\mathscr{C}$, topological
Hochschild homology gives rise to a collection of functors

$\mathrm{THH}:\mathrm{Ho}(\mathscr{M}\mathrm{od})(\mathscr{C},\mathscr{C})=\mathrm{Ho}(\mathscr{M}\mathrm{od}_{(\mathscr{C},\mathscr{C})})\to\mathrm{Ho}(\mathrm{Sp}):\mathscr{D}\mapsto\mathrm{THH}(\mathscr{C};\mathscr{D}),$

satisfying certain properties, which become the axioms in the definition of a
shadow, which we recall now.

######  Definition 2.3 ([CP19, Definition 2.16]).

Let $\mathscr{B}$ be a bicategory with associator $a$, left unit $l$, and
right unit $r$, and let $\mathscr{D}$ be a category. A shadow on $\mathscr{B}$
with values in $\mathscr{D}$ is a functor

$\langle\langle-\rangle\rangle:\coprod_{X\in\mathrm{Obj}(\mathscr{B})}\mathscr{B}(X,X)\to\mathscr{D}$

that satisfies the following conditions.

For every pair of 1-morphisms $F:X\to Y$ and $G:Y\to X$ in $\mathscr{B}$ that
are composable in either order, there is a natural isomorphism

$\theta:\langle\langle FG\rangle\rangle\xrightarrow{\ \cong\ }\langle\langle
GF\rangle\rangle$

such that for all $F:X\to Y$, $G:Y\to Z$, $H:Z\to X$, and $K:X\to X$ the
following diagrams in $\mathscr{D}$ commute up to natural isomorphism.

(2.4) ${\langle\langle
H(GF)\rangle\rangle}$${\langle\langle(GF)H\rangle\rangle}$${\langle\langle
G(FH)\rangle\rangle}$${\langle\langle(HG)F\rangle\rangle}$${\langle\langle
F(HG)\rangle\rangle}$${\langle\langle(FH)G\rangle\rangle}$$\scriptstyle{\theta}$$\scriptstyle{\langle\langle
a\rangle\rangle}$$\scriptstyle{\langle\langle
a\rangle\rangle}$$\scriptstyle{\theta}$$\scriptstyle{\langle\langle
a\rangle\rangle}$$\scriptstyle{\theta}$

(2.5) ${\langle\langle KU_{X}\rangle\rangle}$${\langle\langle
U_{X}K\rangle\rangle}$${\langle\langle KU_{X}\rangle\rangle}$${\langle\langle
K\rangle\rangle}$$\scriptstyle{\theta}$$\scriptstyle{\langle\langle
r\rangle\rangle}$$\scriptstyle{\langle\langle
l\rangle\rangle}$$\scriptstyle{\theta}$$\scriptstyle{\langle\langle
r\rangle\rangle}$

The primary example of a shadow is $\mathrm{THH}$ of spectral categories.

######  Theorem 2.6 ([CP19, Theorem 2.17]).

Topological Hochschild homology of spectral categories is a shadow, i.e., the
family of functors

$\Big{\\{}\mathrm{THH}(\mathscr{C};-):\mathrm{Ho}(\mathscr{M}\mathrm{od}_{(\mathscr{C},\mathscr{C})})\to\mathrm{Ho}(\mathrm{Sp}):\mathscr{D}\mapsto\mathrm{THH}(\mathscr{C};\mathscr{D})\mid\mathscr{C}\in\mathrm{Ob}\,\mathrm{Ho}(\mathscr{M}\mathrm{od})\Big{\\}}$

is such that for all pointwise-cofibrant spectral categories $\mathscr{C}$ and
$\mathscr{C}^{\prime}$, $(\mathscr{C},\mathscr{C}^{\prime})$-bimodules
$\mathscr{D}$, and $(\mathscr{C}^{\prime},\mathscr{C})$-bimodules
$\mathscr{D}^{\prime}$, there is a natural isomorphism

$\theta:\mathrm{THH}(\mathscr{C},\mathscr{D}\wedge_{\mathscr{C}^{\prime}}\mathscr{D}^{\prime})\xrightarrow{\cong}\mathrm{THH}(\mathscr{C}^{\prime},\mathscr{D}^{\prime}\wedge_{\mathscr{C}}\mathscr{D})$

satisfying the conditions of 2.3.

It is common to simplify notation and to write

$\mathrm{THH}(\mathscr{C})=\mathrm{THH}(\mathscr{C};\mathscr{C}),$

where we consider $\mathscr{C}$ as a bimodule over itself in the obvious way.
Note that computing THH of a spectral category yields a spectrum, i.e., an
object in the enriching category.

The axioms of a shadow suffice to prove interesting properties that are known
to hold for $\mathrm{THH}$. For example, Ponto and Campbell prove a general
form of Morita equivalence for shadows [CP19, Proposition 4.6, Proposition
4.8] that implies the classical Morita equivalence for ring spectra [CP19,
Example 5.10].

In 3.19 we show that these properties stem from a deep connection between
shadows and $\mathrm{THH}$. However, in order to understand this relation we
need a higher categorical approach to shadows that moves away from point-set
models. Laying the foundations of such an approach is the goal of the next
subsection.

###  2.2. $\mathrm{THH}$ of enriched $\infty$-categories

In this subsection we review the definition of $\mathrm{THH}$ of an enriched
$\infty$-category. Before we move on to the enriched setting, it is valuable
to explain why we cannot simply generalize 2.2.

An $\infty$-category $\mathscr{C}$ is a simplicial set satisfying additional
conditions, the so-called “inner horn lifting conditions” [Lur09]. These
lifting conditions imply that for any two objects (i.e., $0$-simplices) $x$
and $y$, the simplicial set $\mathrm{Map}(x,y)$ given by the fiber of the map
$\mathscr{C}^{\Delta^{1}}\to\mathscr{C}\times\mathscr{C}$ over the point
$(x,y)$ is a Kan complex and thus a homotopy-meaningful mapping space. There
is no direct composition map, however. Instead, for three objects $x,y,z$,
there is a zig-zag

$\mathrm{Map}(x,y)\times\mathrm{Map}(y,z)\xleftarrow{\simeq}\mathrm{Map}(x,y,z)\to\mathrm{Map}(x,z)$

where $\mathrm{Map}(x,y,z)$ is the fiber of the map
$\mathscr{C}^{\Delta^{2}}\to\mathscr{C}^{3}$ over $(x,y,z)$, and the lifting
conditions imply that the first map is a trivial Kan fibration.

This observation has two unfortunate implications.

  1. (1)

We cannot just define an enriched $\infty$-category in terms of the existence
of composition maps and thus need a more complicated notion of enrichment.

  2. (2)

We cannot just define $\mathrm{THH}$ via the cyclic bar construction, as the
simplicial structure relies on the existence of a direct composition map.

Fortunately, Gepner-Haugseng [GH15] and Hinich [Hin18] have developed notions
of $\infty$-categories enriched in a monoidal $\infty$-category, both of which
are based on an operadic approach to classical enriched category theory,
generalized to the $\infty$-categorical setting. Their constructions are very
powerful and have been used to prove deep results about enriched
$\infty$-categories, but are also very intricate and can be difficult to use
for computations.

As in this article we need a notion of enrichment only to study
$\mathrm{THH}$, we can focus on the case where the enrichment category is not
just monoidal, but actually symmetric monoidal. There is a much more
convenient way of defining enriched $\infty$-categories in the symmetric case,
due to Berman [Ber22]222This approach can be in fact generalized to encompass
the monoidal case, however, we do not require this generality., which also
provides a natural framework for a generalization of $\mathrm{THH}$.

The key idea is that of a bypass operation, which we define next. Recall that
a directed multigraph is a pair of sets $\Gamma=(S,E)$ of vertices and of
directed edges between them. For two multigraphs $\Gamma_{1},\Gamma_{2}$ with
the same set of vertices $S$, there is a bypass operation from $\Gamma_{1}$ to
$\Gamma_{2}$ if $\Gamma_{2}$ can be obtained from $\Gamma_{1}$ by a sequence
of the following combinatorial moves.

  1. (1)

Adding a loop

  2. (2)

Replacing a path of consecutive edges $e_{1},...,e_{n}$ by a single edge $e$.

For a precise definition, see [Ber22, Definition 2.1].

######  Definition 2.7.

Let $S$ be any set. The bypass category on $S$,
$\mathscr{B}\mathrm{ypass}_{S}$, has as objects all directed multigraphs with
fixed vertex set $S$ and a finite set of edges and as morphisms all bypass
operations between such.

Note that $\mathscr{B}\mathrm{ypass}_{S}$ admits a symmetric monoidal product
$\otimes$, where the set of edges of $\Gamma_{1}\otimes\Gamma_{2}$ is the
disjoint union of the sets of edges of $\Gamma_{1}$ and $\Gamma_{2}$, and the
unit is the multigraph with no edges.

######  Definition 2.8.

Let $\mathscr{V}$ be a symmetric monoidal $\infty$-category. A
$\mathscr{V}$-enriched $\infty$-category with set of objects $S$ is a
symmetric monoidal functor of $\infty$-categories

$\mathscr{C}:\mathscr{B}\mathrm{ypass}_{S}\to\mathscr{V}.$

The bypass category is designed to encode the structure of an enriched
category. For example, if $(X_{0},...,X_{n})$ denotes the multigraph with
$n+1$ vertices $X_{0},...,X_{n}$ and a unique edge from $X_{i}$ to $X_{i+1}$
for every $i$, then

$\mathscr{C}(X_{0},...,X_{n})\in\mathscr{V}$

can be viewed as the object in $\mathscr{V}$ of sequences of $n$ composable
maps $X_{0}\to...\to X_{n}$. In particular $\mathscr{C}(X_{0},X_{1})$ plays
the role of the mapping object. We can then use a bypass operation of type (2)
above to get the desired composition map

$\mathscr{C}(X_{0},X_{1},X_{2})\to\mathscr{C}(X_{0},X_{2}).$

The following result should thus not be very surprising.

######  Proposition 2.9 ([Ber22, Proposition 2.7]).

If $\mathscr{V}$ is a symmetric monoidal $\infty$-category, then the
definition of $\mathscr{V}$-enriched $\infty$-category given here agrees with
that in [GH15].

We can use this simplified definition of an enriched $\infty$-category (and in
particular the fact that $\mathscr{B}\mathrm{ypass}_{S}$ is a category) to
define $\mathrm{THH}$ of a $\mathscr{V}$-enriched $\infty$-category as
follows. Let $\Lambda$ be the category with finite cyclically ordered sets
$0<1<...<n<0$, for $n\geq 0$, as objects and maps that respect the cyclic
structure as morphisms. (For more details see [Ber22, Definition 3.3].) Let

$(\mathscr{O}_{hh_{\mathscr{V}}})_{\bullet}:\Lambda\to\mathcal{P}(\mathscr{B}\mathrm{ypass}_{S})$

be the functor that takes a cyclic set $0<1<...<n<0$ to the coproduct of
representables

$\coprod_{X_{0},...,X_{n}\in S}(X_{0},X_{1},...,X_{n},X_{0}),$

and let

$\mathscr{O}_{hh_{\mathscr{V}}}=\mathrm{colim}_{{\mathbbe{\Delta}}^{op}}(\mathscr{O}_{hh_{\mathscr{V}}})_{\bullet}\in\mathcal{P}(\mathscr{B}\mathrm{ypass}_{S}).$

We suppress the choice of the set $S$ of objects from the notation
$(\mathscr{O}_{hh_{\mathscr{V}}})_{\bullet}$ and
$\mathscr{O}_{hh_{\mathscr{V}}}$.

As Berman explains, any enriched $\infty$-category
$\mathscr{C}:\mathscr{B}\mathrm{ypass}_{S}\to\mathscr{V}$ extends to a
colimit-preserving functor

$\mathscr{C}_{*}:\mathcal{P}(\mathscr{B}\mathrm{ypass}_{S})\to\mathscr{V},$

which allows us to formulate the following generalization of $\mathrm{THH}$,
providing a way of associating an object in $\mathscr{V}$ to any
$\mathscr{V}$-enriched $\infty$-category.

######  Definition 2.10 ([Ber22, Definition 4.1]).

Let $\mathscr{C}$ be a $\mathscr{V}$-enriched $\infty$-category. The
$\mathscr{V}$-enriched Hochschild homology of $\mathscr{C}$ is

$\mathrm{HH}_{\mathscr{V}}(\mathscr{C})=\mathscr{C}_{*}(\mathscr{O}_{hh_{\mathscr{V}}})\in\mathrm{Obj}\mathscr{V}.$

######  Remark 2.11.

Notice, $\mathrm{HH}_{\mathscr{V}}$ is simply denoted by $\mathrm{THH}$ in
[Ber22], however, as we will analyze Hochschild homology with several
different enrichments, we have chosen a notation that makes the enrichment
explicit.

Tracing through the definitions, since $\mathscr{C}_{*}$ preserves colimits,
we see that $\mathrm{HH}_{\mathscr{V}}(\mathscr{C})$ is the colimit of a
cyclic diagram that at level $n$ is equivalent to

$\coprod_{X_{0},...,X_{n}\in
S}\mathscr{C}(X_{0},...,X_{n},X_{0})\simeq\coprod_{X_{0},...,X_{n}}\mathscr{C}(X_{0},X_{1})\otimes...\otimes\mathscr{C}(X_{n},X_{0})$

because $\mathscr{C}$ is symmetric monoidal,

######  Example 2.12.

If $\mathscr{V}=(\mathrm{Sp},\wedge)$, the symmetric monoidal
$\infty$-category of spectra with the smash product, then a
$\mathscr{V}$-enriched $\infty$-category $\mathscr{C}$ is a spectral
$\infty$-category, and $\mathrm{HH}_{\mathrm{Sp}}(\mathscr{C})$ is the
$\infty$-categorical analogue of 2.2, and we will hence simply denote it by
$\mathrm{THH}(\mathscr{C})$.

######  Example 2.13.

If $\mathscr{V}=(\mathscr{S},\times)$, the symmetric monoidal
$\infty$-category of spaces, then $\mathscr{V}$-enriched $\infty$-categories
are precisely non-enriched $\infty$-categories. More concretely, we get
complete Segal spaces [Rez01], as follows from [GH15, Theorem 4.4.7, Remark
5.3.10]. We hence denote $\mathrm{HH}_{\mathscr{S}}$ by
$\mathrm{HH}_{\infty}$.

######  Example 2.14.

If $\mathscr{V}=(\mathscr{C}\mathrm{at},\times)$, then we call a
$\mathscr{V}$-enriched $\infty$-category $\mathscr{C}$ a bicategory. Note that
if $\mathscr{C}$ is an bicategory, then
$\mathrm{HH}_{\mathscr{C}\mathrm{at}}(\mathscr{C})$ is a category, and we
hence use the notation $\mathrm{biHH}(\mathscr{C})$.

######  Example 2.15.

If $\mathscr{V}=(\mathscr{C}\mathrm{at}_{\infty},\times)$, then we call a
$\mathscr{V}$-enriched $\infty$-category $\mathscr{C}$ an
$(\infty,2)$-category. Note that if $\mathscr{C}$ is an $(\infty,2)$-category,
then $\mathrm{HH}_{\mathscr{C}\mathrm{at}_{\infty}}(\mathscr{C})$ is an
$\infty$-category, and we hence use the notation
$\mathrm{biHH}_{\infty}(\mathscr{C})$.

######  Remark 2.16.

There are many models of $(\infty,2)$-categories in the literature, such as
$2$-complicial sets [Ver08], $2$-fold complete Segal spaces [Bar05], and
$\Theta_{2}$-spaces [Rez10]. The model of $(\infty,2)$-categories constructed
in 2.15 corresponds to $2$-fold complete Segal spaces, as follows from the
proof in [GH15, Theorem 4.4.7, Remark 5.3.10], with the $\infty$-category of
complete Segal spaces css replacing the $\infty$-category $\mathscr{S}$.

Let $\mathscr{C}$ be a $\mathscr{V}$-enriched $\infty$-category with set of
objects $S$, and let $\tau:\mathrm{HH}_{\mathscr{V}}(\mathscr{C})\to V$ be a
morphism in $\mathscr{V}$. From $\tau$ we can derive a family of morphisms in
$\mathscr{V}$

$\\{\tau_{X}:\mathscr{C}(X,X)\to V\mid X\in S\\}.$

The cyclic structure of $(\mathscr{O}_{hh_{\mathscr{V}}})_{\bullet}$ implies
that

$\textstyle{\mathscr{C}(X_{0},X_{1})\otimes\mathscr{C}(X_{1},X_{0})\simeq\mathscr{C}(X_{0},X_{1},X_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{C}(X_{0},X_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{X_{0}}}$$\textstyle{V}$$\textstyle{\mathscr{C}(X_{1},X_{0})\otimes\mathscr{C}(X_{0},X_{1})\simeq\mathscr{C}(X_{1},X_{0},X_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{C}(X_{1},X_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{X_{1}}}$

commutes for all $X_{0},X_{1}\in S$, where the horizontal “composition” maps
arise from bypasses, and the lefthand vertical arrow is given by the cyclic
structure. In other words, the value of $\tau$ on a composite of two 1-cells
that can be composed in either order is independent of the order of
composition, just as the trace of a product of two matrices that can be
multiplied in either order is independent of the order of multiplication. This
observation justifies the terminology in the next definition.

######  Definition 2.17.

Let $\mathscr{C}$ be a $\mathscr{V}$-enriched $\infty$-category, and let $V$
be an object in $\mathscr{V}$. A trace out of $\mathscr{C}$ into $V$ is a
morphism $\mathrm{HH}_{\mathscr{V}}(\mathscr{C})\to V$ in $\mathscr{V}$.

##  3\. Shadows vs. Traces

In this section we compare the point-set and $\infty$-categorical approaches
to Hochschild homology and deduce an equivalence between shadows on a
bicategory $\mathscr{B}$ with values in a category $\mathscr{D}$ and traces
out of $\mathscr{B}$ into $\mathscr{D}$. To make sense of this comparison, we
need in particular to explain how to define a category that can reasonably be
called the Hochschild homology of $\mathscr{B}$.

######  Remark 3.1.

Motivated by the analysis in Subsection 2.2 and in order to simplify notation,
for a given $\mathscr{V}$-enriched $\infty$-category $\mathscr{C}$, we abuse
notation and write

$\mathscr{C}(X_{0},X_{1},...,X_{n},X_{0})=\mathscr{C}(X_{0},X_{1})\otimes\mathscr{C}(X_{1},X_{2})...\otimes\mathscr{C}(X_{n},X_{0}).$

###  3.1. Hochschild homology of bicategories

Since bicategories are $(2,2)$-categories, i.e., categories enriched in
categories [GH15, Definition 6.1.1], it follows from [Ber22, Proposition 2.7]
that a bicategory is a $\mathscr{C}\mathrm{at}$-enriched category in the sense
of 2.8, where $\mathscr{C}\mathrm{at}$ is equipped with its Cartesian
symmetric monoidal structure. We can therefore apply 2.10 to associate to any
bicategory $\mathscr{B}$ a category $\mathrm{biHH}(\mathscr{B})$, which is its
topological Hochschild homology.

The following, more explicit description of $\mathrm{biHH}(\mathscr{B})$
proves useful in the next subsection.

######  Proposition 3.2.

For every bicategory $\mathscr{B}$, the category $\mathrm{biHH}(\mathscr{B})$
is equivalent to the colimit of

(3.3)
${\displaystyle\coprod_{X_{0}}\mathscr{B}(X_{0},X_{0})}$${\displaystyle\coprod_{X_{0},X_{1}}\mathscr{B}(X_{0},X_{1},X_{0})}$${\displaystyle\coprod_{X_{0},X_{1},X_{2}}\mathscr{B}(X_{0},X_{1},X_{2},X_{0})}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{2}}$$\scriptstyle{d_{0}}$

in $\mathscr{C}\mathrm{at}$.

###### Proof.

Since $\mathscr{C}\mathrm{at}$ is a $(2,1)$-category, for any simplicial
diagram $F:{\mathbbe{\Delta}}^{op}\to\mathscr{C}\mathrm{at}$, there is an
equivalence

$\mathrm{colim}({\mathbbe{\Delta}}^{op}\xrightarrow{F}\mathscr{C}\mathrm{at})\simeq\mathrm{colim}(({\mathbbe{\Delta}}_{\leq
2})^{op}\xrightarrow{i^{op}}{\mathbbe{\Delta}}^{op}\xrightarrow{F}\mathscr{C}\mathrm{at})$

where $i:{\mathbbe{\Delta}}_{\leq 2}\to{\mathbbe{\Delta}}$ is the natural
inclusion map of the full subcategory with objects $[0],[1]$ and $[2]$ [Lur17,
Lemma 1.3.3.10]. ∎

######  Remark 3.4.

While it is not yet proven in full generality that the construction of
$\mathrm{HH}_{\mathscr{V}}$ (2.10) is functorial, the explicit formula in
Diagram 3.3 implies that when $\mathscr{V}=(\mathscr{C}\mathrm{at},\times)$,
the construction of $\mathrm{biHH}$ is indeed functorial.

######  Remark 3.5.

In order to define $\mathrm{biHH}$ of a bicategory, none of the definitions in
Subsection 2.2 is actually needed: we could simply have defined it to be the
colimit of 2.10 However, in Section 6, when we compare $\mathrm{biHH}$ of
bicategories and $\mathrm{biHH}_{\infty}$ of $(\infty,2)$-categories, the more
abstract description proves necessary.

######  Notation 3.6.

Let $\mathscr{B}$ be a bicategory. For objects $B_{1},B_{2}$ chosen among

$\coprod_{X_{0}}\mathscr{B}(X_{0},X_{0}),\coprod_{X_{0},X_{1}}\mathscr{B}(X_{0},X_{1},X_{0}),\coprod_{X_{0},X_{1},X_{2}}\mathscr{B}(X_{0},X_{1},X_{2},X_{0})$

we use $\mathrm{Fun}^{\Delta}(B_{1},B_{2})$ to denote the subcategory of
$\mathrm{Fun}(B_{1},B_{2})$ consisting of functors and natural isomorphisms
generated by those in Diagram 3.3.

Part of the data of a bicategory are explicit composition maps, which one can
view as giving rise to the face maps $d_{i}$ in 2.10. In particular, again
abusing notation somewhat,
$d_{0},d_{1}:\mathscr{B}(X_{0},X_{1},X_{0})\to\mathscr{B}(X_{i},X_{i})$ (where
$i=0,1$) are given by

$d_{0}(F,G)=FG,\ d_{1}(F,G)=GF.$

Moreover, the face maps on $\mathscr{C}(X_{0},X_{1},X_{2},X_{0})$ are given by

$d_{0}(F,G,H)=(FG,H),\ d_{1}(F,G,H)=(F,GH),\ d_{2}(F,G,H)=(HF,G)$

This explicit structure plays an important role in later sections.

It is important to note that the colimit of Diagram 3.3 is evaluated in the
$(2,1)$-category $\mathscr{C}\mathrm{at}$, which differs from the
$1$-categorical colimit in the category of categories, but rather corresponds
to the pseudocolimit [Kel89]. Computing general pseudocolimits can be quite
challenging, although there certain helpful methods via weighted colimits of
enriched categories [Kel05] and homotopy colimits of model categories [Gam08].
The example below illustrates the challenges that arise even in simple
examples.

######  Example 3.7.

Let $(\mathscr{C},\otimes,I)$ be a monoidal category, and let $B\mathscr{C}$
be the bicategory with one object and morphism category given by
$\mathscr{C}$, with composition given by the monoidal product. Interpreting
Diagram 3.3 in this particular case, we see that $\mathrm{biHH}(B\mathscr{C})$
is the colimit of the diagram

${\displaystyle\mathscr{C}}$${\displaystyle\mathscr{C}\times\mathscr{C}}$${\displaystyle\mathscr{C}\times\mathscr{C}\times\mathscr{C}}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{2}}$$\scriptstyle{d_{0}}$

.

Although this diagram is easy to describe explicitly, there is no easy way to
compute its colimit. In general the only description available of the category
$\mathrm{biHH}(B\mathscr{C})$ is formulated in terms of generators and
relations [ML98, Section II.8], which is generally not useful for
computations.

In certain cases we can give a more explicit description of
$\mathrm{biHH}(B\mathscr{C})$, which is the content of Section 5, resulting in
5.15.

One case in which we can more easily compute $\mathrm{biHH}$ of a bicategory,
using homotopy theory, arises when the bicategory is actually a
$(2,0)$-category, i.e., all $1$-cells and $2$-cells are equivalences. We call
such a bicategory a bigroupoid and denote the (3,1)-category of bigroupoids by
$\mathscr{G}\mathrm{rpd}_{2}$. To compute $\mathrm{biHH}$ of a bigroupoid, we
need to understand the relationship between bigroupoids and spaces.

######  Remark 3.8.

Let $\mathscr{S}$ be the $\infty$-category of spaces. By [GH15, Corollary
6.1.10], there is an adjunction of $\infty$-categories

(3.9) ${\mathscr{S}}$${\mathscr{G}\mathrm{rpd},}$$\scriptstyle{\Pi_{\leq
1}}$$\scriptstyle{\bot}$$\scriptstyle{N}$

where $\mathscr{G}\mathrm{rpd}$ denotes the $(2,1)$-category of groupoids, of
which the right adjoint is fully faithful, with essential image given by
$1$-truncated spaces (spaces with trivial homotopy groups above degree $1$).
The right adjoint takes a groupoid $\mathscr{G}$ to a Kan complex of which the
$0$-cells are the objects of $\mathscr{G}$, while for any two objects $X,Y$,
there is an isomorphism of sets
$\mathrm{Map}_{N\mathscr{G}}(X,Y)\cong\mathscr{G}(X,Y)$, where
$\mathrm{Map}_{N\mathscr{G}}$ denotes the path space.

The same corollary ([GH15, Corollary 6.1.10]) implies that there is an
adjunction

(3.10) ${\mathscr{S}}$${\mathscr{G}\mathrm{rpd}_{2}}$$\scriptstyle{\Pi_{\leq
2}}$$\scriptstyle{\bot}$$\scriptstyle{N_{2}}$

of which the right adjoint is fully faithful, with essential image given by
$2$-truncated spaces. By [GH15, Lemma 6.1.9] the functor $N_{2}$ takes a
bigroupoid $\mathscr{G}$ to a Kan complex $N_{2}\mathscr{G}$, of which the
$0$-cells are the objects of $\mathscr{G}$, while for any two objects $X,Y$,
there is an equivalence of groupoids

$\Pi_{1}\mathrm{Map}_{N_{2}\mathscr{G}}(X,Y)\simeq\mathscr{G}(X,Y).$

We can use the adjunction above to compute $\mathrm{THH}$ of a bigroupoid.

######  Example 3.11.

For any bigroupoid $\mathscr{G}$, there is an equivalence

$\mathrm{biHH}(\mathscr{G})=|\coprod_{X_{0},...,X_{n}}\mathscr{G}(X_{0},X_{1},...,X_{0})|\simeq|\coprod_{X_{0},...,X_{n}}\Pi_{1}\mathrm{Map}_{N_{2}\mathscr{G}}(X_{0},X_{1},...,X_{0})|$
$\simeq\Pi_{1}|\coprod_{X_{0},...,X_{n}}\mathrm{Map}_{N_{2}\mathscr{G}}(X_{0},X_{1},...,X_{0})|$

where $\mathrm{Map}_{N_{2}\mathscr{G}}(X_{0},X_{1},...,X_{0})$ denotes the
space of loops through the vertices $X_{0},X_{1},...,X_{n}$ and the
equivalence follows from the previous remark and the fact that $\Pi_{1}$
commutes with colimits. The geometric realization of this diagram of spaces is
known as unstable topological Hochschild homology and has been computed to be
the free loop space $(N_{2}\mathscr{G})^{S^{1}}$ [HS19, Corollary 4, Page
858]. Since $N_{2}$ is fully faithful and $N_{2}B\mathbb{Z}=S^{1}$, it follows
that

$\mathrm{biHH}(\mathscr{G})\simeq(N_{2}\mathscr{G})^{S^{1}}=(N_{2}\mathscr{G})^{N_{2}B\mathbb{Z}}\simeq\mathrm{Fun}(\mathscr{B}\mathbb{Z},\mathscr{G}),$

where the last equivalence follows from the fact that $N_{2}$ is fully
faithful.

Returning to the general case, we can use the explicit colimit presentation of
$\mathrm{biHH}(\mathscr{B})$ (Diagram 3.3) to better describe the category
$\mathrm{Fun}\big{(}\mathrm{biHH}(\mathscr{B}),\mathscr{D}\big{)}$ of traces
out of a bicategory $\mathscr{B}$ into a category $\mathscr{D}$. Let
$\big{(}({\mathbbe{\Delta}}_{\leq 2})^{op}\big{)}^{\rhd}$ be the join of
$({\mathbbe{\Delta}}_{\leq 2})^{op}$ with the terminal category. Intuitively
$\big{(}({\mathbbe{\Delta}}_{\leq 2})^{op}\big{)}^{\rhd}$ is the category
$({\mathbbe{\Delta}}_{\leq 2})^{op}$ equipped with a new terminal object. Note
there are evident inclusion functors $({\mathbbe{\Delta}}_{\leq
2})^{op}\to\big{(}({\mathbbe{\Delta}}_{\leq 2})^{op}\big{)}^{\rhd}$ and
$[0]\to\big{(}({\mathbbe{\Delta}}_{\leq 2})^{op}\big{)}^{\rhd}$, sending the
unique object $0$ to the terminal object we denote by $f$.

######  Definition 3.12.

Let $\mathscr{B}$ be a bicategory and $\mathscr{D}$ a category. Let
$\mathscr{B}_{*}:\mathcal{P}(\mathscr{B}\mathrm{ypass}_{\mathscr{B}_{0}})\to\mathscr{C}\mathrm{at}$
be the colimit-preserving functor extending the functor
$\mathscr{B}\mathrm{ypass}_{\mathscr{B}_{0}}\to\mathscr{C}\mathrm{at}$
encoding the bicategory structure of $\mathscr{B}$.

The category $\mathscr{C}\mathrm{ocone}(\mathscr{B},\mathscr{D})$ of
_$\mathscr{B}$ -cocones taking values in $\mathscr{D}$_ has as objects the
pseudofunctors $(({\mathbbe{\Delta}}_{\leq
2})^{op})^{\rhd}\to\mathscr{C}\mathrm{at}$ that fit into the diagram

${({\mathbbe{\Delta}}_{\leq 2})^{op}}$${(({\mathbbe{\Delta}}_{\leq
2})^{op})^{\rhd}}$${\mathscr{C}\mathrm{at}}$${\\{f\\}}$$\scriptstyle{\mathscr{B}_{*}\circ\mathscr{O}_{bihh}}$$\scriptstyle{\mathscr{D}}$

and pseudonatural transformations as morphisms.

Concretely we can depict an object in
$\mathscr{C}\mathrm{ocone}(\mathscr{B},\mathscr{D})$ as a diagram in
$\mathscr{C}\mathrm{at}$ of the form

(3.13)
${\displaystyle\coprod_{X_{0}}\mathscr{B}(X_{0},X_{0})}$${\displaystyle\coprod_{X_{0},X_{1}}\mathscr{B}(X_{0},X_{1},X_{0})}$${\displaystyle\coprod_{X_{0},X_{1},X_{2}}\mathscr{B}(X_{0},X_{1},X_{2},X_{0})}$${\mathscr{D}}$$\scriptstyle{C_{0}}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{0}}$$\scriptstyle{C_{1}}$$\scriptstyle{d_{2}}$$\scriptstyle{d_{0}}$$\scriptstyle{C_{2}}$

and a morphism as a commutative diagram of natural transformations of the form

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Since the objects in $\mathscr{C}\mathrm{ocone}(\mathscr{B},\mathscr{D})$ are
pseudofunctors, the commutativity in Diagram 3.13 and Diagram 3.14 hold up to
appropriate choices of natural isomorphisms. Understanding the precise data of
objects in $\mathscr{C}\mathrm{ocone}(\mathscr{B},\mathscr{D})$ is an
important aspect of the proof of 3.19. As this description is quite technical,
it has been relegated to Appendix A.

######  Remark 3.15.

For every bicategory $\mathscr{B}$, there is a $\mathscr{B}$-cocone taking
values in $\mathrm{biHH}(\mathscr{B})$, given by the maps into the colimit. As
we observe below, this $\mathscr{B}$-cocone is universal, in the sense that
every other $\mathscr{B}$-cocone factors through it.

The identification below follows immediately from the colimit description of
$\mathrm{biHH}(\mathscr{B})$ (Diagram 3.3) and the universal property of
colimits.

######  Proposition 3.16.

The functor

$\mathrm{Comp}:\mathrm{Fun}\big{(}\mathrm{biHH}(\mathscr{B}),\mathscr{D}\big{)}\xrightarrow{\simeq}\mathscr{C}\mathrm{ocone}(\mathscr{B},\mathscr{D}),$

which composes a functor $\mathrm{biHH}(\mathscr{B})\to\mathscr{D}$ with the
universal $\mathscr{B}$-cocone taking values in $\mathrm{biHH}(\mathscr{B})$,
is an equivalence of categories, which is natural in $\mathscr{B}$ and
$\mathscr{D}$.

In other words, traces out of $\mathscr{B}$ into $\mathscr{D}$ (2.17) are
equivalent to $\mathscr{B}$-cocones taking values in $\mathscr{D}$.

###  3.2. The equivalence between shadows and traces

We are finally ready to establish the main result of this section, that
shadows (Subsection 2.1) are traces of bicategories (Subsection 2.2). To state
the result precisely, we first introduce a notion of morphisms between
shadows.

######  Definition 3.17.

Let $\mathscr{B}$ be a bicategory and $\mathscr{D}$ a category. The category
of shadow functors from $\mathscr{B}$ to $\mathscr{D}$, denoted
$\mathscr{S}\mathrm{ha}(\mathscr{B},\mathscr{D})$, is specified as follows.

  1. (1)

The objects are shadows, i.e., pairs
$\big{(}\langle\langle-\rangle\rangle,\theta\big{)}$ satisfying the conditions
of 2.3.

  2. (2)

Given two shadows
$\big{(}\langle\langle-\rangle\rangle_{1},\theta_{1}\big{)}$,
$\big{(}\langle\langle-\rangle\rangle_{2},\theta_{2}\big{)}$, a morphism
between them consists of a natural transformation

$\alpha_{-}:\langle\langle-\rangle\rangle_{1}\to\langle\langle-\rangle\rangle_{2}$

such that for any pair of 1-morphisms $F:X_{0}\to X_{1}$, $G:X_{1}\to X_{0}$
in $\mathscr{B}$, the diagram

(3.18) ${\langle\langle FG\rangle\rangle_{1}}$${\langle\langle
GF\rangle\rangle_{1}}$${\langle\langle FG\rangle\rangle_{2}}$${\langle\langle
GF\rangle\rangle_{2},}$$\scriptstyle{\theta_{1}}$$\scriptstyle{\alpha_{FG}}$$\scriptstyle{\alpha_{GF}}$$\scriptstyle{\theta_{2}}$

in $\mathscr{D}$ commutes, and $\alpha$ commutes with associators and unitors
as well.

######  Theorem 3.19.

For any bicategory $\mathscr{B}$ and category $\mathscr{D}$, there is an
equivalence of categories

$\mathrm{Fun}\big{(}\mathrm{biHH}(\mathscr{B}),\mathscr{D}\big{)}\xrightarrow{\
\simeq\ }\mathscr{S}\mathrm{ha}(\mathscr{B},\mathscr{D})$

that factors via the functor $\mathrm{Comp}$ (3.16) through the category
$\mathscr{C}\mathrm{ocone}(\mathscr{B},\mathscr{D})$ (3.12) and that is
natural in $\mathscr{B}$ and $\mathscr{D}$. Here
$\mathscr{S}\mathrm{ha}(\mathscr{B},\mathscr{D})$ denotes the category of
shadow functors from $\mathscr{B}$ to $\mathscr{D}$ (3.17).

Before we proceed to the proof, we mention some formal, but valuable,
implications of this theorem.

######  Corollary 3.20.

Let $\mathscr{B}$ be a bicategory. There is a universal shadow functor
$\langle\langle-\rangle\rangle_{u}$ from $\mathscr{B}$ to
$\mathrm{biHH}(\mathscr{B})$ such that for every other shadow functor
$\langle\langle-\rangle\rangle$ from $\mathscr{B}$ to $\mathscr{D}$, there is
a unique functor $F:\mathrm{biHH}(\mathscr{B})\to\mathscr{D}$ and equality
$F_{*}\langle\langle-\rangle\rangle_{u}=\langle\langle-\rangle\rangle$.

###### Proof.

Let $\langle\langle-\rangle\rangle_{u}$ be the shadow that corresponds to the
identity functor $\mathrm{biHH}(\mathscr{B})\to\mathrm{biHH}(\mathscr{B})$
under the equivalence in 3.19. The desired equality now follows from the fact
that the following square commutes by naturality

${\mathrm{Fun}(\mathrm{biHH}(\mathscr{B}),\mathrm{THH}(\mathscr{B}))}$${\mathscr{S}\mathrm{ha}(\mathscr{B},\mathrm{biHH}(\mathscr{B}))}$${\mathrm{Fun}(\mathrm{biHH}(\mathscr{B}),\mathscr{D})}$${\mathscr{S}\mathrm{ha}(\mathscr{B},\mathscr{D})}$$\scriptstyle{\simeq}$$\scriptstyle{F_{*}}$$\scriptstyle{F_{*}}$$\scriptstyle{\simeq}$

,

for any functor $F:\mathrm{biHH}(\mathscr{B})\to\mathscr{D}$. ∎

######  Corollary 3.21.

For any bicategory $\mathscr{B}$, the functor
$\mathscr{S}\mathrm{ha}(\mathscr{B},-):\mathscr{C}\mathrm{at}\to\mathscr{C}\mathrm{at}$
is corepresentable and hence preserves limits.

We now move on to the proof of 3.19.

###### Proof of 3.19.

The proof is quite long, so we break it down into six steps.

Step (I) Reduction to $\mathscr{S}\mathrm{t}$ and $\mathscr{U}\mathrm{n}$: By
3.16 the functor

$\mathrm{Comp}:\mathrm{Fun}\big{(}\mathrm{biHH}(\mathscr{B}),\mathscr{D}\big{)}\to\mathscr{C}\mathrm{ocone}(\mathscr{B},\mathscr{D})$

is a natural equivalence of categories. In order to prove the theorem, we
define two functors natural in $\mathscr{B}$ and $\mathscr{D}$ that we think
of as performing “strictification” and “unstrictification” processes (3.28),

$\mathscr{S}\mathrm{t}:\mathscr{C}\mathrm{ocone}(\mathscr{B},\mathscr{D})\to\mathscr{S}\mathrm{ha}(\mathscr{B},\mathscr{D})$

and

$\mathscr{U}\mathrm{n}:\mathscr{S}\mathrm{ha}(\mathscr{B},\mathscr{D})\to\mathscr{C}\mathrm{ocone}(\mathscr{B},\mathscr{D}),$

and then prove that they are mutually inverse.

Step (II) The functor $\mathscr{S}\mathrm{t}$ on objects: For any cocone
$C=(C_{0},C_{1},C_{2})$ in
$\mathscr{C}\mathrm{ocone}(\mathscr{B},\mathscr{D})$, the functor underlying
$\mathscr{S}\mathrm{t}(C)$ is

$C_{0}:\coprod_{X_{0}}\mathscr{B}(X_{0},X_{0})\to\mathscr{D}.$

The natural isomorphism witnessing the trace-like property of $C_{0}$ is the
composite of the natural isomorphisms witnessing the commutativity of the
lefthand triangles in Diagram 3.13,

$\theta_{C}:C_{0}d_{0}\xrightarrow{\cong}C_{1}\xrightarrow{\cong}C_{0}d_{1}.$

Since this assignment is clearly natural in $\mathscr{B}$ and $\mathscr{D}$,
what remains is to prove that $(C_{0},\theta_{C})$ satisfies the axioms of a
shadow, i.e., that the two diagrams of 2.3 commute. First we show that Diagram
2.4 commutes.

The fact that Diagram 3.13 is a cocone means that the image of the map

$\displaystyle
C_{0}:\mathrm{Fun}^{\Delta}\big{(}\coprod_{X_{0},X_{1},X_{2}}\mathscr{B}(X_{0},X_{1},X_{2},X_{0}),\coprod_{X_{0}}\mathscr{B}(X_{0},X_{0})\big{)}\to\mathrm{Fun}\big{(}\coprod_{X_{0},X_{1},X_{2}}\mathscr{B}(X_{0},X_{1},X_{2},X_{0}),\mathscr{D}\big{)}$

must be contractible (here we are using 3.6). Concretely, this means the image
under postcomposition with $C_{0}$ of any two functors
$\coprod_{X_{0},X_{1},X_{2}}\mathscr{B}(X_{0},X_{1},X_{2},X_{0})\to\coprod_{X_{0}}\mathscr{B}(X_{0},X_{0})$
induced by the simplicial operators must be naturally isomorphic in a unique
way (as a non-unique natural isomorphism would give us a non-trivial loop).

Diagram 3.13 gives rise to the cube below, in which each face commutes up to
natural isomorphism.

${\displaystyle\coprod_{X_{0},X_{1},X_{2}}\mathscr{B}(X_{0},X_{1},X_{2},X_{0})}$${\displaystyle\coprod_{X_{0},X_{1}}\mathscr{B}(X_{0},X_{1},X_{0})}$${\displaystyle\coprod_{X_{0},X_{1}}\mathscr{B}(X_{0},X_{1},X_{0})}$${\displaystyle\coprod_{X_{0}}\mathscr{B}(X_{0},X_{0})}$${\displaystyle\coprod_{X_{0},X_{1}}\mathscr{B}(X_{0},X_{1},X_{0})}$${\displaystyle\coprod_{X_{0}}\mathscr{B}(X_{0},X_{0})}$${\displaystyle\coprod_{X_{0}}\mathscr{B}(X_{0},X_{0})}$${\mathscr{D}}$$\scriptstyle{a}$$\scriptstyle{a}$$\scriptstyle{\theta}$$\scriptstyle{\theta}$$\scriptstyle{d_{2}}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{0}}$$\scriptstyle{a}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{0}}$$\scriptstyle{\theta}$$\scriptstyle{C_{0}}$$\scriptstyle{C_{0}}$$\scriptstyle{C_{0}}$

There are exactly six paths in this cube from the top left corner to the
bottom right corner,

$\displaystyle\coprod_{X_{0},X_{1},X_{2}}\mathscr{B}(X_{0},X_{1},X_{2},X_{0})\to\mathscr{D},$

corresponding to the six objects in Diagram 2.4. The natural isomorphisms on
the faces of the cube correspond to the morphisms between the objects in
Diagram 2.4. As explained above, the diagram of natural isomorphisms must
commute since Diagram 3.13 is a cocone.

It remains to prove that Diagram 2.5 commutes. By symmetry it suffices to
verify commutativity of the left-hand triangle, which we do by an argument
similar to that above. Since Diagram 3.13 is a cocone, the image of

$C_{0}:\mathrm{Fun}^{\Delta}\big{(}\coprod_{X_{0}}\mathscr{B}(X_{0},X_{0}),\coprod_{X_{0}}\mathscr{B}(X_{0},X_{0})\big{)}\to\mathrm{Fun}\big{(}\coprod_{X_{0}}\mathscr{B}(X_{0},X_{0}),\mathscr{D}\big{)}$

must be contractible, which means that any two functors from
$\mathscr{B}(X_{0},X_{0})$ to $\mathscr{D}$ induced by the simplicial diagram
in the domain functor category must be natural isomorphic in a unique manner.

Since the morphisms in the left-hand triangle in Diagram 2.5 are exactly the
natural isomorphisms in the diagram below, we can conclude by the remark above
that the triangle commutes as desired.

${\displaystyle\coprod_{X_{0}}\mathscr{B}(X_{0},X_{0})}$${\displaystyle\coprod_{X_{0},X_{1}}\mathscr{B}(X_{0},X_{1},X_{0})}$${\coprod_{X_{0}}\mathscr{B}(X_{0},X_{0})}$${\displaystyle\coprod_{X_{0}}\mathscr{B}(X_{0},X_{0})}$${\mathscr{D}}$$\scriptstyle{s_{0}}$$\scriptstyle{id}$$\scriptstyle{id}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{0}}$$\scriptstyle{C_{0}}$$\scriptstyle{C_{0}}$$\scriptstyle{r}$$\scriptstyle{l}$$\scriptstyle{\theta}$

Step (III) The functor $\mathscr{S}\mathrm{t}$ on morphisms: We now explain
how to define $\mathscr{S}\mathrm{t}$ on morphisms and show that it is indeed
a functor. Let

$(\alpha_{0},\alpha_{1},\alpha_{2}):(C_{0},C_{1},C_{2})\Rightarrow(C_{0}^{\prime},C_{1}^{\prime},C_{2}^{\prime})$

be a morphism of cocones. We just proved that the functors $C_{0}$ and
$C_{0}^{\prime}$ can be equipped with natural transformations $\theta$ and
$\theta^{\prime}$ with respect to which they are shadows. By definition

$\alpha_{0}:C_{0}\to C_{0}^{\prime}$

is a natural transformation. We need to check that $\alpha_{0}$ satisfies the
conditions formulated in 3.17 with respect to $\theta$ and $\theta^{\prime}$,
allowing us to set
$\mathscr{S}\mathrm{t}(\alpha_{0},\alpha_{1},\alpha_{2})=\alpha_{0}$.
Preservation of composition and of identities by $\mathscr{S}\mathrm{t}$ is
then obvious.

The compatibility of $\alpha_{0}$ and $\alpha_{1}$ with the natural
isomorphisms in the cocones $C$ and $C^{\prime}$ implies that the diagram

${C_{0}d_{0}}$${C_{1}}$${C_{0}d_{1}}$${C_{0}^{\prime}d_{0}}$${C_{1}^{\prime}}$${C_{0}^{\prime}d_{1}}$$\scriptstyle{\theta}$$\scriptstyle{\alpha_{0}}$$\scriptstyle{\alpha_{1}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\alpha_{0}}$$\scriptstyle{\theta^{\prime}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

commutes and hence that Diagram 3.18 commutes. It remains to show that
$\alpha_{0}$ commutes with associators and unitors, which follows from the
diagrams below, by standard whiskering arguments in bicategories, again using,
as in the previous step, that $(C_{0},C_{1},C_{2})$ and
$(C^{\prime}_{0},C^{\prime}_{1},C^{\prime}_{2})$ are cocones.

${\displaystyle\coprod_{X_{0},X_{1},X_{0}}\mathscr{B}(X_{0},X_{1},X_{0})}$${\displaystyle\coprod_{X_{0},X_{1},X_{2}}\mathscr{B}(X_{0},X_{1},X_{2},X_{0})}$${\displaystyle\coprod_{X_{0}}\mathscr{B}(X_{0},X_{0})}$${\mathscr{D}}$${\displaystyle\coprod_{X_{0},X_{1},X_{0}}\mathscr{B}(X_{0},X_{1},X_{0})}$$\scriptstyle{d_{0}}$$\scriptstyle{a}$$\scriptstyle{d_{0}}$$\scriptstyle{d_{1}}$$\scriptstyle{C_{0}}$$\scriptstyle{C_{0}^{\prime}}$$\scriptstyle{d_{0}}$$\scriptstyle{\alpha_{0}}$

${\displaystyle\coprod_{X_{0},X_{1}}\mathscr{B}(X_{0},X_{1},X_{0})}$${\displaystyle\coprod_{X_{0}}\mathscr{B}(X_{0},X_{0})}$${\displaystyle\coprod_{X_{0}}\mathscr{B}(X_{0},X_{0})}$${\mathscr{D}}$${\displaystyle\coprod_{X_{0},X_{1}}\mathscr{B}(X_{0},X_{1},X_{0})}$$\scriptstyle{d_{0}}$$\scriptstyle{\mathrm{Id}}$$\scriptstyle{s_{0}}$$\scriptstyle{s_{0}}$$\scriptstyle{C_{0}}$$\scriptstyle{C_{0}^{\prime}}$$\scriptstyle{d_{1}}$$\scriptstyle{r}$$\scriptstyle{l}$$\scriptstyle{\alpha_{0}}$

Thus $\alpha_{0}$ is indeed a morphism of shadows.

Step (IV) The functor $\mathscr{U}\mathrm{n}$ on objects: Given any shadow
functor $\big{(}\langle\langle-\rangle\rangle,\theta\big{)}$ from
$\mathscr{B}$ to $\mathscr{D}$, we construct a $\mathscr{B}$-cocone taking
values in $\mathscr{D}$, using Lemma A.18 in the case where the target
$2$-category is $\mathscr{C}\mathrm{at}$, and the functor $F$ is
$\mathscr{O}_{bihh}$. We need thus to choose an appropriate category, functor,
and natural isomorphism that satisfy the conditions stated in Lemma A.18.

The extra condition on the extension in the definition of
$\mathscr{B}$-cocones (3.12) implies that the category chosen as the image of
the terminal object must be $\mathscr{D}$. The functor we choose as the image
of $t_{0}:0\to f$ is
$\langle\langle-\rangle\rangle:\coprod_{X_{0}}\mathscr{B}(X_{0},X_{0})\to\mathscr{D}$,
while the natural isomorphism chosen should clearly be that given as part of
the shadow structure, $\theta:\langle\langle
FG\rangle\rangle\cong\langle\langle GF\rangle\rangle$.

We need to prove that the two equalities of 2-cells stated in Lemma A.18 hold
for the choices that we have made.

The first condition in Lemma A.18 translates to the following diagram

(3.22) $\leavevmode\hbox to137.68pt{\vbox
to70.77pt{\pgfpicture\makeatletter\hbox{\hskip
68.83766pt\lower-35.38692pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-68.83766pt}{-35.38692pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
25.38509pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-21.07954pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\langle\langle
d_{0}s_{0}(-)\rangle\rangle}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
25.38509pt\hfil&\hfil\hskip 61.52008pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-21.07954pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${\langle\langle
d_{1}s_{0}(-)\rangle\rangle}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
25.38509pt\hfil\cr\vskip 36.135pt\cr\hfil\hskip
0.0pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 49.885pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-9.44446pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\langle\langle-\rangle\rangle}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
13.75pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} {}{ {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-17.8675pt}{20.5675pt}\pgfsys@lineto{17.46754pt}{20.5675pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.66753pt}{20.5675pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.64305pt}{22.92027pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{\theta}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-29.05312pt}{11.7078pt}\pgfsys@lineto{29.16193pt}{-24.09828pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.85178}{-0.5239}{0.5239}{0.85178}{29.33226pt}{-24.20305pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-21.03618pt}{-16.15277pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\langle\langle r\rangle\rangle}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{43.45259pt}{11.7078pt}\pgfsys@lineto{43.45259pt}{-23.62724pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{43.45259pt}{-23.82722pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{45.80536pt}{-8.6597pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\langle\langle l\rangle\rangle}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$

which commutes by Diagram 2.5.

The second condition that the natural isomorphisms in Lemma A.18 must satisfy
translates to the commutativity of the following diagram.

(3.23) ${\langle\langle d_{0}d_{0}(-)\rangle\rangle}$${\langle\langle
d_{1}d_{0}(-)\rangle\rangle}$${\langle\langle
d_{0}d_{2}(-)\rangle\rangle}$${\langle\langle
d_{0}d_{1}(-)\rangle\rangle}$${\langle\langle
d_{1}d_{1}(-)\rangle\rangle}$${\langle\langle
d_{1}d_{2}(-)\rangle\rangle}$$\scriptstyle{\theta}$$\scriptstyle{\langle\langle
a\rangle\rangle}$$\scriptstyle{\langle\langle
a\rangle\rangle}$$\scriptstyle{\theta}$$\scriptstyle{\langle\langle
a\rangle\rangle}$$\scriptstyle{\theta}$

Since the commutativity of this diagram is exactly Diagram 2.4 in the
definition of a shadow, we can conclude.

Step (V) The functor $\mathscr{U}\mathrm{n}$ on morphisms: We now extend
$\mathscr{U}\mathrm{n}$ to a functor. Let

$\alpha:\langle\langle-\rangle\rangle_{1}\to\langle\langle-\rangle\rangle_{2}$

be a morphism in $\mathscr{S}\mathrm{ha}(\mathscr{B},\mathscr{D})$, i.e., a
natural transformation such that the diagrams in 3.17 commute. Our goal is it
to construct a natural transformation from
$\mathscr{U}\mathrm{n}(\langle\langle-\rangle\rangle_{1})\to\mathscr{U}\mathrm{n}(\langle\langle-\rangle\rangle_{2})$,
which informally should have following shape.

${\displaystyle\coprod_{X_{0}}\mathscr{B}(X_{0},X_{0})}$${\displaystyle\coprod_{X_{0},X_{1}}\mathscr{B}(X_{0},X_{1},X_{0})}$${\displaystyle\coprod_{X_{0},X_{1},X_{2}}\mathscr{B}(X_{0},X_{1},X_{2},X_{0}).}$${\mathscr{D}}$$\scriptstyle{\langle\langle-\rangle\rangle_{2}}$$\scriptstyle{\langle\langle-\rangle\rangle_{1}}$$\scriptstyle{d_{1}}$$\scriptstyle{d_{0}}$$\scriptstyle{\langle\langle
d_{0}(-)\rangle\rangle_{2}}$$\scriptstyle{\langle\langle
d_{0}(-)\rangle\rangle_{1}}$$\scriptstyle{d_{2}}$$\scriptstyle{d_{0}}$$\scriptstyle{\langle\langle
d_{0}d_{0}(-)\rangle\rangle_{2}}$$\scriptstyle{\langle\langle
d_{0}d_{0}(-)\rangle\rangle_{1}}$$\scriptstyle{\alpha}$$\scriptstyle{\alpha
d_{0}}$$\scriptstyle{\alpha d_{0}d_{0}}$

More precisely, a natural transformation
$\mathscr{U}\mathrm{n}(\langle\langle-\rangle\rangle_{1})\to\mathscr{U}\mathrm{n}(\langle\langle-\rangle\rangle_{2})$
is a pseudofunctor

$\mathscr{U}\mathrm{n}\alpha:(({\mathbbe{\Delta}}_{\leq
2})^{op})^{\rhd}\to\mathrm{Fun}([1],\mathscr{C}\mathrm{at})$

that fits into the following diagram.

(3.24) ${\mathscr{C}\mathrm{at}}$${({\mathbbe{\Delta}}_{\leq
2})^{op}}$${(({\mathbbe{\Delta}}_{\leq
2})^{op})^{\rhd}}$${\mathrm{Fun}([1],\mathscr{C}\mathrm{at})}$${\mathscr{C}\mathrm{at}}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{\mathscr{U}\mathrm{n}\alpha}$$\scriptstyle{\mathscr{U}\mathrm{n}\langle\langle-\rangle\rangle_{1}}$$\scriptstyle{\mathscr{U}\mathrm{n}\langle\langle-\rangle\rangle_{2}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{2}}$

Here, the arrow labeled $\mathrm{id}$ corresponds the identity natural
transformation on the functor $\mathscr{B}_{*}\circ\mathscr{O}_{bihh}$ (see
3.12), which is the common value of the restrictions of
$\mathscr{U}\mathrm{n}\langle\langle-\rangle\rangle_{1}$ and
$\mathscr{U}\mathrm{n}\langle\langle-\rangle\rangle_{2}$ to
$({\mathbbe{\Delta}}_{\leq 2})^{op}$. Our goal is thus to construct such a
functor $\mathscr{U}\mathrm{n}\alpha$.

To lift the functor $\mathrm{id}:({\mathbbe{\Delta}}_{\leq
2})^{op}\to\mathrm{Fun}([1],\mathscr{C}\mathrm{at})$, we again use Lemma A.18.
We proceed now to make the necessary choices and verify the necessary
conditions. For the remainder of the proof, it is helpful to keep in mind that
$\mathrm{Fun}([1],\mathscr{C}\mathrm{at})$ is a bicategory with functors
$[1]\to\mathscr{C}\mathrm{at}$ as objects and squares that commute up to
natural isomorphisms as morphisms.

Considering the hypotheses of Lemma A.18, this time with
$\mathrm{Fun}([1],\mathscr{C}\mathrm{at})$ as target bicategory, we choose the
image of the final object to be the identity functor on $\mathscr{D}$ and the
image of the 1-cell $t_{0}:0\to f$ to be $\alpha$ itself, seen as filling in a
square where the composites along the sides are
$\mathrm{id}_{\mathscr{D}}\circ\langle\langle-\rangle\rangle_{1}$ and
$\langle\langle-\rangle\rangle_{2}\circ\mathrm{id}_{\coprod\mathscr{B}(X_{0},X_{0})}$.
Given these choices, we must now find an appropriate natural isomorphism
between $\alpha d_{0}$ and $\alpha d_{1}$, for which we select the diagram

(3.25) ${\langle\langle d_{0}(-)\rangle\rangle_{1}}$${\langle\langle
d_{1}(-)\rangle\rangle_{1}}$${\langle\langle
d_{0}(-)\rangle\rangle_{2}}$${\langle\langle
d_{1}(-)\rangle\rangle_{2}}$$\scriptstyle{\theta_{1}}$$\scriptstyle{\alpha
d_{0}}$$\scriptstyle{\alpha d_{1}}$$\scriptstyle{\theta_{2}}$

which commutes by the conditions of a morphism of shadows (3.17).

We now need to verify the two conditions imposed on the natural isomorphisms
from Lemma A.18. The first one requires that the following diagram commute

(3.26) $\leavevmode\hbox to299.92pt{\vbox
to156pt{\pgfpicture\makeatletter\hbox{\hskip 149.96109pt\lower-78.00085pt\hbox
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}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 64.59152pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-16.92398pt}{0.0pt}\pgfsys@invoke{
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d_{0}s_{0}\rangle\rangle_{1}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
21.22952pt\hfil&\hfil\hskip 43.362pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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whereas the second condition requires the diagram below to commute.

(3.27) ${\langle\langle d_{0}d_{0}(-)\rangle\rangle_{1}}$${\langle\langle
d_{1}d_{0}(-)\rangle\rangle_{1}}$${\langle\langle
d_{0}d_{2}(-)\rangle\rangle_{1}}$${\langle\langle
d_{0}d_{0}(-)\rangle\rangle_{2}}$${\langle\langle
d_{1}d_{0}(-)\rangle\rangle_{2}}$${\langle\langle
d_{0}d_{2}(-)\rangle\rangle_{2}}$${\langle\langle
d_{0}d_{1}(-)\rangle\rangle_{1}}$${\langle\langle
d_{1}d_{1}(-)\rangle\rangle_{1}}$${\langle\langle
d_{1}d_{2}(-)\rangle\rangle_{1}}$${\langle\langle
d_{0}d_{1}(-)\rangle\rangle_{2}}$${\langle\langle
d_{1}d_{1}(-)\rangle\rangle_{2}}$${\langle\langle
d_{1}d_{2}(-)\rangle\rangle_{2}}$$\scriptstyle{\alpha
d_{0}d_{0}}$$\scriptstyle{\theta_{1}}$$\scriptstyle{\langle\langle
a\rangle\rangle_{1}}$$\scriptstyle{\alpha
d_{1}d_{0}}$$\scriptstyle{\langle\langle
a\rangle\rangle_{1}}$$\scriptstyle{\alpha
d_{0}d_{2}}$$\scriptstyle{\theta_{2}}$$\scriptstyle{\langle\langle
a\rangle\rangle_{2}}$$\scriptstyle{\theta_{1}}$$\scriptstyle{\alpha
d_{0}d_{1}}$$\scriptstyle{\langle\langle
a\rangle\rangle_{1}}$$\scriptstyle{\alpha
d_{1}d_{1}}$$\scriptstyle{\theta_{1}}$$\scriptstyle{\alpha
d_{1}d_{2}}$$\scriptstyle{\theta_{2}}$$\scriptstyle{\langle\langle
a\rangle\rangle_{2}}$$\scriptstyle{\theta_{2}}$$\scriptstyle{\langle\langle
a\rangle\rangle_{2}}$

To establish the commutativity of these two diagrams, we use that the outer
squares and triangles in the diagram commute (by 3.17), whence the inside
diagram commutes as well, since categories are $2$-coskeletal.

Applying Lemma A.18, we to deduce that we have indeed defined a functor
$\mathscr{U}\mathrm{n}\alpha:(({\mathbbe{\Delta}}_{\leq
2})^{op})^{\rhd}\to\mathrm{Fun}([1],\mathscr{C}\mathrm{at})$, which fits by
construction into Diagram 3.24, since
$\pi_{1}\mathscr{U}\mathrm{n}(\alpha)=\mathscr{U}\mathrm{n}(\langle\langle-\rangle\rangle_{1})$
and
$\pi_{2}\mathscr{U}\mathrm{n}(\alpha)=\mathscr{U}\mathrm{n}(\langle\langle-\rangle\rangle_{2})$.

Step (VI): $\mathscr{S}\mathrm{t}$ and $\mathscr{U}\mathrm{n}$ are mutually
inverse: We now prove that $\mathscr{S}\mathrm{t}$ and $\mathscr{U}\mathrm{n}$
are inverse to each other. If
$\big{(}\langle\langle-\rangle\rangle,\theta\big{)}$ is a shadow, then

$\mathscr{U}\mathrm{n}\Big{(}\langle\langle-\rangle\rangle,\theta\Big{)}=\Big{(}\langle\langle-\rangle\rangle,\langle\langle
d_{0}(-)\rangle\rangle,\langle\langle d_{0}d_{0}(-)\rangle\rangle\Big{)}$

and so

$\mathscr{S}\mathrm{t}\mathscr{U}\mathrm{n}\big{(}\langle\langle-\rangle\rangle,\theta\big{)}=\big{(}\langle\langle-\rangle\rangle,\theta\big{)},$

i.e., $\mathscr{S}\mathrm{t}\mathscr{U}\mathrm{n}$ is the identity functor.

On the other hand, let $C=(C_{0},C_{1},C_{2})$ be an arbitrary cocone. By A.17
this cocone (which is by definition a pseudofunctor) is naturally equivalent
to a strict cocone, which, by Lemma A.18 is of the form
$(C_{0},C_{0}d_{0},C_{0}d_{0}d_{0})$. By functoriality $\mathscr{S}\mathrm{t}$
and $\mathscr{U}\mathrm{n}$ preserve natural isomorphisms, so it suffices to
prove that $\mathscr{U}\mathrm{n}\mathscr{S}\mathrm{t}$ takes this particular
cocone to itself.

This follows by direct computation. Indeed
$\mathscr{S}\mathrm{t}(C_{0},C_{0}d_{0},C_{0}d_{0}d_{0})$ is a shadow with
shadow functor $C_{0}$ and $\mathscr{U}\mathrm{n}$, by its very definition,
takes this shadow to the cocone $(C_{0},C_{0}d_{0},C_{0}d_{0}d_{0})$. ∎

######  Remark 3.28.

The proof above can be thought of as a “strictification argument”. We
strictify a general diagram such that the total amount of data reduces to that
of a shadow. The proof shows that no data is lost in this strictification
process. A shadow can thus be characterized as the minimal amount of data
required to describe a cocone.

######  Remark 3.29.

Applying 3.19 to the shadow on the bicategory
$\mathrm{Ho}(\mathscr{M}\mathrm{od})$ of 2.6 gives rise to a trace (2.17) out
of $\mathrm{Ho}(\mathscr{M}\mathrm{od})$ into $\mathrm{Ho}(\mathrm{Sp})$,

$\mathrm{THH}:\mathrm{biHH}\big{(}\mathrm{Ho}(\mathscr{M}\mathrm{od})\big{)}\to\mathrm{Ho}(\mathrm{Sp}).$

We generalize this result to $\infty$-categorical bimodules in 6.11.

##  4\. Morita Invariance of Shadows via Hochschild Homology

An important property satisfied by $\mathrm{THH}$ of ring spectra is Morita
invariance: if two ring spectra $R$ and $R^{\prime}$ are Morita equivalent
(i.e., their model categories of modules are Quillen equivalent), then
$\mathrm{THH}(R)\simeq\mathrm{THH}(R^{\prime})$ [BM12]. In line with the idea
that a shadow is an axiomatic, bicategorical generalization of $\mathrm{THH}$,
Campbell and Ponto proved that shadows satisfied a natural generalization of
Morita invariance [CP19, Proposition 4.6]. In this section we recover Morita
invariance of shadows using the machinery we developed in the previous
sections, which enables us to define a functorial lift of the useful Euler
characteristic construction of Campbell and Ponto, whose work we begin by
reviewing.

Let $\mathscr{B}$ be a bicategory equipped with a shadow functor
$\langle\langle-\rangle\rangle$ taking values in a category $\mathscr{D}$.
Given an adjunction in $\mathscr{B}$, i.e., a diagram

(4.1)
${C}$${D}$$\scriptstyle{F}$$\scriptstyle{GF}$$\scriptstyle{G}$$\scriptstyle{FG}$$\scriptstyle{u}$$\scriptstyle{c}$