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Stream: theory: category theory

Topic: properties of free categories

Mike Shulman (Jul 20 2023 at 01:10):

The free category $F[X]$ on a directed graph doesn't have very many limits and colimits or other categorical properties, but here are some interesting ones it does have:

Every morphism is both monic and epic.
Every commutative square $hf = kg$ has a diagonal filler in one direction or another, i.e. either an $i$ such that $hi = k$ and $f = ig$ , or a $j$ such that $kj = h$ and $g = jf$ . Whichever exists is unique, and if both exist then both are identities.
If there exists a commutative square $hf=kg$ , then the span $(f,g)$ has a pushout.

Property (3) follows from (1) and (2), but properties (1) and (3) have the nice property that they are preserved by products of categories, so that categories of the form $F[X] \times F[Y]$ also have them.

Are there any other interesting properties like this? Does anyone know a reference for any of them?

Todd Trimble (Jul 28 2023 at 03:56):

For what interest this has: $F[X]$ has a terminal object iff $X$ is a rooted tree (directed toward the root as terminal object).

Mike Shulman (Jul 28 2023 at 03:57):

Ah yes. Nice!

Morgan Rogers (he/him) (Jul 28 2023 at 08:47):

The only idempotent endomorphisms are identities. Similarly for isomorphisms, although that's not equivalence-invariant.
Property 2 that you observed is known as equidivisibility for monoids. Apparently that plus grading (that every morphism has a well-defined length) characterizes free monoids, so maybe you can characterize free categories that way.

Mike Shulman (Jul 28 2023 at 17:37):

Thanks for that link! As stated there, equidivisibility doesn't include the uniqueness of " $i$ or $j$ ". And with it stated that way at least, something must be wrong with that characterization, because any group with everything in degree 0 would satisfy it. The page Levi's lemma states the characterization as (non-unique) equidivisibilty plus a $\mathbb{N}$ -grading such that only the identity has degree 0, which seems more plausible.

Mike Shulman (Jul 28 2023 at 17:43):

Oh, I see that apparently some people say "graded monoid" to mean a connected graded monoid, i.e. such that the identity is the only element of degree 0. That's confusing...

Morgan Rogers (he/him) (Jul 28 2023 at 17:47):

Hmm so greater precision is needed, but I expect the proof lifts reasonably directly?