Category = Groupoid x Poset?

Is it possible to split a given category $C$ up into its groupoid of isomorphisms and a category that resembles a poset?

"Splitting up" should be that $C$ can be expressed as some kind of extension of a groupoid $G$ by a poset $P$ (or "directed category" $P$ the only epimorphisms in $P$ are the identities, all isomorphisms in $P$ are identities).

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I don't see why you should, without some kind of "acyclicity" condition on your category. Consider the monoid of natural numbers as a one-object category, for instance -- what should this splitting be? –  Dan Petersen Mar 24 '10 at 14:58
The requirement that the only epimorphisms in $P$ are identities is not satisfiable for general categories; consider the following counter example: f 1 -----> 2 ^ -----> ^ |\ g |\ - - The morphisms f,g are both epi, but not identities. If you replace 'epimorphisms' with isomorphisms, then the construction I outlined below should work. –  Mikola Mar 24 '10 at 15:06
I'm no category theorist but -- consider the special case of the monoid $M = R - {0}$ under multiplication, where R is an integral domain. Then we have $U = R^{\times}$, the group of units of the monoid, and $M/U$ has a natural partial ordering induced by the divisibility relation. I wonder whether there is a generalization of this to (some more, not all) categories? –  Pete L. Clark Mar 24 '10 at 21:10
Perhaps this should be posted as a separate question, but will this splitting up work if we allow monoids? That is, can a category be split up into posets, groupoids and monoids? –  Colin Tan Apr 20 '12 at 16:04
You seem to mean that any category without isomorphisms is directed or a poset. The category with only one object whose endomorphism monoid is the natural numbers is not. –  Fernando Muro May 17 at 7:07

5 Answers

I am also looking forward to answers to your question. Meanwhile here is something pointing roughly into that direction:

One can study a category $C$ through its set-valued functor category $Set^C$. By the Yoneda lemma, $C$ sits as a full subcategory inside this functor category, and from it one can reconstruct something close to $C$ (I think the idempotent completion of $C$). But non-equivalent categories can give rise to equivalent functor categories, e.g. category $C$ in which not every idempotent splits and its idempotent completion, i.e. the category made from $C$ by adjoining objects such that each idempotent becomes a composition of projection to and inclusion of a subobject and thus splits. One calls such categories Morita-equivalent.

Now $Set^C$ is a Grothendieck topos (:=category of sheaves on a site, in this case with trivial topology) and there is the following theorem about those:

A locale is a distributive lattice closed under meets and finite joins, just like the lattice of open sets of a topological space, so it is a particular poset. The theorem of Joyal and Tierney, from their monograph "An extension of the Galois theory of Grothendieck", states that every Grothendieck topos is equivalent to the category of $G$-equivariant sheaves on a groupoid object in locales - see e.g. here.

Well at least it is a statement which separates a category into a groupoid and a poset part. So if you look from very far and take it with a boulder of salt you could read this as saying that every category is "Morita-equivalent" (not really!) to a groupoid internal to posets (it makes some intuitive sense to see this as an extension).

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I agree that considering idempotents is important. To focus the issue: take the category C with only one object and only one non-identity morphism, which is idempotent. The original question seems to founder on this example; how do you "split it up"? Peter reaches a positive answer by blurring this issue out (which probably has to be done if you do want a positive answer). –  Tom Leinster Mar 25 '10 at 4:00

One type of category that factors nicely is called an EI category. The definition is that every Endomorphism is an Isomorphism. After taking the quotient by the groupoid, every endomorphism is the identity. But it is still not a poset. It could be something like the category of two parallel arrows, where Mor(A,B) has two elements, Mor(B,A) none, and the endomorphisms of each object are only the identity. This has a further poset quotient $A\to B$, but it isn't there yet.

So groupoid and poset are only two kinds of behavior in categories. Monoids are a third that have been mentioned before. In particular, idempotents, as in the monoid {0,1} under multiplication, do not embed in any group. And the two parallel arrows category is yet a fourth.

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this reminds me of a characterization of the graphs of dynamical systems –  Joey Hirsh Jul 31 '12 at 5:23

Unless I totally misunderstand the question, this doesn't even work for categories with one object, i.e. monoids (which are not groups), does it?

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Given any locally small category, $C$, the collection of all isomorphisms forms a subgroupoid, $G \subseteq C$, where $Ob(G) = Ob(C)$ and $Hom_G(A,B) = \left ( f \in Hom_C(A,B) : \exists g, h \in Hom_C(B,A) g \circ f = id_A, f \circ h = id_B \right )$.

Because $G$ is a groupoid, it determines an equivalence relation, $R$ on the objects and morphisms of $C$ such for $A, B \in Ob(C)$:

$A \equiv_R B \Longleftrightarrow Hom_G(A,B) \neq \emptyset$

And for $f, g \in Hom_C(A,B)$:

$f \equiv_{R_{A,B}} g \Longleftrightarrow \exists h_B \in Hom_G(B,B), h_A \in Hom_G(A,A) : h_B \circ f = g \circ h_A$

If I understand what you are asking, then the quotient $C/R$ should be the 'poset' you want.

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*Subject to the substitution epi -> iso as clarified in the comments, as otherwise this is not possible. –  Mikola Mar 24 '10 at 16:27

In the finite, one-object case, Krohn-Rhodes theory gives a way to decompose a semigroup as a wreath product of finite simple groups and aperiodic semigroups, which are in some sense "as non-grouplike as possible", though in a different way from posets. There is an extension of the theory to categories due to Wells. I don't know much about this, but it is similar to the sort of decomposition you're asking for.

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