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I'm reading and struggling with bits and pieces of the book Mal'cev, Protomodular, Homological, and Semi-Abelian categories by Borceux and Bourn. At the moment I'm having trouble with:

Theorem 1.3.22 Let $\mathsf C$ be a unital category. For all objects $X,Y\in \mathsf C$, the set $\boldsymbol{\mathsf Z}(X,Y)$ of central morphisms acts transitively on $\mathsf C(X,Y)$, where the action is given by addition.

The only bit of the proof (page 38) which seems relevant is the observation that given $z\in \boldsymbol{\mathsf Z}(X,Y),f\in \mathsf C(X,Y)$ the cooperator $\phi_{z,f}$ is the composite $\phi _z\circ (1_X \times f)$. How does this show the action is transitive?

The transitivity of this action seems strange to me in concrete terms - it's saying every two arrows $X\rightrightarrows Y$ have a "difference" which is almost well defined, and I don't see what's going on here unless $\boldsymbol{\mathsf Z}(X,Y)$ is a group... What am I seeing wrong?

Added:

Definition. A unital category is a finitely complete pointed category such that the unit injections $j_i:A_i\to A_1\times A_2$ given by $j_1=(1,0),j_2=(0,1)$ where $0,1$ are the zero and identity arrows are jointly extremally epimorphic.

Example. The unital algebraic varieties are unital categories.

Definition. In a unital category, arrows $A\overset{f}{\to}C\overset{g}{\leftarrow}B$ are said to cooperate if there's a unique arrow $A\times B\to C$ making the obvious diagram with the $j_i$ commute.

Example. Monoid homomorphisms cooperate iff their images commute pointwise, i.e $f(x)g(y)=g(y)f(x)$.

Definition. An arrow is central if it cooperates with the identity on its codomain.

Example. A submonoid is central iff its contained in the center of the monoid.

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    $\begingroup$ Could you please add the definitions of the less standard terms? unital category, central morphism, cooperator. $\endgroup$
    – HeinrichD
    Commented Nov 16, 2016 at 14:59
  • $\begingroup$ @HeinrichD done. $\endgroup$
    – Arrow
    Commented Nov 16, 2016 at 15:32

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The action is definitely not transitive. Consider for example the category of groups, which is unital. Here, the statement would be that for two groups $X,Y$ the group $\hom(X,Z(Y))$ (where $Z(Y)$ denotes the center of $Y$) acts transitively on the set $\hom(X,Y)$ via pointwise multiplication. But the orbit of the trivial homomorphism $X \to Y$ is just the set of homomorphisms $X \to Z(Y)$. Also, if $X=\mathbb{Z}$, we simply have the action of $Z(Y)$ on the underlying set of $Y$, whose orbit set can be quite large.

Perhaps the authors meant something different with transitive here, in particular because $\mathsf{Z}(X,Y)$ in general is just a commutative monoid?

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    $\begingroup$ PS: It seems that the theorem is used a couple of times in the book, but only the statement that addition is an action. So it's not so bad if the transitivity fails. $\endgroup$
    – HeinrichD
    Commented Nov 17, 2016 at 10:34
  • $\begingroup$ At the end of section 1.4, on page 49, the authors state that in the case of pointed sets, the axioms for transivity for a monoid $M$ on a subset $R\subseteq X\times M$ reduce to $(x,y)(y'\star y'')=((x,y)\cdot y')\cdot y''$, $(x,y)\cdot 1=(x,y)$. So it seems that by "transitive" action they mean a morphism $M\times X\to X$ (the action of $M$ on $X$) that is compatible with the unital magma structure on $M$ (i.e. $m_1(m_2x)=(m_1m_2)x)$ and the unit $ex=x$). $\endgroup$
    – Vladimir Sotirov
    Commented Jan 21, 2022 at 19:08

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