AMSa"79

# Twisted Commutators and
Internal Crossed Modules

Bo Shan Deval and  Tim Van der Linden<EMAIL_ADDRESS><EMAIL_ADDRESS>Institut de Recherche en Mathématique et
Physique, Université catholique de Louvain, chemin du cyclotron 2 bte
L7.01.02, B–1348 Louvain-la-Neuve, Belgium  Dedicated to Hvedri Inassaridze on
the occasion of his 90th birthday

###### Abstract.

We introduce a notion of _relative commutator_ —an important special case
being _commutators twisted by an action_ —as a straightforward modification of
the definition of the _Higgins commutator_ , establish its relation with a new
notion of commutativity—also obtained as a modification of the usual
notion—and show how we can use it to characterise _internal crossed modules_
in the context of a semi-abelian category.

###### Key words and phrases:

Commutator; internal action; internal crossed module; semi-abelian category

###### 2020 Mathematics Subject Classification:

18D40, 18E13, 18G45

Both authors were funded by the Fonds de la Recherche Scientifique–FNRS: the
first through a FRIA Doctoral Grant, the second as a Senior Research Associate

##  1\. Introduction

The aim of this article is to initiate a theory of _commutativity twisted by
an action_. The wish for such a development arose in a recent investigation in
semi-abelian cohomology (on the relation between [RVdL16] and [SVdL21]) where
in order to better understand a certain higher-dimensional commutator, a
decomposition into simpler commutators would be helpful. The situation there
does, however, involve non-trivial actions, which calls for an appropriate
categorical “twisted commutator theory”. Our purpose here is not to go into
those more involved applications, but rather to develop basic foundations,
which we then apply to a simple test case: we characterise internal crossed
modules in a semi-abelian category (in the sense of [JMT02] and [Jan03]) by
means of twisted commutators.

The definition of a _twisted Higgins commutator_ is a straightforward
modification of the usual definition [Hig56, MM10a] and agrees with it in the
case of a trivial action. In a fixed semi-abelian category, we let an object
$B$ act on an object $X$ via $\xi\colon B\flat X\to X$. (Details on how such a
map codifies an action are recalled in Section 2; essential for us here is the
equivalence between actions and split extensions via a semi-direct product
construction.) We consider the induced split short exact sequence (SSES)

${0}$${X}$${X\rtimes_{\xi}B}$${B}$${0}$$\scriptstyle{k}$$\scriptstyle{d}$$\scriptstyle{e}$
(*)

and the following couniversal property: we may ask under which conditions two
given morphisms $f$ and $g$ as in the diagram

${X}$${X\rtimes_{\xi}B}$${B}$${Y}$$\scriptstyle{k}$$\scriptstyle{f}$$\scriptstyle{\varphi_{{f},{g}}}$$\scriptstyle{e}$$\scriptstyle{g}$

induce a morphism $\varphi_{{f},{g}}$ such that $\varphi_{{f},{g}}\circ k=f$
and $\varphi_{{f},{g}}\circ e=g$. It is clear that this condition—which agrees
with the condition that $f$ and $g$ commute in the ordinary sense of [Huq68,
Bou02] when $\xi$ is a trivial action—may be characterised in terms of
equivariance of $f$ and $g$ with respect to the action $\xi$ and the
conjugation action of $X$ on itself (see, for instance, [HL13, Proposition
4.4]), but we here follow an alternative approach.

We may mimic the definition of the Higgins commutator as follows (though the
Huq commutator would work equally well). We consider the solid part of the
diagram below, where the top short exact sequence is induced by taking the
kernel of the map $\langle k,e\rangle$ out of the coproduct of $X$ and $B$.

${0}$${X\diamond_{\xi}B}$${X+B}$${X\rtimes_{\xi}B}$${0}$${[{f},{g}]_{\xi}}$${Y}$$\scriptstyle{\iota_{{k},{e}}}$$\scriptstyle{\langle
k,e\rangle}$$\scriptstyle{\langle
f,g\rangle}$$\scriptstyle{\varphi_{{f},{g}}}$

This kernel $X\diamond_{\xi}B$ is the _twisted cosmash product_ of $X$ and
$B$, induced by $\xi$; it “consists of” twisted formal commutator words in $X$
and $B$. The dotted arrows form the image factorisation of the composite
$\langle f,g\rangle\circ\iota_{{k},{e}}$, so that the dashed arrow
$\varphi_{{f},{g}}$ exists and satisfies $\varphi_{{f},{g}}\circ k=f$ and
$\varphi_{{f},{g}}\circ e=g$ if and only if the thus defined commutator
$[{f},{g}]_{\xi}$ vanishes.

In Section 2, we investigate this definition in detail, recalling known
definitions and results, proving fundamental properties, comparing with
related results in the literature. This is then applied in Section 3 where we
characterise Janelidze’s internal crossed modules [Jan03] in terms of twisted
commutators. Our main result here is Corollary 3.19 which states that in a
semi-abelian category satisfying a certain condition denoted (SH), if
$\partial\colon X\to B$ is an arrow and $\xi$ an action of $B$ on $X$ inducing
the SSES (* ‣ 1), then the couple $(\partial,\xi)$ is an internal crossed
module if and only if the twisted commutators $[{\partial},{1_{B}}]_{\xi}$ and
$[{k},{e\circ\partial}]_{c^{{X},{X}}}$ vanish.

##  2\. Relative Commutativity and Commutators

In this section, we work towards the definition of a _commutator twisted by an
action_. We start by extending the definition of a commuting pair of arrows to
commutativity relative to a chosen cospan and analysing the commutator that
results from this. We then focus on the special case of the cospan of
monomorphisms in a split extension, which will provide us with the commutator
twisted by the action corresponding, through the semi-direct product
equivalence, to the given split extension.

###  2.1. Relative Commutativity

In a pointed category with finite products, we say that two coterminal arrows
$f\colon X\to Y$ and $g\colon B\to Y$ _(Huq-)commute_ [Huq68] or _cooperate_
[Bou02, Definition 4.10]

${X}$${X\times
B}$${B}$${Y}$$\scriptstyle{(1_{X},0)}$$\scriptstyle{f}$$\scriptstyle{\varphi_{{f},{g}}}$$\scriptstyle{(0,1_{B})}$$\scriptstyle{g}$

whenever there exists an arrow ${\varphi_{{f},{g}}\colon X\times B\to Y}$,
called a _cooperator of $f$ and $g$_, such that
$\varphi_{{f},{g}}\circ(1_{X},0)=f$ and $\varphi_{{f},{g}}\circ(0,1_{B})=g$.

The goal of this subsection is to generalise this notion of commutativity by
replacing the arrows $(1_{X},0)$ and $(0,1_{B})$ in the definition above by an
arbitrary cospan. We will also show that we keep some of the properties of the
classical situation. Note that the following notion has already been explored
by Martins-Ferreira under the name of _admissible pair_ in [MF10].

######  Definition 2.2.

Consider a cospan ${X}$${A}$${B}$$\scriptstyle{k}$$\scriptstyle{s}$ in an
arbitrary category. We say that two arrows ${f\colon X\to Y}$ and $g\colon
B\to Y$ with the same codomain _$(k,s)$ -commute_ or _commute relatively to
$(k,s)$_ whenever there exists an arrow $\varphi_{{f},{g}}\colon A\to Y$ such
that $\varphi_{{f},{g}}\circ k=f$ and $\varphi_{{f},{g}}\circ s=g$.

${X}$${A}$${B}$${Y}$$\scriptstyle{k}$$\scriptstyle{f}$$\scriptstyle{\varphi_{{f},{g}}}$$\scriptstyle{s}$$\scriptstyle{g}$

This arrow $\varphi_{{f},{g}}$ is called a _$(k,s)$ -cooperator of $f$ and
$g$_. When there is no ambiguity, we will drop the $(k,s)$.

######  Examples 2.3.

Here are some examples of commutativity relative to some particular cospans.

  * —

If $A$ is a zero object, then two arrows commute relatively to the cospan
${X}$${0}$${B}$ if and only if both of them are trivial arrows.

${X}$${0}$${B}$${Y}$$\scriptstyle{f}$$\scriptstyle{g}$

  * —

Let $B=A=X$ and $k=s=1_{X}$. Then two arrows $(k,s)$-commute if and only if
they are equal.

${A}$${A}$${A}$${Y}$$\scriptstyle{f}$$\scriptstyle{\varphi_{{f},{g}}}$$\scriptstyle{g}$

  * —

_Subtraction structures_ were introduced in [BJ09] and we will show here that
this notion can be expressed as a commutativity condition. More generally, we
say that an arrow $f\colon X\to Y$ in a pointed category _admits a subtractor
along $g\colon Y\to Z$_ [Sha23, Section 1] whenever there is an arrow
$\varphi\colon Y\times X\to Z$, called a _subtractor of $f$ along $g$_, making

${Y}$${Y\times
X}$${X}$${Z}$$\scriptstyle{(1_{Y},0)}$$\scriptstyle{g}$$\scriptstyle{\varphi}$$\scriptstyle{(f,1_{X})}$$\scriptstyle{0}$

commute—i.e. if $0$ and $g$ commute relatively to ${Y}$${Y\times
X}$${X}$$\scriptstyle{(1_{Y},0)}$$\scriptstyle{(f,1_{X})}$ with $\varphi$
being a cooperator. If $g$ is the identity on $Y$, we simply say that
$\varphi$ is a _subtractor of $f$_.

######  Remark 2.4.

In what follows, we will generally work in a finitely complete context and
consider the commutativity relative to an _extremally epic_ cospan, i.e. a
cospan ${X}$${A}$${B}$$\scriptstyle{k}$$\scriptstyle{s}$ such that, if there
exists a factorisation

${M}$${X}$${A}$${B}$$\scriptstyle{m}$$\scriptstyle{k}$$\scriptstyle{k^{\prime}}$$\scriptstyle{s}$$\scriptstyle{s^{\prime}}$

of this cospan through a monomorphism $m\colon M\to A$, then this $m$ is
necessarily an isomorphism. In particular, this implies that this cospan is
epic [BB04, Proposition A.4.3] so, in this case, the cooperator is necessarily
unique. (For example, this is the case for the classical commutativity in a
_unital_ context, i.e. in a pointed finitely complete category where all
cospans of the form ${X}$${X\times
B}$${B}$$\scriptstyle{(1_{X},0)}$$\scriptstyle{(0,1_{B})}$ are extremally
epic.)

Before going to the next subsection where we introduce commutators, let us
show some composition and cancellation properties of this relative
commutativity (see [BB04, Section 1.3] for analogous results for classical
commutativity).

######  Proposition 2.5.

Consider the following diagram.

${X}$${A}$${B}$${Y}$${Z}$$\scriptstyle{k}$$\scriptstyle{f}$$\scriptstyle{h\circ
f}$$\scriptstyle{s}$$\scriptstyle{g}$$\scriptstyle{h\circ g}$$\scriptstyle{h}$

If $f$ and $g$ commute relatively to $(k,s)$, then $h\circ f$ and $h\circ g$
also commute. Moreover, the converse holds in a finitely complete context if
$h$ is monic and the cospan is extremally epic. The link between their
respective cooperators is given by $\varphi_{{h\circ f},{h\circ
g}}=h\circ\varphi_{{f},{g}}$.

###### Proof.

The first statement is easily proved by checking the given formula for the
cooperators. Thus, let us show the second statement. Assume that $h$ is monic
and $h\circ f$ and $h\circ g$ $(k,s)$-commute. We will show that the
cooperator $\varphi_{{h\circ f},{h\circ g}}$ factorises through $h$ as
$\varphi_{{h\circ f},{h\circ g}}=h\circ\varphi$. Since $h$ is monic, this will
imply that $\varphi$ is a cooperator for $f$ and $g$, finishing the proof.
Consider the following commutative squares.

${X}$${A}$${B}$${A}$${Y}$${Z}$${Y}$${Z}$$\scriptstyle{k}$$\scriptstyle{f}$$\scriptstyle{\varphi_{{h\circ
f},{h\circ
g}}}$$\scriptstyle{s}$$\scriptstyle{g}$$\scriptstyle{\varphi_{{h\circ
f},{h\circ g}}}$$\scriptstyle{h}$$\scriptstyle{h}$

By [BB04, Proposition A.4.3], there is a diagonal arrow $\varphi\colon A\to Y$
making the four triangles commute, which is the cooperator of $f$ and $g$ as
desired. ∎

######  Proposition 2.6.

Consider the following diagram.

${X^{\prime}}$${A^{\prime}}$${B^{\prime}}$${X}$${A}$${B}$${Y}$$\scriptstyle{k^{\prime}}$$\scriptstyle{x}$$\scriptstyle{s^{\prime}}$$\scriptstyle{y}$$\scriptstyle{k}$$\scriptstyle{f}$$\scriptstyle{s}$$\scriptstyle{g}$

If $k\circ x$ and $s\circ y$ commute relatively to $(k^{\prime},s^{\prime})$
and $f$ and $g$ commute relatively to $(k,s)$, then $f\circ x$ and $g\circ y$
commute relatively to $(k^{\prime},s^{\prime})$ with the cooperator
$\varphi_{{f\circ x},{g\circ y}}=\varphi_{{f},{g}}\circ\varphi_{{k\circ
x},{s\circ y}}$.

${X^{\prime}}$${A^{\prime}}$${B^{\prime}}$${X}$${A}$${B}$${Y}$$\scriptstyle{k^{\prime}}$$\scriptstyle{x}$$\scriptstyle{\varphi_{{k\circ
x},{s\circ
y}}}$$\scriptstyle{s^{\prime}}$$\scriptstyle{y}$$\scriptstyle{k}$$\scriptstyle{f}$$\scriptstyle{\varphi_{{f},{g}}}$$\scriptstyle{s}$$\scriptstyle{g}$

###### Proof.

Again, this follows directly from the formula for the cooperators. ∎

######  Remark 2.7.

Let us remark that, in the classical case, the Huq-commutativity condition for
$k\circ x$ and $s\circ y$ is automatically satisfied, their cooperator being
the induced arrow $x\times y\colon X^{\prime}\times B^{\prime}\to X\times B$.

######  Remark 2.8.

A left-cancellation property analogous to [BB04, Proposition 1.6.4] might also
make sense, but the necessary assumptions are less straightforward and we will
not need such a result in this paper.

###  2.9. Relative Commutators

We will now define a notion of _relative commutator_ which, as in the
classical case, is a tool that allows us to detect when two arrows commute in
the relative sense, or more generally measure “how close they are to
commuting” when this is not the case. In this subsection, we work in the
context of a _normal_ category $\mathscr{C}$, i.e. a pointed regular category
where every regular epimorphism is a normal epimorphism. We will construct the
relative commutator via an image factorisation, so we need the regularity,
while normality is needed to show that a certain regular epimorphism is the
cokernel of its kernel, in order to ensure the correspondence between
commutators and commutativity (see Proposition 2.13).

As a reminder, in a unital normal category, the _(Higgins) commutator
$[{f},{g}]$_ of two arrows $f\colon X\to Y$, $g\colon B\to Y$ with the same
codomain [Definition 5.1]MM10bHig56 is the image

${0}$${X\diamond B}$${X+B}$${X\times
B}$${[{f},{g}]}$${Y}$$\scriptstyle{\iota_{{X},{B}}}$$\scriptstyle{\langle(1_{X},0),(0,1_{B})\rangle}$$\scriptstyle{\langle
f,g\rangle}$

of the composite $\langle f,g\rangle\circ\iota_{{X},{B}}$ where
$\iota_{{X},{B}}$ is the kernel of the canonical arrow
$\langle(1_{X},0),(0,1_{B})\rangle\colon X+B\to X\times B$. This commutator
was designed to vanish exactly when the arrows $f$ and $g$ commute.

We will use a similar procedure to define the commutator relative to a
cospan—which we will require to be extremally epic, in order to recover the
detection of commutativity; note that it was for the same reason that we gave
the definition of the Higgins commutator in a unital context. Similarly to
Definition 2.2, the idea is to replace the arrows $(1_{X},0)$ and $(0,1_{B})$
in the diagram above by an arbitrary extremally epic cospan.

######  Definition 2.10.

Let ${X}$${A}$${B}$$\scriptstyle{k}$$\scriptstyle{s}$ be an extremally epic
cospan and ${f\colon X\to Y}$, ${g\colon B\to Y}$ two arrows with the same
codomain. The _$(k,s)$ -commutator $[{f},{g}]_{k,s}$_ of $f$ and $g$ is the
image of the composite $\langle f,g\rangle\circ\iota_{{k},{s}}$ where
$\iota_{{k},{s}}$ is the kernel of the arrow $\langle k,s\rangle\colon X+B\to
A$. By convention and by coherence with the classical situation, we write
$\iota_{{X},{B}}$ for $\iota_{{(1_{X},0)},{(0,1_{B})}}$. The kernel object of
$\langle k,s\rangle\colon X+B\to A$ is called the _$(k,s)$ -cosmash product_
of $X$ and $B$ and is denoted $X\diamond_{k,s}B$.

${0}$${X\diamond_{k,s}B}$${X+B}$${A}$${[{f},{g}]_{k,s}}$${Y}$$\scriptstyle{\iota_{{k},{s}}}$$\scriptstyle{\langle
k,s\rangle}$$\scriptstyle{\langle f,g\rangle}$

######  Remark 2.11.

The twisted cosmash product appears in independent recent work of Duvieusart:
see [Duv23].

Before we show that these relative commutators detect the relative
commutativity, here is a result that tells us that the commutators are
preserved by direct images. For an analogous result for the classical
commutator, see [Sha17, Lemma 3.4]. The proof given there is less simple than
ours, since it concerns commutators of subobjects, which apparently introduces
some technicalities.

######  Proposition 2.12.

Given the diagram

${X}$${A}$${B}$${Y}$${Z}$$\scriptstyle{k}$$\scriptstyle{f}$$\scriptstyle{s}$$\scriptstyle{g}$$\scriptstyle{h}$

with ${X}$${A}$${B}$$\scriptstyle{k}$$\scriptstyle{s}$ extremally epic, we
have the formula

$[{h\circ f},{h\circ g}]_{k,s}=h([{f},{g}]_{k,s})$

where the latter object is the direct image of $[{f},{g}]_{k,s}$ along $h$.

###### Proof.

The formula follows from the diagram

${X\diamond_{k,s}B}$${X+B}$${[{f},{g}]_{k,s}}$${Y}$${h([{f},{g}]_{k,s})}$${Z}$$\scriptstyle{\langle
f,g\rangle}$$\scriptstyle{h}$

—where the dotted and dashed arrows are constructed as the image factorisation
of the corresponding squares—by the essential uniqueness of the image
factorisation and the fact that, in a regular context, regular epimorphisms
compose [BB04, Corollary A.5.4]. ∎

Let us observe how these commutators detect commutativity relative to an
extremally epic cospan.

######  Proposition 2.13.

Two arrows $f\colon X\to Y$ and $g\colon B\to Y$ commute relatively to an
extremally epic cospan ${X}$${A}$${B}$$\scriptstyle{k}$$\scriptstyle{s}$ if
and only if their $(k,s)$-commutator $[{f},{g}]_{k,s}$ is trivial.

###### Proof.

First, since $k$ and $s$ are jointly extremally epic, $\langle k,s\rangle$ is
extremally epic so it is a regular epimorphism by regularity [BB04,
Proposition A.4.3, Corollary A.5.4]. Thus, since we are in a normal context,
this arrow is a normal epimorphism; in particular, it is the cokernel of its
kernel $\iota_{{k},{s}}$. What we need to prove now follows from the universal
property of this cokernel $\langle k,s\rangle$.

${0}$${X\diamond_{k,s}B}$${X+B}$${A}$${0}$${[{f},{g}]_{k,s}}$${Y}$$\scriptstyle{\iota_{{k},{s}}}$$\scriptstyle{\langle
k,s\rangle}$$\scriptstyle{\langle
f,g\rangle}$$\scriptstyle{\varphi_{{f},{g}}}$

If $f$ and $g$ commute relatively to $(k,s)$, we have an arrow
$\varphi_{{f},{g}}$ such that

$\varphi_{{f},{g}}\circ\langle k,s\rangle=\langle\varphi_{{f},{g}}\circ
k,\varphi_{{f},{g}}\circ s\rangle=\langle f,g\rangle\text{.}$

Therefore, we have

$\langle f,g\rangle\circ\iota_{{k},{s}}=(\varphi_{{f},{g}}\circ\langle
k,s\rangle)\circ\iota_{{k},{s}}=\varphi_{{f},{g}}\circ 0=0$

so its image $[{f},{g}]_{k,s}$ is trivial.

Conversely, if $[{f},{g}]_{k,s}=0$, then the composite $\langle
f,g\rangle\circ\iota_{{k},{s}}$ is trivial, so we can apply the universal
property of $\langle k,s\rangle$ to obtain an arrow $\varphi_{{f},{g}}\colon
A\to Y$ such that $\varphi_{{f},{g}}\circ\langle k,s\rangle=\langle
f,g\rangle$. Then $\varphi_{{f},{g}}\circ k=f$ and $\varphi_{{f},{g}}\circ
s=g$, which finishes the proof. ∎

######  Examples 2.14.

Let us examine the commutators associated to some of the cospans of 2.3.

  * —

If $A$ is a zero object, then we can choose $\iota_{{k},{s}}=1_{X+B}$ so our
commutator $[{f},{g}]_{k,s}$ is simply the image $\operatorname{Im}\langle
f,g\rangle$ of $\langle f,g\rangle$.

${0}$${X+B}$${X+B}$${0}$${\operatorname{Im}\langle
f,g\rangle}$${Y}$$\scriptstyle{\langle f,g\rangle}$

  * —

Let $B=A=X$ and $k=s=1_{X}$. The $(1_{X},1_{X})$-cosmash product is called the
_difference object $D(X)$_ of $X$ and its inclusion into $X+X$ is denoted
$\delta_{X}$. We have that $[{f},{g}]_{1_{X},1_{X}}$ is trivial if and only if
$f=g$—i.e. if they $(1_{X},1_{X})$-commute, as follows from Proposition 2.13.
So this commutator can be seen as a measure of the difference between $f$ and
$g$.

${0}$${D(X)}$${X+X}$${X}$${[{f},{g}]_{1_{X},1_{X}}}$${Y}$$\scriptstyle{\delta_{X}}$$\scriptstyle{\langle
1_{X},1_{X}\rangle}$$\scriptstyle{\langle f,g\rangle}$

###  2.15. Commutators Twisted by an Action

A special case of these relative commutators which we will be especially
interested in consists of those commutators relative to a cospan induced by an
internal action.

In what follows, whenever we consider internal actions, our category
$\mathscr{C}$ will always be semi-abelian in the sense of [JMT02]. Such a
category is indeed normal [BB04, Proposition 3.1.23]. Further note that any
semi-abelian category is _homological_ in the sense of [BB04, Definition
4.1.1]—see [BB04, Proposition 5.1.2]—and unital [BB04, Proposition 3.1.18,
Proposition 1.8.4]—cf. Remark 2.4.

Let us recall what is an internal action. An _(internal) action_ of an object
$B$ on an object $X$ is an arrow $\xi\colon B\flat X\to X$ making certain
diagrams commute, where $B\flat X$ is constructed by choosing a coproduct
$B+X$ of $B$ and $X$ then taking a kernel $\kappa_{{B},{X}}\colon B\flat X\to
B+X$ of the arrow $\langle 1_{B},0\rangle\colon B+X\to B$ [Sections
3.2–3.3]BJK05BJ98. (This construction gives us a bifunctor
$\flat\colon\mathscr{C}\times\mathscr{C}\to\mathscr{C}$.) Such an internal
action induces a split short exact sequence (SSES)

${0}$${X}$${X\rtimes_{\xi}B}$${B}$${0}$$\scriptstyle{k}$$\scriptstyle{p}$$\scriptstyle{s}$
(†)

as in the commutative diagram

${0}$${B\flat
X}$${B+X}$${B}$${0}$${X}$${X\rtimes_{\xi}B}$${B}$${0}$$\scriptstyle{\kappa_{{B},{X}}}$$\scriptstyle{\xi}$$\scriptstyle{\langle
1_{B},0\rangle}$$\scriptstyle{q}$$\scriptstyle{\iota_{1}}$$\scriptstyle{k=q\circ\iota_{2}}$$\scriptstyle{\iota_{2}}$$\scriptstyle{p}$$\scriptstyle{s=q\circ\iota_{1}}$
(‡)

where $\iota_{1}$, $\iota_{2}$ are the inclusions in the coproduct $B+X$ and
$q$ is constructed as a coequaliser of $\kappa_{{B},{X}}$ and
$\iota_{2}\circ\xi$. This construction is part of an equivalence of categories
between internal actions and SSESs [BJ98, Section 3]. (In fact, even if [BJ98]
discusses an equivalence for each fixed acting object $B$, these equivalences
for every $B$ induce an equivalence on the level of all internal actions and
all SSESs.)

######  Example 2.16.

An action $\xi$ whose induced SSES is of the form

${0}$${X}$${X\times
B}$${B}$${0}$$\scriptstyle{(1_{X},0)}$$\scriptstyle{\pi_{2}}$$\scriptstyle{(0,1_{B})}$

(where $\pi_{2}$ is the projection of $X\times B$ on $B$) is said to be a
_trivial action_. More explicitly, we have $\xi=\langle
0,1_{X}\rangle\circ\kappa_{{B},{X}}$.

######  Example 2.17.

The _conjugation action_ $c^{{X},{X}}$ of an object $X$ on itself is defined
as the action corresponding to the SSES

${0}$${X}$${X\times
X}$${X}$${0}$$\scriptstyle{(1_{X},0)}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\Delta_{X}}$

where $\pi_{2}$ is the projection on the second factor and
$\Delta_{X}\coloneqq(1_{X},1_{X})$ is the _diagonal_ of $X$. Again, we have an
explicit formula $c^{{X},{X}}=\langle
1_{X},1_{X}\rangle\circ\kappa_{{X},{X}}$.

Let us now define commutators twisted by an action. In order to apply the
definition of $(k,s)$-commutators (Definition 2.10), we only need that the
cospan ${X}$${X\rtimes_{\xi}B}$${B}$$\scriptstyle{k}$$\scriptstyle{s}$ is
extremally epic, which is indeed the case since $\mathscr{C}$ is a
protomodular category (see [JMT02, Section 2.4]). Let us remark that for a
cospan $(k,s)$ to be induced by an action is a property rather than additional
structure. Indeed, whenever it exists, an arrow $p\colon A\to B$ turning this
cospan into a SSES is uniquely determined by the identities $p\circ k=0$ and
$p\circ s=1_{B}$ since $k$ and $s$ are jointly epic. If now $k$ is the kernel
of $p$, then $p$ is necessarily its cokernel. Note that such an arrow $p$ is a
$(k,s)$-cooperator of $0$ and $1_{B}$ so the condition that the cospan
${X}$${A}$${B}$$\scriptstyle{k}$$\scriptstyle{s}$ is induced by an action
involves the $(k,s)$-commutativity of $0$ and $1_{B}$.

######  Definitions 2.18.

Let $\xi\colon B\flat X\to X$ be an action and († ‣ 2.15) its induced SSES.
Let $f\colon X\to Y$, $g\colon B\to Y$ be two arrows with the same codomain.

  * —

We say that $f$ and $g$ _commute relatively to $\xi$_ or _$\xi$ -commute_ when
they $(k,s)$-commute. Their $(k,s)$-cooperator is called the _$\xi$
-cooperator of $f$ and $g$_ (see Definition 2.2).

  * —

The _$\xi$ -commutator of $f$ and $g$_ (or _commutator twisted by the action
$\xi$_) is the $(k,s)$-commutator of $f$ and $g$ (see Definition 2.10). We
denote it by $[{f},{g}]_{\xi}$.

######  Remark 2.19.

If $\xi$ is a trivial action, then we regain the definition of the classical
commutator: see Example 2.16.

######  Example 2.20.

Let us recall that an arrow $f\colon X\to Y$ _admits a subtractor $\varphi$
along $g\colon Y\to Z$_ (see 2.3) whenever $g$ and $0\colon X\to Z$ commute
relatively to ${Y}$${Y\times
X}$${X}$$\scriptstyle{(1_{Y},0)}$$\scriptstyle{(f,1_{X})}$.

${Y}$${Y\times
X}$${X}$${Z}$$\scriptstyle{(1_{Y},0)}$$\scriptstyle{g}$$\scriptstyle{\varphi}$$\scriptstyle{(f,1_{X})}$$\scriptstyle{0}$

In fact, we can turn this cospan into the SSES

${0}$${Y}$${Y\times
X}$${X}$${0}$$\scriptstyle{(1_{Y},0)}$$\scriptstyle{\pi_{2}}$$\scriptstyle{(f,1_{X})}$

so, in a semi-abelian context, this is an example of commutativity twisted by
an action. In particular, if $Y=X$ and $f$ is the identity on $X$, then we
find the SSES associated with the conjugation action on $X$: see Example 2.17.

We end this section with an interpretation of commutativity twisted by an
action in terms of equivariance. In the case of commutators twisted by an
action $\xi$, the relative commutativity of two arrows $f$ and $g$ (Definition
2.2) can also be expressed as the equivariance of $f$ and $g$ with respect to
$\xi$ and the conjugation action $c^{{Y},{Y}}$ of $Y$ on itself. Note that
this corresponds to [Jan03, Theorem 1.3] (since $c^{{Y},{Y}}\circ(g\flat
f)=\langle g,f\rangle\circ\kappa_{{B},{X}}$ by definition of $c^{{Y},{Y}}$)
which provides an alternate proof of this fact. Recall that
$Y\rtimes_{c^{{Y},{Y}}}Y=Y\times Y$: see Example 2.17.

######  Lemma 2.21.

The arrows $f\colon X\to Y$ and $g\colon B\to Y$ commute relatively to $\xi$
if and only if they are equivariant with respect to $\xi$ and the conjugation
action $c^{{Y},{Y}}$ of $Y$ on itself. The latter condition means that the
square

${B\flat X}$${X}$${Y\flat Y}$${Y}$$\scriptstyle{\xi}$$\scriptstyle{g\flat
f}$$\scriptstyle{f}$$\scriptstyle{c^{{Y},{Y}}}$

commutes or, in terms of SSESs via the equivalence mentioned in the start of
the subsection, there exists an arrow $\psi\colon X\rtimes_{\xi}B\to Y\times
Y$ such that $(f,\psi,g)$

${0}$${X}$${X\rtimes_{\xi}B}$${B}$${0}$${0}$${Y}$${Y\times
Y}$${Y}$${0}$$\scriptstyle{k}$$\scriptstyle{f}$$\scriptstyle{p}$$\scriptstyle{\psi}$$\scriptstyle{s}$$\scriptstyle{g}$$\scriptstyle{(1_{Y},0)}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\Delta_{Y}}$

is a morphism of SSESs.

###### Proof.

We will show this result from the point of view of SSESs.

If $(f,g)$ is equivariant, we easily check that
$\varphi_{{f},{g}}=\pi_{1}\circ\psi$ satisfies the conditions of Definition
2.2.

Conversely, let us check that $\psi=(\varphi_{{f},{g}},g\circ p)$ induces a
morphism $(f,\psi,g)$ of SSESs, where $\varphi_{{f},{g}}$ is given by the
$(k,s)$-commutativity of $f$ and $g$. The equalities $\psi\circ
k=(1_{Y},0)\circ f$ and $g\circ p=\pi_{2}\circ\psi$ are immediate and, for the
last one, we compute

$\psi\circ s=\bigl{(}\varphi_{{f},{g}}\circ s,(g\circ p)\circ
s\bigr{)}=(g,g)=\Delta_{Y}\circ g\text{,}$

which finishes the proof. ∎

We can also obtain a kind of converse of this result, i.e. a commutativity
condition to characterise the equivariance of a pair of arrows. For that, we
will need a lemma about internal actions, which tells us that, if we restrict
the conjugation action of a semi-direct product to its factors, we recover the
original action.

######  Lemma 2.22 ([HL13, Corollary 4.5]).

Consider an internal action $\xi\colon B\flat X\to X$ and

${0}$${X}$${X\rtimes_{\xi}B}$${B}$${0}$$\scriptstyle{k}$$\scriptstyle{d}$$\scriptstyle{e}$

its induced SSES. Then the square

${B\flat
X}$${X}$${(X\rtimes_{\xi}B)\flat(X\rtimes_{\xi}B)}$${X\rtimes_{\xi}B}$$\scriptstyle{\xi}$$\scriptstyle{e\flat
k}$$\scriptstyle{k}$$\scriptstyle{c^{{X\rtimes_{\xi}B},{X\rtimes_{\xi}B}}}$

is commutative or, equivalently, there exists $\psi\colon
X\rtimes_{\xi}B\to(X\rtimes_{\xi}B)\times(X\rtimes_{\xi}B)$ such that

${0}$${X}$${X\rtimes_{\xi}B}$${B}$${0}$${0}$${X\rtimes_{\xi}B}$${(X\rtimes_{\xi}B)\times(X\rtimes_{\xi}B)}$${Y}$${0}$$\scriptstyle{k}$$\scriptstyle{k}$$\scriptstyle{d}$$\scriptstyle{\psi}$$\scriptstyle{e}$$\scriptstyle{e}$$\scriptstyle{(1_{X\rtimes_{\xi}B},0)}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\Delta_{X\rtimes_{\xi}B}}$

is a morphism of SSESs.

###### Proof.

Using the formula of Example 2.17, we compute

$\begin{split}c^{{X\rtimes_{\xi}B},{X\rtimes_{\xi}B}}\circ(e\flat k)&=(\langle
1_{X\rtimes_{\xi}B},1_{X\rtimes_{\xi}B}\rangle\circ\kappa_{{X\rtimes_{\xi}B},{X\rtimes_{\xi}B}})\circ(e\flat
k)\\\ &=\langle
1_{X\rtimes_{\xi}B},1_{X\rtimes_{\xi}B}\rangle\circ\bigl{(}(e+k)\circ\kappa_{{B},{X}}\bigr{)}\\\
&=\langle e,k\rangle\circ\kappa_{{B},{X}}=k\circ\xi\end{split}$

as wanted, where the last equality follows from the construction of the
induced SSES: see Equation ‡. (And the second equality follows from the
construction of the bifunctor $\flat$.) ∎

######  Corollary 2.23.

Let $\xi\colon B\flat X\to X$, $\xi^{\prime}\colon B^{\prime}\flat
X^{\prime}\to X^{\prime}$ be two actions and

${0}$${X}$${X\rtimes_{\xi}B}$${B}$${0\
\text{,}}$${0}$${X^{\prime}}$${X^{\prime}\rtimes_{\xi^{\prime}}B^{\prime}}$${B^{\prime}}$${0}$$\scriptstyle{k}$$\scriptstyle{p}$$\scriptstyle{s}$$\scriptstyle{k^{\prime}}$$\scriptstyle{p^{\prime}}$$\scriptstyle{s^{\prime}}$

their induced SSESs. Two arrows $f\colon X\to X^{\prime}$ and $g\colon B\to
B^{\prime}$ are equivariant with respect to $\xi$ and $\xi^{\prime}$ if and
only if $k^{\prime}\circ f\colon X\to
X^{\prime}\rtimes_{\xi^{\prime}}B^{\prime}$ and $s^{\prime}\circ g\colon B\to
X^{\prime}\rtimes_{\xi^{\prime}}B^{\prime}$ commute relatively to $\xi$.

###### Proof.

By Lemma 2.21, the commutativity of $k^{\prime}\circ f$ and $s^{\prime}\circ
g$ relative to $\xi$ is equivalent to the commutativity of the square below.

${B\flat
X}$${X}$${(X^{\prime}\rtimes_{\xi^{\prime}}B^{\prime})\flat(X^{\prime}\rtimes_{\xi^{\prime}}B^{\prime})}$${X^{\prime}\rtimes_{\xi^{\prime}}B^{\prime}}$$\scriptstyle{\xi}$$\scriptstyle{(s^{\prime}\circ
g)\flat(k^{\prime}\circ f)}$$\scriptstyle{k^{\prime}\circ
f}$$\scriptstyle{c^{{X^{\prime}\rtimes_{\xi^{\prime}}B^{\prime}},{X^{\prime}\rtimes_{\xi^{\prime}}B^{\prime}}}}$

Now, consider the following diagram.

${B\flat X}$${X}$${B^{\prime}\flat
X^{\prime}}$${X^{\prime}}$${(X^{\prime}\rtimes_{\xi^{\prime}}B^{\prime})\flat(X^{\prime}\rtimes_{\xi^{\prime}}B^{\prime})}$${X^{\prime}\rtimes_{\xi^{\prime}}B^{\prime}}$$\scriptstyle{\xi}$$\scriptstyle{g\flat
f}$$\scriptstyle{f}$$\scriptstyle{\xi^{\prime}}$$\scriptstyle{s^{\prime}\flat
k^{\prime}}$$\scriptstyle{k^{\prime}}$$\scriptstyle{c^{{X^{\prime}\rtimes_{\xi^{\prime}}B^{\prime}},{X^{\prime}\rtimes_{\xi^{\prime}}B^{\prime}}}}$

Since $k^{\prime}$ is (split) monic and the bottom square always commutes by
Lemma 2.22, the commutativity of the outer rectangle (which is the square
above by functoriality of $\flat$) is equivalent to the commutativity of the
top square, which correspond to the equivariance of $f$ and $g$, finishing the
proof. ∎

######  Remark 2.24.

This corollary implies that, whenever we consider cospans induced by an action
in Proposition 2.6, we can reformulate the
$(k^{\prime},s^{\prime})$-commutativity condition of $k\circ x$ and $s\circ y$
as the equivariance of $x$ and $y$ with respect to the two actions involved.

##  3\. Internal Crossed Modules

The concept of an _internal crossed module_ extends the idea behind a crossed
module from the category of groups Grp to arbitrary semi-abelian categories.
This hinges upon the classical equivalence—see for instance [BS76,
ML98]—between the category of crossed modules (of groups) and the category of
internal categories in Grp. In [Jan03], _internal crossed_ and _precrossed
modules_ were defined as couples $(\partial,\xi)$ where $\partial\colon X\to
B$ is an arrow and $\xi\colon B\flat X\to X$ is an action of $B$ on $X$,
satisfying certain compatibility conditions ensuring that the category of
internal crossed modules (resp. precrossed modules) would be equivalent to the
category of internal groupoids (resp. reflexive graphs). The aim of this
section is to express those conditions in terms of twisted commutators.

###  3.1. Definitions

We make the definitions explicit (rephrasing (2.2) and (3.15) in [Jan03]) by
means of the equivalence mentioned in Section 2.15. Let $\partial\colon X\to
B$ be an arrow and $\xi\colon B\flat X\to X$ an action with induced SSES

${0}$${X}$${X\rtimes_{\xi}B}$${B}$${0}$$\scriptstyle{k}$$\scriptstyle{d}$$\scriptstyle{e}$.

######  Definition 3.2.

A couple $(\partial,\xi)$ as above is an _internal precrossed module_ whenever
there exists an arrow $c\colon X\rtimes_{\xi}B\to B$ such that $c\circ
e=1_{B}$ and $c\circ k=\partial$.

${0}$${X}$${X\rtimes_{\xi}B}$${B}$${0}$$\scriptstyle{k}$$\scriptstyle{d}$$\scriptstyle{c}$$\scriptstyle{e}$

######  Remark 3.3.

Since the arrow $k$ can be recovered as a kernel of $d$, an internal
precrossed module is essentially just a _reflexive graph_
$(X\rtimes_{\xi}B,B,d,c,e)$: a diagram of the form
${C_{1}}$${C_{0}}$$\scriptstyle{d}$$\scriptstyle{c}$$\scriptstyle{e}$ where
$d\circ e=1_{C_{0}}$ and $c\circ e=1_{C_{0}}$.

######  Definition 3.4.

An internal precrossed module is an _internal crossed module_ if, when we see
it as a reflexive graph (see Remark 3.3 above), it admits an internal groupoid
structure.

Before proceeding with the characterisations of precrossed and crossed modules
in terms of twisted commutators, in the next subsection we first present some
definitions and results about reflexive graphs.

###  3.5. Reflexive Graphs

In what follows, unless otherwise stated, we will work in a pointed finitely
complete category $\mathscr{C}$. We recall the definition of the
_normalisation_ of a reflexive graph as well as some relations between a given
reflexive graph and its normalisation.

######  Definition 3.6.

The _normalisation_ of a reflexive graph
${C_{1}}$${C_{0}}$$\scriptstyle{d}$$\scriptstyle{c}$$\scriptstyle{e}$ is the
arrow $c\circ k$ where $k$ is a kernel of $d$.

Originally, we considered the normalisation of effective equivalence relations
and, in this case, this normalisation is a normal monomorphism, whence the
terminology. It is well known that, in a semi-abelian category, a reflexive
graph is an equivalence relation if and only if its normalisation is a
(normal) monomorphism. Let us recall the proof:

######  Lemma 3.7.

If a reflexive graph
${C_{1}}$${C_{0}}$$\scriptstyle{d}$$\scriptstyle{c}$$\scriptstyle{e}$ is a
_reflexive relation_ , i.e. if $d$ and $c$ are jointly monic, then its
normalisation $c\circ k$, where $k\colon X\to C_{1}$ is a kernel of $d$, is
monic. Moreover, if we are in a pointed protomodular context, then the
converse is also true.

###### Proof.

Let us first assume that we have a reflexive relation. Then $d\circ k$ and
$c\circ k$ are jointly monic since $k$ is a monomorphism. Moreover, $d\circ k$
is the trivial arrow because $k$ is a kernel of $d$ so the previous sentence
implies that $c\circ k$ is monic.

Now, let us assume that our category $\mathscr{C}$ is protomodular and that
$c\circ k$ is monic. Consider the following diagram where all the squares
except the bottom right one are pullbacks.

${X\cap\operatorname{Ker}(c)}$${\operatorname{Ker}(c)}$${0}$${X}$${C_{1}}$${C_{0}}$${0}$${C_{0}}$${C_{0}\times
C_{0}}$$\AMSlrcorner$$\AMSlrcorner$$\scriptstyle{\ker(c)}$$\AMSlrcorner$$\scriptstyle{k}$$\scriptstyle{c}$$\scriptstyle{d}$$\scriptstyle{(d,c)}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\pi_{1}}$

We check that $X\cap\operatorname{Ker}(c)$ is a kernel of $(d,c)$ (using the
universal properties of the two kernels and the pullback involved) and we will
show that this object is trivial, which implies that $(d,c)$ is monic (meaning
that $d$ and $c$ are jointly monic) as needed for [BB04, Proposition 3.1.21].
Since the two top squares are pullbacks, so is the top rectangle. Therefore,
$X\cap\operatorname{Ker}(c)$ is a kernel of $c\circ k$. So it is trivial since
$c\circ k$ is monic, which finishes the proof. ∎

Our second lemma is a characterisation of connected graphs, which is an
instance of [Bou01, Proposition 17]; see also [RVdL10].

######  Lemma 3.8.

In a homological category $\mathscr{C}$, a reflexive graph
${C_{1}}$${C_{0}}$$\scriptstyle{d}$$\scriptstyle{c}$$\scriptstyle{e}$ is
_connected_ , i.e. the arrow $(d,c)\colon C_{1}\to C_{0}\times C_{0}$ is a
regular epimorphism, if and only if its normalisation $c\circ k$, where
$k\colon X\to C_{1}$ is a kernel of $d$, is a regular epimorphism.

###### Proof.

Let $(d,c)$ be a regular epimorphism. We check that the following square is a
pullback since $k$ is a kernel of $d$.

${X}$${C_{1}}$${C_{0}}$${C_{0}\times
C_{0}}$$\AMSlrcorner$$\scriptstyle{k}$$\scriptstyle{c\circ
k}$$\scriptstyle{(d,c)}$$\scriptstyle{(0,1_{C_{0}})}$

Therefore, since $(d,c)$ is a regular epimorphism and, in a regular category,
regular epimorphisms are pullback-stable, this diagram tells us that $c\circ
k$ is also a regular epimorphism.

For the other direction, consider the following diagram where $k$ and
$(0,1_{C_{0}})$ are kernels of $d$ and $\pi_{1}$, respectively. Here $d$ is a
cokernel of its kernel $k$, since it is split by $e$.

${0}$${X}$${C_{1}}$${C_{0}}$${0}$${0}$${C_{0}}$${C_{0}\times
C_{0}}$${C_{0}}$$\scriptstyle{k}$$\scriptstyle{c\circ
k}$$\scriptstyle{d}$$\scriptstyle{(d,c)}$$\scriptstyle{(0,1_{C_{0}})}$$\scriptstyle{\pi_{1}}$

We may check that this diagram commutes. Then, by [BB04, Lemma 4.2.5], the
morphism $(d,c)$ is a regular epimorphism, since we have a regular epimorphism
on the left and an isomorphism on the right. ∎

Before going to the last definitions of this section, here is an alternative
characterisation for connected graphs in an exact Mal’tsev category.

######  Lemma 3.9.

In an exact Mal’tsev category $\mathscr{C}$, a reflexive graph
${C_{1}}$${C_{0}}$$\scriptstyle{d}$$\scriptstyle{c}$$\scriptstyle{e}$ is such
that the arrow $(d,c)\colon C_{1}\to C_{0}\times C_{0}$ is an epimorphism if
and only if the pushout $\pi_{0}(C)$ of $d$ and $c$ is trivial. Moreover, when
this is the case, the epimorphism $(d,c)$ is regular so this reflexive graph
is connected in the sense of the previous lemma.

###### Proof.

First, note that the pushout of $d$ and $c$ exists by [CKP93, Theorem 5.7]
since $d$ and $c$ are split epimorphisms.

Now, assume that $(d,c)$ is an epimorphism and consider the pushout

${C_{1}}$${C_{0}}$${C_{0}}$${\pi_{0}(C)}$$\scriptstyle{c}$$\scriptstyle{d}$$\scriptstyle{i_{2}}$$\scriptstyle{i_{1}}$

$\AMSlrcorner$

of $d$ and $c$. Let $\pi_{1}$, $\pi_{2}$ be the two projections of the product
$C_{0}\times C_{0}$. We compute

$(i_{1}\circ\pi_{1})\circ(d,c)=i_{1}\circ d=i_{2}\circ
c=(i_{2}\circ\pi_{2})\circ(d,c)$

so $i_{1}\circ\pi_{1}$ and $i_{2}\circ\pi_{2}$ are equal since $(d,c)$ is
epic.

${C_{0}\times
C_{0}}$${C_{0}}$${C_{0}}$${0}$${\pi_{0}(C)}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{i_{2}}$$\scriptstyle{i_{1}}$

$\AMSlrcorner$

Therefore, $i_{1}$ and $i_{2}$ factor through the pushout of $\pi_{1}$ and
$\pi_{2}$, which is trivial since it is the pushout of a product, so $i_{1}$
and $i_{2}$ are trivial arrows. Finally, since $i_{1}$ and $i_{2}$ are jointly
epic as they form a pushout, this implies that $\pi_{0}(C)$ is trivial—a
cospan of trivial arrows is jointly epic only if their codomain is
trivial—which finishes the proof of this implication.

For the other direction, since $\pi_{0}(C)$ is trivial, $C_{0}\times C_{0}$ is
a pullback of $i_{1}$ and $i_{2}$ so $(d,c)$ is the comparison map between the
pushout and this pullback in the diagram below.

${C_{1}}$${C_{0}\times
C_{0}}$${C_{0}}$${C_{0}}$${\pi_{0}(C)=0}$$\scriptstyle{c}$$\scriptstyle{(d,c)}$$\scriptstyle{d}$$\AMSlrcorner$$\scriptstyle{\pi_{2}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{i_{2}}$$\scriptstyle{i_{1}}$

Therefore, by [CKP93, Theorem 5.7] and the fact that $d$ and $c$ are regular
epimorphisms as they are split, $(d,c)$ is a regular epimorphism as announced.
∎

Let us now introduce multiplicative and star-multiplicative graphs [MM10,
Definitions 4.1 and 5.1], needed for Theorem 3.16 below. This theorem is
relevant for our characterisation of crossed modules since, in a semi-abelian
category, groupoids are exactly multiplicative graphs [Theorem
2.2]CPP92[Proposition 5.1.2]BB04.

######  Definitions 3.10.

Consider a reflexive graph
${C_{1}}$${C_{0}}$$\scriptstyle{d}$$\scriptstyle{c}$$\scriptstyle{e}$.

  * —

The reflexive graph $(C_{1},C_{0},d,c,e)$ is _multiplicative_ when there is an
arrow $m\colon C_{1}\times_{C_{0}}C_{1}\to C_{1}$ such that
$m\circ(1_{C_{1}},e\circ d)=1_{C_{1}}$ and $m\circ(e\circ
c,1_{C_{1}})=1_{C_{1}}$

${C_{1}}$${C_{1}\times_{C_{0}}C_{1}}$${C_{1}}$${C_{1}}$$\scriptstyle{(1_{C_{1}},e\circ
d)}$$\scriptstyle{m}$$\scriptstyle{(e\circ c,1_{C_{1}})}$

where $C_{1}\times_{C_{0}}C_{1}$ is constructed as the pullback

${C_{1}\times_{C_{0}}C_{1}}$${C_{1}}$${C_{1}}$${C_{0}}$$\AMSlrcorner$$\scriptstyle{p_{1}}$$\scriptstyle{p_{0}}$$\scriptstyle{c}$$\scriptstyle{d}$

and $(1_{C_{1}},e\circ d)$, $(e\circ c,1_{C_{1}})$ are the arrows induced by
the universal property of this pullback. This arrow $m$ is called the
_multiplication_ of $(C_{1},C_{0},d,c,e)$.

  * —

Let $k\colon X\to C_{1}$ be a kernel of $d$. The reflexive graph
$(C_{1},C_{0},d,c,e)$ is _star-multiplicative_ when there is an arrow
$\zeta\colon C_{1}\times_{C_{0}}X\to X$ such that $\zeta\circ(k,0)=1_{X}$ and
$\zeta\circ(e\circ c\circ k,1_{X})=1_{X}$

${X}$${C_{1}\times_{C_{0}}X}$${X}$${X}$$\scriptstyle{(k,0)}$$\scriptstyle{\zeta}$$\scriptstyle{(e\circ
c\circ k,1_{X})}$ (‡)

where $C_{1}\times_{C_{0}}X$ is constructed as the pullback

${C_{1}\times_{C_{0}}X}$${X}$${C_{1}}$${C_{0}}$$\AMSlrcorner$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{0}}$$\scriptstyle{c\circ
k}$$\scriptstyle{d}$

and $(k,0)$, $(e\circ c\circ k,1_{X})$ are the arrows induced by the universal
property of this pullback. This arrow $\zeta$ is called the _star-
multiplication_ of $(C_{1},C_{0},d,c,e)$.

######  Remark 3.11.

Every multiplicative graph is star-multiplicative; the idea is that the star-
multiplication is a kind of restriction of the multiplication: see [MFVdL12]
for further details. Consider the arrow $m\circ\mu$ where $\mu\colon
C_{1}\times_{C_{0}}X\to C_{1}\times_{C_{0}}C_{1}$ is induced in the diagram
below by the universal property of the pullback.

${C_{1}\times_{C_{0}}X}$${X}$${C_{1}}$${C_{1}\times_{C_{0}}C_{1}}$${C_{1}}$${C_{1}}$${C_{0}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\mu}$$\scriptstyle{\pi_{0}}$$\scriptstyle{k}$$\AMSlrcorner$$\scriptstyle{p_{1}}$$\scriptstyle{p_{0}}$$\scriptstyle{c}$$\scriptstyle{d}$

We will show that $m\circ\mu$ factors through $X$ and that this factorisation
is the wanted star-multiplication. Since $X$ is a kernel of $d$, we must show
that $d\circ(m\circ\mu)=0$. In order to do so, we will precompose this arrow
with $(k,0)$ and $(e\circ c\circ k,1_{X})$, which will let us conclude since
these two arrows are jointly epic (as $(k,0)$ is a kernel of the splitting
$\pi_{1}$ of $(e\circ c\circ k,1_{X})$). By uniqueness, we have
$\mu=(1_{C_{1}},e\circ d)\circ\pi_{0}=(e\circ c,1_{C_{1}})\circ k\circ\pi_{1}$
so that

$d\circ(m\circ\mu)\circ(k,0)=d\circ\pi_{0}\circ(k,0)=d\circ k=0$

and

$d\circ(m\circ\mu)\circ(e\circ c\circ k,1_{X})=d\circ
k\circ\pi_{1}\circ(e\circ c\circ k,1_{X})=d\circ k=0$

as expected. Therefore, the universal property of $k$ gives us an arrow from
$C_{1}\times_{C_{0}}X$ to $X$ that factorises $m\circ\mu$ through $k$ and this
is this arrow that we choose to be the star-multiplication $\zeta$. To
conclude, we must show that this $\zeta$ makes

${X}$${C_{1}\times_{C_{0}}X}$${X}$${X}$$\scriptstyle{(k,0)}$$\scriptstyle{\zeta}$$\scriptstyle{(e\circ
c\circ k,1_{X})}$

commute and, since $k$ is monic, we can reduce these two equations by
postcomposing them by $k$. But these two reduced equations correspond exactly
to the computations we made just above, the composition with $d$ intervening
only on the last step in both cases, which finishes the proof.

######  Remark 3.12.

Given a reflexive graph $(C_{1},C_{0},d,c,e)$, let us note that the arrow
$\pi_{1}\colon C_{1}\times_{C_{0}}X\to X$ is at the same time a cokernel of
$(k,0)\colon X\to C_{1}\times_{C_{0}}X$ and a splitting of $(e\circ c\circ
k,1_{X})\colon X\to C_{1}\times_{C_{0}}X$. The fact that $\pi_{1}\circ(e\circ
c\circ k,1_{X})=1_{X}$ is immediate. To show that $\pi_{1}$ is a cokernel of
$(k,0)$, let us show that $(k,0)$ is kernel of it, the conclusion following
since $\pi_{1}$ is split. Let $f\colon A\to C_{1}\times_{C_{0}}X$ such that
$\pi_{1}\circ f=0$. Then, by the definition of the pullback, $f$ is of the
form $f=(g,0)$ with $g\colon A\to C_{1}$ such that $d\circ g=(c\circ k)\circ
0=0$ and, since $k$ is a kernel of $d$, we also have a factorisation
$g=k\circ\varphi$. Finally, we remark that this factorisation $\varphi\colon
A\to X$ is the unique arrow such that $(k,0)\circ\varphi=f$, showing that
$(k,0)$ is a kernel of $\pi_{1}$.

${A}$${X}$${C_{1}\times_{C_{0}}X}$${X}$$\scriptstyle{f}$$\scriptstyle{\exists!\varphi}$$\scriptstyle{(k,0)}$$\scriptstyle{\pi_{1}}$

Therefore, $(k,0)$ and $(e\circ c\circ k,1_{X})$ is the cospan associated to
an action $\xi\colon X\to X$ and we can view Equation ‡ as the commutativity
of $1_{X}$ with itself, twisted by this action $\xi$. Finally, by Proposition
2.13, this means that the star-multiplicativity of a reflexive graph
$(C_{1},C_{0},d,c,e)$ is characterised by the commutator condition
$[{1_{X}},{1_{X}}]_{\xi}=0$ where $\xi\colon X\to X$ is the internal action
associated to the SSES

${0}$${X}$${C_{1}\times_{C_{0}}X}$${X}$${0}$$\scriptstyle{(k,0)}$$\scriptstyle{\pi_{1}}$$\scriptstyle{(e\circ
c\circ k,1_{X})}$.

(Let us also note that the multiplicativity of a reflexive graph can also be
characterised by a commutator condition, but this commutator has no reason to
be twisted by an action.)

###  3.13. Characterisation in Terms of Twisted Commutators

We can now state and prove our characterisations of precrossed and crossed
modules in terms of twisted commutators. These are variations of the results
of [MM10] so the goal of this section is mainly to reformulate them in terms
of twisted commutators via Lemma 2.21 (and provide an alternative proof in the
case of Theorem 3.17). In what follows, in a semi-abelian category, we
consider an arrow $\partial\colon X\to B$ and an action $\xi\colon B\flat X\to
X$ with induced SSES (* ‣ 1).

Let us begin with the characterisation of internal precrossed modules, which
is a straightforward consequence of Proposition 2.13.

######  Theorem 3.14.

In a semi-abelian category, let $\partial\colon X\to B$ be an arrow and $\xi$
an action of $B$ on $X$. The couple $(\partial,\xi)$ is an internal precrossed
module if and only if it satisfies the so-called _Precrossed Module Condition
(PCM)_ , which is the equality $[{\partial},{1_{B}}]_{\xi}=0$.

###### Proof.

By Proposition 2.13, the triviality of the commutator
$[{\partial},{1_{B}}]_{\xi}$ is equivalent to the existence of an arrow
$\varphi\colon X\rtimes_{\xi}B\to B$

${X}$${X\rtimes_{\xi}B}$${B}$${B}$$\scriptstyle{k}$$\scriptstyle{\partial}$$\scriptstyle{\varphi}$$\scriptstyle{e}$$\scriptstyle{1_{B}}$

such that $\varphi\circ k=\partial$ and $\varphi\circ e=1_{B}$, which are
exactly the conditions the arrow $c$ in Definition 3.2 must satisfy. ∎

The characterisation of internal crossed modules is less simple. In fact, we
will even need an additional assumption: we ask that our semi-abelian category
$\mathscr{C}$ satisfies the _Smith-is-Huq Condition(SH)_ [BG02a].

######  Definition 3.15 ([MFVdL12, Section 1]).

A semi-abelian category $\mathscr{C}$ satisfies the _Smith-is-Huq
Condition(SH)_ whenever two effective equivalence relations in $\mathscr{C}$
commute if and only if their normalisations commute.

Examples include any Moore category in the sense of [Rod04] and, more
generally, any _algebraically coherent_ semi-abelian category [CGVdL15]. We
will not use this definition directly but the following characterisation
instead.

######  Theorem 3.16 ([MFVdL12, Theorem 3.8]).

For a semi-abelian category, the following conditions are equivalent:

  1. (SM)

every star-multiplicative graph is multiplicative,

  2. (SH)

two (effective) equivalence relations commute if and only if their
normalisations commute.∎

Indeed, since an internal crossed module is an internal precrossed module that
is also a groupoid and, as said earlier, in a semi-abelian context,
multiplicative graphs coincide with groupoids, we see that the condition (SM)
might be useful for our desired characterisation. We now have all the
necessary tools for its proof.

######  Theorem 3.17.

In a semi-abelian category, let the couple $(\partial,\xi)$ be an internal
precrossed module with (* ‣ 1) the SSES associated to $\xi$. Then
$(\partial,\xi)$ satisfies the so-called _Peiffer Condition (PFF)_ , which is
the equality $[{k},{e\circ\partial}]_{c^{{X},{X}}}=0$, if and only if the
reflexive graph $(X\rtimes_{\xi}B,B,d,c,e)$ is star-multiplicative.

In particular, by Remark 3.11 and Theorem 3.16, this implies that, if
$(\partial,\xi)$ is an internal crossed module, then (PFF) is satisfied and,
conversely, if the category $\mathscr{C}$ satisfies (SH), then the condition
(PFF) implies that $(\partial,\xi)$ is an internal crossed module.

###### Proof.

Let us recall that

${0}$${X}$${X\times
X}$${X}$${0}$$\scriptstyle{(1_{X},0)}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\Delta_{X}}$,

where $\pi_{2}$ is the projection on the second factor and $\Delta_{X}$ is the
diagonal of $X$, is the SSES associated to $c^{{X},{X}}$.

Let us start with $[{k},{e\circ\partial}]_{c^{{X},{X}}}=0$. By Proposition
2.13, this gives us an arrow $\varphi\colon X\times X\to X\rtimes_{\xi}B$ such
that

${X}$${X\times
X}$${X}$${X\rtimes_{\xi}B}$$\scriptstyle{(1_{X},0)}$$\scriptstyle{k}$$\scriptstyle{\varphi}$$\scriptstyle{\Delta_{X}}$$\scriptstyle{e\circ\partial}$

commutes. Then this arrow $\varphi$ makes

${0}$${X}$${X\times
X}$${X}$${0}$${0}$${X}$${X\rtimes_{\xi}B}$${B}$${0}$$\scriptstyle{(1_{X},0)}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\varphi}$$\scriptstyle{\partial}$$\scriptstyle{\Delta_{X}}$$\scriptstyle{k}$$\scriptstyle{d}$$\scriptstyle{e}$

commute. Indeed, we must just check that $d\circ\varphi=\partial\circ\pi_{2}$
and we see that this is the case by precomposing with $(1_{X},0)$ and
$\Delta_{X}$ which are jointly epic. Therefore, the right-hand square whose
the horizontal arrows point to the right is a pullback, since we have an
isomorphism on the left [BB04, Lemma 4.2.5] so this square is the pullback
mentioned in the definition of a star-multiplicative graph (see 3.10). (Note
that, by definition of precrossed module, we do indeed have $\partial=c\circ
k$.) Finally, we check that $\pi_{1}\colon X\times X\to X$ is the needed star-
multiplication: we have the commutative diagram

${X}$${X\times
X}$${X}$${X}$$\scriptstyle{(1_{X},0)}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\Delta_{X}}$

and we easily check that $(1_{X},0)$ and $\Delta_{X}$ are the required arrows
(induced by the universal property of the pullback) in Equation ‡.

Conversely, if we have a star-multiplication
$\zeta\colon(X\rtimes_{\xi}B)\times_{B}X\to X$ on the reflexive graph
associated to our precrossed module, with $(X\rtimes_{\xi}B)\times_{B}X\to X$
being constructed as the pullback

${(X\rtimes_{\xi}B)\times_{B}X}$${X}$${X\rtimes_{\xi}B}$${C_{0}}$$\AMSlrcorner$$\scriptstyle{p_{1}}$$\scriptstyle{p_{0}}$$\scriptstyle{c\circ
k=\partial}$$\scriptstyle{d}$,

then, by Proposition 2.13, we must show that
$[{k},{e\circ\partial}]_{c^{{X},{X}}}=0$, i.e. that there is an arrow
$\varphi\colon X\times X\to X\rtimes_{\xi}B$ that makes

${X}$${X\times
X}$${X}$${X\rtimes_{\xi}B}$$\scriptstyle{(1_{X},0)}$$\scriptstyle{k}$$\scriptstyle{\varphi}$$\scriptstyle{\Delta_{X}}$$\scriptstyle{e\circ\partial}$
(§)

commute. As a reminder, the star-multiplication $\zeta$ is defined as the
arrow making the following diagram commute.

${X}$${(X\rtimes_{\xi}B)\times_{B}X}$${X}$${X}$$\scriptstyle{(k,0)}$$\scriptstyle{\zeta}$$\scriptstyle{(e\circ\partial,1_{X})}$

The first implication of this proposition tells us that, if we have a crossed
module, then the product of $X$ with itself coincides (up to isomorphism) with
the pullback $(X\rtimes_{\xi}B)\times_{B}X$ (with different projections) and
that the wanted arrow $\varphi$ is the first projection of this pullback, so
let us show that it is also the case here. Consider the following diagram.

${X}$${(X\rtimes_{\xi}B)\times_{B}X}$${X}$${X}$${X\times
X}$${X}$$\scriptstyle{(k,0)}$$\scriptstyle{p_{1}}$$\scriptstyle{(\zeta,p_{1})}$$\scriptstyle{(e\circ\partial,1_{X})}$$\scriptstyle{(1_{X},0)}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\Delta_{X}}$

We check that this diagram commutes and that both rows are (split) exact
sequences. The arrow $(k,0)$ is indeed a kernel of $p_{1}$ since $k$ is the
kernel of $d$ and $p_{1}$ is the pullback of $d$. Therefore, we can apply the
(Split) Short Five Lemma [BB04, Theorem 4.1.10] to deduce that $(\zeta,p_{1})$
is an isomorphism. Finally, we see that $\varphi=p_{0}\circ(\zeta,p_{1})^{-1}$
is the wanted arrow making Equation § commute, which finishes the proof. ∎

######  Remark 3.18.

Let us justify our terminology by remarking that, by Corollary 2.23 and
Proposition 2.13, the Peiffer Condition
$[{k},{e\circ\partial}]_{c^{{X},{X}}}=0$ is equivalent to the commutativity of
the square

${X\flat X}$${X}$${B\flat
X}$${X}$$\scriptstyle{c^{{X},{X}}}$$\scriptstyle{\partial\flat
1_{X}}$$\scriptstyle{\xi}$

which (via the equivalence between actions and SSESs) corresponds to the
definition of Peiffer graph of [MM10, Definition 5.2]. Moreover, it was
already shown in [MM10, Theorem 5.3] that Peiffer graphs and star-
multiplicative graphs coincide, so this gives an alternative proof for Theorem
3.17.

These two theorems combine to:

######  Corollary 3.19.

In a semi-abelian category satisfying (SH), let $\partial\colon X\to B$ be an
arrow and $\xi$ an action of $B$ on $X$ inducing the SSES (* ‣ 1). The couple
$(\partial,\xi)$ is an internal crossed module if and only if the twisted
commutators $[{\partial},{1_{B}}]_{\xi}$ and
$[{k},{e\circ\partial}]_{c^{{X},{X}}}$ vanish.∎

###  3.20. Two Special Cases of Crossed Modules

Let us see what this characterisation amounts to in two well-known special
types of crossed modules: those $(\partial,\xi)$ where $\partial$ is a
monomorphism and the ones where $\partial$ is a regular epimorphism.

#### Monic $\partial$

We start with the case where $\partial$ is monic. Then $(\partial,\xi)$ is a
crossed module if and only if $\partial\colon X\to B$ is the inclusion of a
normal subobject $X$ of $B$ and ${\xi\colon B\flat X\to X}$ is the action of
conjugation of $B$ on $X$, i.e. the unique action making

${B\flat X}$${X}$${B\flat
B}$${B}$$\scriptstyle{\xi}$$\scriptstyle{B\flat\partial}$$\scriptstyle{\partial}$$\scriptstyle{c^{{B},{B}}}$
(¶)

commute [JMU07, Proposition 7.2]. Note that the uniqueness follows from the
fact that $\partial$ is monic. The given reference states the equivalence for
precrossed modules but, as we will see below, any precrossed module
$(\partial,\xi)$ with $\partial$ monic is automatically a crossed module.

Let us first assume that $(\partial,\xi)$ is just a precrossed module. By
Theorem 3.14, this is equivalent to the triviality of the commutator
$[{\partial},{1_{B}}]_{\xi}$. Moreover, by Proposition 2.13 and Lemma 2.21, we
can reformulate this condition as the equivariance of $\partial$ and $1_{B}$
with respect to $\xi$, which means exactly that $\xi$ is the conjugation
action of $B$ on $X$ (see (¶ ‣ 3.20)). Finally, the existence of such a
conjugation action is equivalent to the normality of the subobject
$\partial\colon X\to B$ ([JMU09, Corollary 2.3] combined with [JMT02,
Proposition p. 382]—as a reminder, semi-abelian categories are normal so we
can indeed apply [JMT02, Proposition p. 382] on the square of [JMU09,
Corollary 2.3.(b)]), which is what was announced in the previous paragraph.

Thus we already know that a couple $(\partial,\xi)$ with $\partial$ monic is a
precrossed module if and only if $\partial$ is the inclusion of a normal
subobject. Now, let us see what tells us the Peiffer Condition, i.e. the
triviality of $[{k},{e\circ\partial}]_{c^{{X},{X}}}$ (see Theorem 3.17). Using
again Proposition 2.13 and Lemma 2.21, this condition is equivalent to the
commutativity of the diagram

${X\flat
X}$${X}$${(X\rtimes_{\xi}B)\flat(X\rtimes_{\xi}B)}$${X\rtimes_{\xi}B}$$\scriptstyle{c^{{X},{X}}}$$\scriptstyle{(e\circ\partial)\flat
k}$$\scriptstyle{k}$$\scriptstyle{c^{{X\rtimes_{\xi}B},{X\rtimes_{\xi}B}}}$

which can be decomposed as

${X\flat X}$${X}$${B\flat
X}$${X}$${(X\rtimes_{\xi}B)\flat(X\rtimes_{\xi}B)}$${X\rtimes_{\xi}B}$$\scriptstyle{c^{{X},{X}}}$$\scriptstyle{\partial\flat
1_{X}}$$\scriptstyle{\xi}$$\scriptstyle{e\flat
k}$$\scriptstyle{k}$$\scriptstyle{c^{{X\rtimes_{\xi}B},{X\rtimes_{\xi}B}}}$.

Since we already showed that $\xi$ is a conjugation action, the commutativity
of the top square is quite expected. More formally, let us show that these two
arrows from $X\flat X$ to $X$ are equal by postcomposing them with $\partial$,
which is monic by assumption. Using the fact that $\xi$ is the conjugation
action of $B$ on $X$ as well as the explicit formula for conjugation actions
from Example 2.17, we compute

$\begin{split}\partial\circ\bigl{(}\xi\circ(\partial\flat
1_{X})\bigr{)}&=\bigl{(}c^{{B},{B}}\circ(B\flat\partial)\bigr{)}\circ(\partial\flat
1_{X})=c^{{B},{B}}\circ(\partial\flat\partial)\\\ &=(\langle
1_{B},1_{B}\rangle\circ\kappa_{{B},{B}})\circ(\partial\flat\partial)=\langle
1_{B},1_{B}\rangle\circ\bigl{(}(\partial+\partial)\circ\kappa_{{X},{X}}\bigr{)}\\\
&=\langle\partial,\partial\rangle\circ\kappa_{{X},{X}}=\partial\circ(\langle
1_{X},1_{X}\rangle\circ\kappa_{{X},{X}})=\partial\circ c^{{X},{X}}\end{split}$

as wanted. Finally, the bottom square commutes by Lemma 2.22. In conclusion,
we showed that, if $\partial$ is a monomorphism, precrossed modules and
crossed modules coincide and correspond to the normal subobjects. (We need the
condition (SH) in order to apply Theorem 3.17 for the crossed modules but this
characterisation of normal subobjects remains true in an arbitrary semi-
abelian category. Indeed, if we have a precrossed module with $\partial\colon
X\to B$ monic, the corresponding reflexive graph is a reflexive relation by
Lemma 3.7 so it is an equivalence relation since our category is Mal’tsev
[BB04, Proposition 5.1.2]. Therefore, it is transitive and this gives us a
multiplication, making our precrossed module an internal groupoid, hence a
crossed module.)

#### Regularly epic $\partial$

For our second example, let us assume that $\partial$ is a regular
epimorphism. In this case, it is known [BG02, Corollary 3.1] (combined with
[BG02, Remark 3.1], [GVdL08, Proposition 2.2] and Lemma 3.8) that these
crossed modules correspond to _central extensions_. Such is a regular
epimorphism $\partial\colon X\to B$ whose kernel $\kappa\colon K\to X$ Huq-
commutes with its domain $1_{X}$ or, in other words, for which there exists an
arrow $\psi\colon K\times X\to X$ making the diagram below commute.

${K}$${K\times
X}$${X}$${X}$$\scriptstyle{(1_{K},0)}$$\scriptstyle{\kappa}$$\scriptstyle{\psi}$$\scriptstyle{(0,1_{X})}$

This time, we will need (SH) for our proof—note, however, that this result
nevertheless holds in an arbitrary semi-abelian category, see the references
mentioned above.

Let us now examine how the characterisation mentioned above express itself in
terms of commutator conditions. First, the fact that a crossed module
$(\partial,\xi)$ with $\partial$ being a regular epimorphism is a central
extension follows directly from the following lemma.

######  Lemma 3.21.

For a crossed module $(\partial,\xi)$, the kernel $\kappa\colon K\to X$ and
the domain $1_{X}$ of $\partial$ commute (i.e. $\kappa$ is central).

###### Proof.

Let us consider the following commutative diagram

${K\flat X}$${X}$${X\flat
X}$${X}$${(X\rtimes_{\xi}B)\flat(X\rtimes_{\xi}B)}$${(X\rtimes_{\xi}B)}$$\scriptstyle{\zeta}$$\scriptstyle{\kappa\flat
1_{X}}$$\scriptstyle{c^{{X},{X}}}$$\scriptstyle{(e\circ\partial)\flat
k}$$\scriptstyle{k}$$\scriptstyle{c^{{X\rtimes_{\xi}B},{X\rtimes_{\xi}B}}}$

where $\zeta\coloneqq c^{{X},{X}}\circ(\kappa\flat 1_{X})$ is the induced
action of $K$ on $X$ and the bottom square commutes by Theorem 3.17 combined
with Lemma 2.21. We will show that $\zeta$ is the trivial action of $K$ on
$X$, which will finish the proof since the commutativity of the top square
then means that $\kappa$ and $1_{X}$ are equivariant with respect to the
trivial action $\zeta$ and the conjugation action $c^{{X},{X}}$, i.e. that
$\kappa$ and $1_{X}$ commute by Lemma 2.21 (remember that Huq-commutativity
corresponds to commutativity relative to the trivial action: see Remark 2.19).

We compute

$\begin{split}k\circ\zeta&=c^{{(X\rtimes_{\xi}B)},{(X\rtimes_{\xi}B)}}\circ\Bigl{(}\bigl{(}(e\circ\partial)\flat
k\bigr{)}\circ(\kappa\flat 1_{X})\Bigr{)}\\\ &=(\langle
1_{X\rtimes_{\xi}B},1_{X\rtimes_{\xi}B}\rangle\circ\kappa_{{X\rtimes_{\xi}B},{X\rtimes_{\xi}B}})\circ(0\flat
k)\\\ &=\langle
1_{X\rtimes_{\xi}B},1_{X\rtimes_{\xi}B}\rangle\circ\bigl{(}(0+k)\circ\kappa_{{K},{X}}\bigr{)}=\langle
0,k\rangle\circ\kappa_{{K},{X}}\\\ &=k\circ(\langle
0,1_{X}\rangle\circ\kappa_{{K},{X}})\text{,}\end{split}$

where we used that $\kappa$ is a kernel of $\partial$ for the second equality,
so $\zeta=\langle 0,1_{X}\rangle\circ\kappa_{{K},{X}}$ since $k$ is monic,
which is indeed the trivial action of $K$ on $X$ as announced (see Example
2.16). ∎

######  Remark 3.22.

Let us present an alternative proof for this lemma using [Sha23, Theorem 4.1]
which gives a characterisation of central arrows (note that a semi-abelian
category is indeed regular—by definition—and subtractive—see [Jan05,
Proposition 3] and [BB04, Proposition 3.1.18]). Consider the following
commutative diagram

${X}$${X\times K}$${K}$${X}$${X\times
X}$${X}$${X\rtimes_{\xi}B}$$\scriptstyle{(1_{X},0)}$$\scriptstyle{1_{X}\times\kappa}$$\scriptstyle{(\kappa,1_{K})}$$\scriptstyle{\kappa}$$\scriptstyle{(1_{X},0)}$$\scriptstyle{k}$$\scriptstyle{\varphi}$$\scriptstyle{\Delta_{X}}$$\scriptstyle{e\circ\partial}$

where the dotted arrow $\varphi$ is given by the Peiffer Condition
$[{k},{e\circ\partial}]_{c^{{X},{X}}}=0$. By Proposition 2.6, this means that
$0=(e\circ\partial)\circ\kappa$ and $k$ commute relatively to the cospan
${K}$${X\times K}$${X}$$\scriptstyle{(\kappa,1_{K})}$$\scriptstyle{(1_{X},0)}$
so, by Proposition 2.5, it is also the case for $0$ and $1_{X}$ since $k$ is
monic, which lets us conclude that $\kappa$ is central by [Sha23, Theorem 4.1]
(see 2.3).

Moreover, let us remark that, for these two proofs of Lemma 3.21, we only
needed the Peiffer Condition and not the Precrossed Module Condition.

Now, we consider a central extension $\partial\colon X\to B$ and will see how
it naturally has a (unique) structure of crossed module. By Remark 3.18, the
action $\xi$ of this crossed module $(\partial,\xi)$ must make the square

${X\flat X}$${X}$${B\flat
X}$${X}$$\scriptstyle{c^{{X},{X}}}$$\scriptstyle{\partial\flat
1_{X}}$$\scriptstyle{\xi}$ (∥)

commute. Using the centrality of $\partial$ and the fact that it is a regular
epimorphism, one can show that such an arrow $\xi\colon B\flat X\to X$ exists
[HVdL11, Example 6.2]. (For example, in a variety of algebras, $\partial$ is a
quotient and this diagram tells us that the action by an element $b\in B$ is
given by choosing a representative $x$ of $b$ (i.e. an element $x\in X$ such
that $\partial(x)=b$) and acting by conjugation with this $x$, the centrality
of $\partial$ assuring that this procedure is well-defined.) Once we know the
existence of this arrow, we can show that it is indeed an action using the
fact that $c^{{X},{X}}$ is an action and that $\partial\flat 1_{X}$ is
(regularly) epic [dMVdL20, Lemma 2.6]. Let us now check the two commutator
conditions. By Remark 3.18, the square we used to define $\xi$ is equivalent
to the condition (PFF). For (PCM), consider the following diagram.

${X\flat X}$${X}$${B\flat X}$${X}$${B\flat
B}$${B}$$\scriptstyle{c^{{X},{X}}}$$\scriptstyle{\partial\flat
1_{X}}$$\scriptstyle{\xi}$$\scriptstyle{B\flat\partial}$$\scriptstyle{\partial}$$\scriptstyle{c^{{B},{B}}}$

By Lemma 2.21, we want to show that the bottom square is commutative. Since
the top square is commutative by construction of $\xi$ and the arrow
$\partial\flat 1_{X}$ is epic, we must just check that the outer rectangle is
also commutative, which is indeed the case since

$\begin{split}c^{{B},{B}}\circ\bigl{(}(B\flat\partial)\circ(\partial\flat
1_{X})\bigr{)}&=(\langle
1_{B},1_{B}\rangle\circ\kappa_{{B},{B}})\circ(\partial\flat\partial)=\langle
1_{B},1_{B}\rangle\circ\bigl{(}(\partial+\partial)\circ\kappa_{{X},{X}}\bigr{)}\\\
&=\langle\partial,\partial\rangle\circ\kappa_{{X},{X}}=\partial\circ(\langle
1_{X},1_{X}\rangle\circ\kappa_{{X},{X}})=\partial\circ
c^{{X},{X}}\text{.}\end{split}$

#### Conclusion

To summarise, when $\partial\colon X\to B$ is a monomorphism, the Precrossed
Module Condition corresponds exactly to the existence of a conjugation action
of $B$ on $X$, which is a characterisation of normal subobjects in a semi-
abelian context. Moreover, when it is the case, the Peiffer Condition follows
automatically. For a regular epimorphism $\partial\colon X\to B$, the
condition (PFF) implies that $\partial$ is central (even in a more general
case without any assumption on $\partial$) so we have a central extension.
Conversely, given a central extension $\partial\colon X\to B$, the condition
(PFF) means that the action $\xi$ makes Equation ∥ commute and, using the
centrality of $\partial$, we may show that such an action exists. Then, once
we have this action $\xi$ making Equation ∥ commute, the condition (PCM) comes
for free.

##  4\. Further work

The theory of twisted commutators can be further developed in several
different ways. We name a few.

First of all, more examples of twisted commutators can be found “in nature”;
we are expecting that, often, the existence of an arrow may be seen as twisted
commutativity. One situation which we would like to better understand from our
perspective is the so-called _Peiffer commutator_ of [CMM17]. In fact, there
are several instances of twisted commutativity in that article.

Another way to extend the current theory would be beyond the condition (SH).
In the case of the classical commutator, this involves ternary commutators, as
explained in [HVdL13]. So, is there an appropriate concept of _higher-order
twisted cosmash product_? And a corresponding _higher-order twisted
commutator_? In the classical case, there is a join decomposition formula for
subobjects $K$, $L$, $M\leq X$:

$[K,L\vee M]=[K,L]\vee[K,M]\vee[K,L,M]\text{,}$

which explains the sudden appearance of a ternary commutator. Does a similar
decomposition make sense for twisted commutators? Can we perhaps decompose a
twisted commutator in terms of ordinary commutators, or decompose ordinary
commutators in terms of the twisted commutator? The same question arises for
more exotic types of commutators, such as, in particular, the ones considered
in [SVdL21]—which is what initially motivated us to start these
investigations.

Finally, we believe it would be interesting to explore relations with
categorical Galois theory [BJ01], with the aim of extending the use of Galois
theory to the study of abelian non-central extensions.

## Acknowledgements

Many thanks to Nelson Martins-Ferreira for fruitful discussions on the subject
of the article and, in particular, for suggesting the level of generality in
Definition 2.2.

## References

  * [BB04] Francis Borceux and Dominique Bourn  “Mal’cev, Protomodular, Homological and Semi-Abelian Categories”, Math. Appl. 566  Kluwer Academic Publishers, 2004  DOI: 10.1007/978-1-4020-1962-3
  * [BJ01] Francis Borceux and George Janelidze  “Galois Theories”, Camb. Stud. Adv. Math. 72  Cambridge University Press, 2001  DOI: 10.1017/cbo9780511619939
  * [BJK05] Francis Borceux, George Janelidze and Gregory Maxwell Kelly  “Internal object actions”  In _Comment. Math. Univ. Carolin._ 46.2, 2005, pp. 235–255  URL: https://dml.cz/handle/10338.dmlcz/119522
  * [Bou01] Dominique Bourn  “$3\times 3$ Lemma and Protomodularity”  In _J. Algebra_ 236.2, 2001, pp. 778–795  DOI: 10.1006/jabr.2000.8526
  * [Bou02] Dominique Bourn  “Intrinsic centrality and associated classifying properties”  In _J. Algebra_ 256.1, 2002, pp. 126–145  DOI: 10.1016/s0021-8693(02)00149-7
  * [BG02] Dominique Bourn and Marino Gran  “Central extensions in semi-abelian categories”  In _J. Pure Appl. Algebra_ 175.1–3, 2002, pp. 31–44  DOI: 10.1016/s0022-4049(02)00127-5
  * [BG02a] Dominique Bourn and Marino Gran  “Centrality and Normality in Protomodular Categories”  In _Theory Appl. Categ._ 9.8, 2002, pp. 151–165  URL: http://www.tac.mta.ca/tac/volumes/9/n8/9-08abs.html
  * [BJ98] Dominique Bourn and George Janelidze  “Protomodularity, Descent, and Semidirect Products”  In _Theory Appl. Categ._ 4.2, 1998, pp. 37–46  URL: http://www.tac.mta.ca/tac/volumes/1998/n2/4-02abs.html
  * [BJ09] Dominique Bourn and Zurab Janelidze  “Subtractive Categories and Extended Subtractions”  In _Appl. Categ. Structures_ 17.4, 2009, pp. 317–343  DOI: 10.1007/s10485-008-9182-z
  * [BS76] Ronald Brown and Christopher B. Spencer  “$G$-Groupoids, Crossed Modules and the Fundamental Groupoid of a Topological Group”  In _Indag. Math. (Proc.)_ 79.4, 1976, pp. 296–302  DOI: 10.1016/1385-7258(76)90068-8
  * [CKP93] Aurelio Carboni, Gregory Maxwell Kelly and Maria Cristina Pedicchio  “Some Remarks on Maltsev and Goursat Categories”  In _Appl. Categ. Structures_ 1.4, 1993, pp. 385–421  DOI: 10.1007/bf00872942
  * [CPP92] Aurelio Carboni, Maria Cristina Pedicchio and N. Pirovano  “Internal Graphs and Internal Groupoids in Mal’cev Categories”  In _Category Theory 1991_ , CMS Conf. Proc. 13  American Mathematical Society, 1992, pp. 97–109
  * [CGVdL15] Alan S. Cigoli, James R.. Gray and Tim Van der Linden  “Algebraically Coherent Categories”  In _Theory Appl. Categ._ 30.54, 2015, pp. 1864–1905  URL: http://www.tac.mta.ca/tac/volumes/30/54/30-54abs.html
  * [CMM17] Alan S. Cigoli, Sandra Mantovani and Giuseppe Metere  “Peiffer Product and Peiffer Commutator for Internal Pre-Crossed Modules”  In _Homology Homotopy Appl._ 19.1, 2017, pp. 181–207  DOI: 10.4310/hha.2017.v19.n1.a10
  * [dMVdL20] Davide Micco and Tim Van der Linden  “Compatible Actions in Semi-Abelian Categories”  In _Homology Homotopy Appl._ 22.2, 2020, pp. 221–250  DOI: 10.4310/hha.2020.v22.n2.a14
  * [Duv23] Arnaud Duvieusart  “Ternary semidirect products in semiabelian categories” poster at the International Category Theory Conference CT2023, Louvain-la-Neuve, 2023
  * [GVdL08] Marino Gran and Tim Van der Linden  “On the second cohomology group in semi-abelian categories”  In _J. Pure Appl. Algebra_ 212.3, 2008, pp. 636–651  DOI: 10.1016/j.jpaa.2007.06.009
  * [HL13] Manfred Hartl and Bruno Loiseau  “On Actions and Strict Actions in Homological Categories”  In _Theory Appl. Categ._ 27.15, 2013, pp. 347–392  URL: http://www.tac.mta.ca/tac/volumes/27/15/27-15abs.html
  * [HVdL11] Manfred Hartl and Tim Van der Linden  “The ternary commutator obstruction for internal crossed modules”, 2011  DOI: 10.48550/arXiv.1107.0954
  * [HVdL13] Manfred Hartl and Tim Van der Linden  “The ternary commutator obstruction for internal crossed modules”  In _Adv. Math._ 232.1, 2013, pp. 571–607  DOI: 10.1016/j.aim.2012.09.024
  * [Hig56] Philip J. Higgins  “Groups with Multiple Operators”  In _Proc. Lond. Math. Soc. (3)_ 6.3, 1956, pp. 366–416  DOI: 10.1112/plms/s3-6.3.366
  * [Huq68] Syed A. Huq  “Commutator, Nilpotency, and Solvability in Categories”  In _Q. J. Math._ 19.1, 1968, pp. 363–389  DOI: 10.1093/qmath/19.1.363
  * [Jan03] George Janelidze  “Internal Crossed Modules”  In _Georgian Math. J._ 10.1, 2003, pp. 99–114  DOI: 10.1515/GMJ.2003.99
  * [JMT02] George Janelidze, László Márki and Walter Tholen  “Semi-abelian categories”  In _J. Pure Appl. Algebra_ 168.2–3, 2002, pp. 367–386  DOI: 10.1016/S0022-4049(01)00103-7
  * [JMU07] George Janelidze, László Márki and Aldo Ursini  “Ideals and clots in universal algebra and in semi-abelian categories”  In _J. Algebra_ 307.1, 2007, pp. 191–208  DOI: 10.1016/j.jalgebra.2006.05.022
  * [JMU09] George Janelidze, László Márki and Aldo Ursini  “Ideals and Clots in Pointed Regular Categories”  In _Appl. Categ. Structures_ 17.4, 2009, pp. 345–350  DOI: 10.1007/s10485-008-9135-6
  * [Jan05] Zurab Janelidze  “Subtractive Categories”  In _Appl. Categ. Structures_ 13.4, 2005, pp. 343–350  DOI: 10.1007/s10485-005-0934-8
  * [ML98] Saunders Mac Lane  “Categories for the Working Mathematician”, Grad. Texts Math. 5  Springer, 1998  DOI: 10.1007/978-1-4757-4721-8
  * [MM10] Sandra Mantovani and Giuseppe Metere  “Internal Crossed Modules and Peiffer Condition”  In _Theory Appl. Categ._ 23.6, 2010, pp. 113–135  URL: http://www.tac.mta.ca/tac/volumes/23/6/23-06abs.html
  * [MM10a] Sandra Mantovani and Giuseppe Metere  “Normalities and commutators”  In _J. Algebra_ 324.9, 2010, pp. 2568–2588  DOI: 10.1016/j.jalgebra.2010.07.043
  * [MF10] Nelson Martins-Ferreira  “Star-Multiplicative Graphs in Pointed Protomodular Categories”  In _Theory Appl. Categ._ 23.9, 2010, pp. 170–198  URL: http://www.tac.mta.ca/tac/volumes/23/9/23-09abs.html
  * [MFVdL12] Nelson Martins-Ferreira and Tim Van der Linden  “A Note on the “Smith is Huq” Condition”  In _Appl. Categ. Structures_ 20.2, 2012, pp. 175–187  DOI: 10.1007/s10485-010-9231-2
  * [Rod04] Diana Rodelo  “Moore Categories”  In _Theory Appl. Categ._ 12.6, 2004, pp. 237–247  URL: http://www.tac.mta.ca/tac/volumes/12/6/12-06abs.html
  * [RVdL10] Diana Rodelo and Tim Van der Linden  “The Third Cohomology Group Classifies Double Central Extensions”  In _Theory Appl. Categ._ 23.8, 2010, pp. 150–169  URL: http://www.tac.mta.ca/tac/volumes/23/8/23-08abs.html
  * [RVdL16] Diana Rodelo and Tim Van der Linden  “Higher central extensions and cohomology”  In _Adv. Math._ 287, 2016, pp. 31–108  DOI: 10.1016/j.aim.2015.09.023
  * [Sha17] Vaino Tuhafeni Shaumbwa  “A Characterization of the Huq Commutator”  In _Theory Appl. Categ._ 32.46, 2017, pp. 1588–1600  URL: http://www.tac.mta.ca/tac/volumes/32/46/32-46abs.html
  * [Sha23] Vaino Tuhafeni Shaumbwa  “Commutative Objects, Central Morphisms and Subtractors in Subtractive Categories”  In _Appl. Categ. Structures_ 31.2, 2023  DOI: 10.1007/s10485-023-09715-2
  * [SVdL21] Cyrille Sandry Simeu and Tim Van der Linden  “On the Ternary Commutator, I: Exact Mal’tsev Categories”  In _Theory Appl. Categ._ 36.15, 2021, pp. 379–422  URL: http://www.tac.mta.ca/tac/volumes/36/15/36-15abs.html