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: learning: questions

Topic: finitely presentable objects in presheaves

Jan Pax (Aug 26 2022 at 12:34):

How can be described finitely presentable objects in ${\mathbf{Set}^{\cal A}}^{op}$ for $\cal A$ small ?

Nathanael Arkor (Aug 26 2022 at 12:47):

They're precisely the finite colimits of representables: see this MathOverflow question for instance.

Jan Pax (Aug 26 2022 at 17:22):

I would add that they are split subobjects of the finite colimits of representables. Do you agree with me ?

Nathanael Arkor (Aug 26 2022 at 18:12):

Retracts of finite colimits of representables are finite colimits of representables, so you don't need to add anything else.

Jan Pax (Aug 26 2022 at 18:51):

This fact is known to me. However I would like to see a proof. Could you please give me a link to one ? Or is it completely obvious ?

Morgan Rogers (he/him) (Aug 26 2022 at 19:59):

You can deduce it from point three of Tim Campion's answer and the fact that a split subobject is the coequalizer of the identity (on the "finite colimit of representables") and the idempotent endomorphism induced by the split subobject.

Zhen Lin Low (Aug 26 2022 at 23:44):

I think in practice one does not need the (a priori) stronger fact that the finitely presentable objects are finite colimits of representables, but only the fact that the class of finitely presentable objects is the smallest class containing the representables and also closed under finite colimits.

Jan Pax (Aug 27 2022 at 17:22):

@Zhen Lin Low Could you please explain to me the difference between weaker "in the closure of finite colimits of hom functors" and the stronger "finite colimits of representables" ? They seem to me to be the very same things.

Reid Barton (Aug 27 2022 at 17:25):

It's not obvious that a finite colimit of (finite colimits of representables) is a finite colimit of representables. And other statements like this are false.

John Baez (Aug 28 2022 at 20:42):

Right. Jan: if you read the MathOverflow question (which I asked), and Tim Campion's initial response:

Hang on - this shows that X is a finite colimit of (finite colimits of representables) -- a "2-fold" finite colimit of representables. But how does one turn this into an actual finite colimit of representables?

and then his later answer, it should help.

Jan Pax (Aug 29 2022 at 17:33):

@John Baez Right. BTW he writes: "Hom(x,-) commutes with colimits for x representable." Reading this, I have in mind something slightly different: that hom(x,-) preserves directed colimits. What's the difference between mine and his approach ? What does Djament mean by x representable ? There is yet a third possiiblity that x is a hom-functor but now we have already hom outside x.

John Baez (Aug 29 2022 at 22:37):

Sorry, I don't have the energy to think about this stuff now. Djament's answer is at least a bit mixed up, so if I were you I'd focus on Tim Campion's answer.

Morgan Rogers (he/him) (Sep 02 2022 at 11:06):

@Jan Pax I'll sketch out a comprehensive answer so that we can restrict the conversation to here.
The representables in $\mathcal{E} := [\mathcal{A}^{\mathrm{op}},\mathrm{Set}]$ are those functors (isomorphic to ones) of the form $y(X) = \mathrm{Hom}_{\mathcal{A}}(X,-)$, for $X$ an object of $\mathcal{A}$. One consequence of the Yoneda lemma is that $\mathrm{Hom}_{\mathcal{E}}(y(X),P) \cong P(X)$ for any presheaf $P$, and since colimits of presheaves are computed pointwise, the functor $\mathrm{Hom}_{\mathcal{E}}(y(X),-)$ preserves small colimits; the representables have the property of being "indecomposable projective", or "supercompact projective", and conversely any object with these properties is a retract of a representable.
Now, if I want to identify the objects which merely preserve filtered colimits, then this will include all of the representables, but also some objects constructed from them; the result is that they are precisely the finite colimits of representables. The proof that finite colimits of representables are finitely presentable uses the fact that filtered colimits commute with finite limits in $\mathrm{Set}$. Going the other way, we take advantage of the fact that every object in a presheaf topos is a small colimit of representables, and that any small colimit can be expressed as a filtered colimit of finite colimits. Let $P$ be a finitely presentable object, and write $P \cong \mathrm{colim}_{i \in I} \mathrm{colim}_{j \in J_i} y(X_{i,j})$ with $I$ filtered and $J_i$ finite. Then $\mathrm{id}_{P} \in \mathrm{Hom}_{\mathcal{E}}(P,P) \cong \mathrm{colim}_{i \in I} \mathrm{Hom}_{\mathcal{E}}(P,\mathrm{colim}_{j \in J_i} y(X_{i,j})),$ where the other direction of the isomorphism is by composition with the legs of the colimit cone (this is a detail of what "preserving colimits" means which is not always spelled out). This means that there must be a morphism $P \to \mathrm{colim}_{j \in J_i} y(X_{i,j})$ for some index $i \in I$ which is split by the corresponding leg of the colimit cone (a filtered colimit of empty sets is empty); this is the monomorphism that Djament mentions. Thus $P$ is a retract of a finite colimit of representables, and we can use Tim Campion's answer to deduce that $P$ itself can be presented as a finite limit of representables too.