# Decomposing a large colimit as a pushout of smaller colimits

I would like to find a reference in the literature for the following result. I have it on high authority that it isn't in 'Categories for the Working Mathematician' and I can't find it in Borceux's handbook. It's a result that I'm confident is true (at least when stated correctly) and is probably second nature to category theorists. I however am writing for group theorists and so want to reference results thoroughly.

I have a functor $F:\mathcal{C} \rightarrow \mathcal{D}$. The target category $\mathcal{D}$ is cocomplete. The source category $\mathcal{C}$ is finite and can be decomposed as the pushout of smaller categories $\mathcal{C}_1\leftarrow\mathcal{C}_0\rightarrow\mathcal{C}_2$. The functors from these into $\mathcal{D}$ are denoted $F_1,F_0$ and $F_2$ respectively.

I need to take the colimit of $F$ and I think that it can be taken to be the pushout of

$\text{colim}F_1\leftarrow\text{colim}F_0\rightarrow\text{colim}F_2$.

Obviously if $\mathcal{C}$ were constructed from a different colimit rather than a pushout one might expect an analogous result.

Giving a proof is an option, but would be out of context with the rest of the paper and probably consigned to an unread appendix. Or I could just quote it without proof. Help, or just opinions would be very welcome.

• You should probably clarify exactly what you mean by "can be decomposed as a pushout of smaller categories ..." Also, tthe finiteness of C seems unnecessary. (Incidentally, I am never sure what people mean by a "finite" category. Finite number of (objects and) morphisms? Nerve is a finite simplicial set? Each of these can be a useful hypothese in working with (co)limits; so can "finitely generated". But I don't you need any of them for this.) – Tom Goodwillie Jul 14 '10 at 14:49
• To respond to Tom Goodwillie's remark about the meaning of "finite" in category theory in his comment, I think the standard usage is that "finite" means finitely many objects and finitely many arrows. This should be compared with the usage of "small" and "locally small" (e.g., "locally finite" means that the hom sets are all finite). – Michael A Warren Jul 14 '10 at 22:37
• Thank you for the point Tom, I thought briefly about writing small and probably should have. Of course by finite I did mean finitely many objects and finitely many arrows as Michael suggested. – James Griffin Jul 16 '10 at 8:41
• I thought that was what you meant. The stronger condition "the nerve is a finite simplicial set" does come up in working with homotopy limits and colimits sometimes. And the weaker one -- generated by a finite set of morphisms -- comes up in at least one context: limit over such a category commutes with filtered colimits. – Tom Goodwillie Jul 20 '10 at 16:41

I'd agree that the result is true, and “well-known” to category-theorists! Unfortunately I don't know a specific reference, but my best guess would be something like Kelly’s “Elements of Enriched CT”, which proves lots of useful things about (co)ends and weighted (co)limits, which specialise to lots of useful things about limits. It's also hard not to wonder about the legendary treatise of Chevalley on “all possible properties of limits” that got lost in the mail...

If you can't find a reference, though, the proof can certainly be made pretty short — I used 6 well-spaced or 2 cramped lines, and it can probably be compressed further...

[this was meant to be just a comment, but it got a bit too long]

$\newcommand{\C}{\mathbf{C}} \newcommand{\D}{\mathbf{D}} \DeclareMathOperator{\colim}{colim}$ Edit: For clarification, the precise statement I had in mind is that $$\colim_{I}\ (\colim_{\C_i}\ F_i)\ \cong\ \colim_{\left( \colim_{I} \C_i \right)} [F_i]_{i \in I}$$ where $I$ is a small cat, $\C_i$ is an $I$-indexed diagram of small cats, $F_i : \C_i \to \D$ is a co-cone of functors, $[F\_i]\_{i \in I}$ denotes the induced cotuple functor $\colim_{I} \C\_i \to \D$, and all the relevant colimits exist in $\D$.

• Hi Peter, long time no 'see', I will investigate that reference. Perhaps if my search fails I should just include a short proof, 6 well-spaced lines sounds acceptable. – James Griffin Jul 16 '10 at 8:55
• Oh, ha — I hadn't taken in the name on the question, so hadn't realised it was you! Long time indeed — great to see you here :-) Good luck with the reference-hunting, in any case… – Peter LeFanu Lumsdaine Jul 18 '10 at 9:27
• So… in strange-coincidence-of-the-week, I've just found myself needing to use exactly the same fact in writing up a bit of the background in my thesis. (Maybe this was unconsciously at the back of my mind when I answered your question, although it's something I hadn't thought about consciously in months or even years!) So if you do find a reference, I'd be very grateful for it too! :-) – Peter LeFanu Lumsdaine Jul 20 '10 at 14:42
• Every lost piece of writing by a Bourbaki member is a small tragedy in and of itself =(. – Harry Gindi Jul 20 '10 at 15:18
• Just a note that (10 years later) Zhen Lin gives a nice, formal proof of this statement here. Other answers to that question give an $\infty$-categorical generalization. – Tim Campion Sep 3 '20 at 23:32