Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: theory: category theory

Topic: Colimits of finite poset-shaped diagrams

Amar Hadzihasanovic (Jan 26 2024 at 08:49):

I'm in the following situation: I have a category with

• a strict initial object, and
• pushouts of spans of monomorphisms, and monomorphisms are stable under these pushouts.
  These also imply that I have finite coproducts, and coproduct injections are mono.

I would like to deduce the following:
Every diagram of monomorphisms whose domain is a finite poset has a colimit.

The proof I have in mind goes a bit like the proof of Newman's lemma in rewriting, you create the colimit cone by induction on the Hasse diagram of the poset, closing every "bifurcation" $x \leftarrow y \rightarrow z$ locally with a pushout, and finally taking a coproduct of all the colimits of connected components.

(Also, this doesn't really seem to have anything to do with monomorphisms, you could replace "monomorphisms" with any class of morphisms which is stable under pushouts in the sense above, and such that the unique maps from the initial object belong to it.)

Does anyone know a reference for this, or some other result that it can be deduced from?

Morgan Rogers (he/him) (Jan 26 2024 at 09:10):

It doesn't really have to be a poset; I think you could mimic a proof that initial + pushouts implies all finite colimits, just adding an argument for the existence of the colimits involved at each stage.

Amar Hadzihasanovic (Jan 26 2024 at 09:17):

Unfortunately no, that argument involves forming pushouts that use some universal maps out of a coproduct, which are not guaranteed to be monomorphisms/belong to the specified class. Indeed in the example I'm interested in, I do not have coequalisers.

Morgan Rogers (he/him) (Jan 26 2024 at 09:23):

Ah I misread the pushouts as the more typical "pushouts of monos" (i.e. along arbitrary morphisms).
Okay, another suggestion would be to present this as the existence of a Kan extension along the inclusion of the finite poset into its ideal completion... although I can still only imagine proving that by induction as you suggested. I'll shut up and see if anyone provides a reference.

Reid Barton (Jan 27 2024 at 01:11):

This seems unlikely to me. Here is something which is almost and probably in fact is a counterexample, though it seems fiddly to prove the required colimits actually don't exist:
In the category of simplicial sets, consider the full subcategory of all those objects with the weak homotopy type of a set. This is closed under the formation of the initial object (which is strict) and pushouts of spans of monos (which are again monos), but it isn't closed under forming colimits of general diagrams of monos indexed by a poset.
For instance, with such a limit I can form two parallel 1-simplices with their endpoints identified, which has the homotopy type of a circle.

Morgan Rogers (he/him) (Jan 27 2024 at 05:29):

Isn't your last example the pushout of the inclusion of two 0-simplices into the 1-simplex along itself? That's a mono, so if the pushout doesn't exist then the hypotheses fails.

Amar Hadzihasanovic (Jan 27 2024 at 07:11):

While I am also not sure about the counterexample, I did realise that “poset-shaped” is too general. For example I can obtain the coequaliser of two monomorphisms $f, g: a \to b$ by the following poset-shaped diagram: the poset is $\{0, 0', 1, 1'\}$ with $0, 0' < 1, 1'$, and I map $0 < 1$ to $f$, $0' < 1$ to $g$, and the other two to identities on $a$.

Amar Hadzihasanovic (Jan 27 2024 at 07:14):

The problem seems to arise when I have two different, non-joinable “forks” under the same pair of maximal elements of the poset, because in that case I can only join one of them with a pushout, and the other would need to be “coequalised” instead.

Amar Hadzihasanovic (Jan 27 2024 at 07:17):

A possible revision of this conjecture would be requiring the domain to be a meet-semilattice instead of just a poset. Then any such pairs of forks could be "merged" into the single fork from their meet. Of course, the poset from my counterexample does not have meets.

Reid Barton (Jan 28 2024 at 17:49):

Yeah, the example I gave doesn't quite work, but I agree the original version of the conjecture is not plausible.

Reid Barton (Jan 29 2024 at 16:47):

How would you approach a colimit over a cube $\{0 < 1\}^3$ with its terminal object removed? That's a meet semilattice.

Amar Hadzihasanovic (Jan 29 2024 at 17:15):

In no way, I guess. On further thought it doesn't seem that my original conjecture has any valid non-trivial restriction.

Morgan Rogers (he/him) (Jan 30 2024 at 09:30):

Well that was fun :cowboy: