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: Trouble with Yoneda not preserving colimits

John Onstead (May 06 2025 at 11:24):

As I've been learning about presheaves and the co-Yoneda lemma, I've realized it's set up an interesting dichotomy between two ways of gluing objects in a category $C$. You can either take a colimit in $C$ to get another object of $C$ that acts like the objects in the diagram "glued together", or you could take the colimit in the presheaf category. This comes with the intuition of building up a space "locally modeled on" the objects of $C$ that might not live in $C$ itself. The reason this seems to be a dichotomy is because Yoneda embedding does not preserve colimits, so these two ways of gluing do not, on the surface, seem to correspond.

John Onstead (May 06 2025 at 11:25):

The reason I'm struggling with this is because there are situations where you have to choose between these options to do some sort of gluing you want, and it's not always obvious which choice to make. My question is: is there any general rule connecting these two ideas, and if so, what would be the conceptual underpinning (IE, in terms of what gluing is doing what). Maybe there's some formula to convert a colimit diagram in one setting into an "equivalent" one in the other setting?

Nathan Corbyn (May 06 2025 at 11:26):

This problem is one way to motivate the study of sheaves. Have a look at this article.

James Deikun (May 06 2025 at 11:43):

To expand on this: subcanonical Grothendieck topologies on the category $C$ present ways of gluing objects of $C$ that interpolate between the two above methods, with the trivial Grothendieck topology representing the presheaf category and its colimits and the canonical Grothendieck topology representing, I believe, a completion of $C$ that preserves any [[universal colimits]] it already has.

And for your question as to a formula to convert: you convert a colimit in $\widehat C$ to one in $C$ by taking the [[weighted colimit]] of the identity functor in $C$ weighted by the presheaf. This weighted colimit will exist precisely if taking the colimit in $C$ in the first place would have.

Mike Shulman (May 06 2025 at 14:49):

More generally, for any collection $\Phi$ of colimits that exist in $C$, you can consider the full subcategory of the presheaf category $\widehat{C}$ consisting of those presheaves that preserve the colimits in $\Phi$ (that is, take them to limits in Set). This is (assuming $C$ and $\Phi$ are small) a reflective subcategory of $\widehat{C}$, hence complete and cocomplete, and the Yoneda embedding factors through it by a functor that preserves $\Phi$-colimits. It has a universal property as the free cocompletion of $C$ subject to preservation of $\Phi$-colimits.

Ivan Di Liberti (May 06 2025 at 15:13):

A very general version of this story appears as Section 4 of my paper with Lobbia.

Vincent Moreau (May 06 2025 at 17:41):

I did not know that! Is there a description of $\widehat{C}$ when $\Phi$ is the collection of all colimits that exist in $C$?

John Baez (May 06 2025 at 17:46):

Do you mean $\widehat{C}$, which doesn't depend on $\Phi$, or the full subcategory of the presheaf category $\widehat{C}$ consisting of those presheaves that preserve the colimits in $\Phi$?

Vincent Moreau (May 06 2025 at 17:48):

Sorry, I meant a description of the full subcategory of $\widehat{C}$ whose objects are presheaves preserving all colimits

Mike Shulman (May 06 2025 at 17:55):

Older references include Kennison "On limit-preserving functors", and Freyd-Kelly "Categories of continuous functors I".

Mike Shulman (May 06 2025 at 17:55):

I don't think there is a description of $\widehat{C}_\Phi$ other than its definition, in general.

Mike Shulman (May 06 2025 at 17:56):

(Although I'm not sure what sort of "description" you wanted.)

Vincent Moreau (May 06 2025 at 18:28):

Thanks for the references! I was wondering if these categories had a universal property like the fact that the whole category of presheaves is the free cocompletion (putting aside size issues). In the article "On limit-preserving functors", Kennison writes:

As Lambek points out this result implies that [A, Ens]inf is sup-complete and can be regarded as a nicely behaved completion of A°, the dual or opposite category of A.

Is this category [A, Ens]inf, which I think is $\widehat{C}_\Phi$ for $C$ the opposite of A and $\Phi$ the collection of all colimits, a completion with a universal property, for instance?

Mike Shulman (May 06 2025 at 18:30):

Mike Shulman said:

It has a universal property as the free cocompletion of $C$ subject to preservation of $\Phi$-colimits.

Vincent Moreau (May 06 2025 at 18:32):

I see, thanks

John Onstead (May 06 2025 at 19:46):

Nathan Corbyn said:

This problem is one way to motivate the study of sheaves. Have a look at this article.

James Deikun said:

To expand on this: subcanonical Grothendieck topologies on the category $C$ present ways of gluing objects of $C$ that interpolate between the two above methods

That's cool... I'd heard of sheaves in this use before but hadn't thought too much about it prior to now!

John Onstead (May 06 2025 at 19:49):

Mike Shulman said:

More generally, for any collection $\Phi$ of colimits that exist in $C$, you can consider the full subcategory of the presheaf category $\widehat{C}$ consisting of those presheaves that preserve the colimits in $\Phi$ (that is, take them to limits in Set). This is (assuming $C$ and $\Phi$ are small) a reflective subcategory of $\widehat{C}$

But as a reflective subcategory of a presheaf category, isn't this just the same thing as sheafification? Unless somehow the resulting category isn't a topos, or the reflection isnt left exact, etc.

John Onstead (May 06 2025 at 19:56):

James Deikun said:

And for your question as to a formula to convert: you convert a colimit in $\widehat C$ to one in $C$ by taking the [[weighted colimit]] of the identity functor in $C$ weighted by the presheaf. This weighted colimit will exist precisely if taking the colimit in $C$ in the first place would have.

That's interesting... my follow up to this would be about going the other way. Is there a weighted colimit formula to express a colimit in $C$ within $\widehat C$?

Mike Shulman (May 06 2025 at 20:39):

John Onstead said:

the resulting ... reflection isn't left exact

Bingo.

Mike Shulman (May 06 2025 at 20:39):

(And therefore the resulting category isn't a topos.)

John Onstead (May 07 2025 at 11:54):

Mike Shulman said:

Bingo.

Ah ok, that makes sense!

John Onstead (May 07 2025 at 11:55):

But as a left adjoint, the reflection from presheaves into this subcategory preserves colimits. So that leads to an interesting question. Let's say there was an object $X$ of $C$ that had a colimit representation in both $C$ and $\widehat{C}$ (obviously different, since Yoneda does not preserve colimits). By the universal property of $\widehat{C}_{\Phi}$, as well as the fact that left adjoints preserve colimits, both of these colimit representations of $X$ have incarnations in $\widehat{C}_{\Phi}$.

John Onstead (May 07 2025 at 11:55):

My question is then: what, if any, is the connection between them- do they ever converge, or are they seemingly parallel and independent colimit representations of $X$ in this category?

Mike Shulman (May 07 2025 at 15:06):

Generally they'll be different.

Mike Shulman (May 07 2025 at 15:10):

Of course it depends on what "representations" you pick. For instance, there are in fact a few colimits that Yoneda does preserve, namely the [[absolute colimits]]. And if you choose a $\Phi$-colimit in $C$, then it would be preserved in $\widehat{C}_\Phi$. But if you choose an arbitrary colimit in $C$ and an arbitrary one in $\widehat{C}_\Phi$ they might have totally different shapes even if they were both $\Phi$-colimits.

John Onstead (May 07 2025 at 19:41):

I see. It's certainly interesting that there's a setting where both kinds of colimit exist, but don't necessarily "talk to each other"!

John Onstead (May 07 2025 at 21:23):

I think I now have a good handle on how to "resolve" the fact that Yoneda does not preserve colimits on a mathematical level (IE, take the $\Phi$ subcategory), and I even know how to go from colimits in the presheaf category back to a weighted colimit in the original category. But I still feel as if I am missing sort of a conceptual picture for this situation. That is, what, on a conceptual level, is the fundamental difference between gluing in a category versus in its presheaf category, especially if we are meant to interpret the representables in a presheaf category as the objects of the original category via the Yoneda embedding?

Mike Shulman (May 07 2025 at 21:28):

I don't know if this helps, but one analogy you can think about is the free group $FG$ generated by the underlying set of a group $G$. There's an injection $X\to FX$ for any set $X$, and so we can if we wish think of the generators inside $FG$ as "being" the elements of $G$. But the multiplication in the group $FG$ has nothing to do with the original multiplication in $G$; in particular the product of two generators is no longer a generator.

John Onstead (May 08 2025 at 01:14):

Yes, this analogy helps! Especially factoring in a discussion I had a while ago on here about the analogy between free algebraic objects on generators and the idea of free cocompletion.