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: equivalences of presheaf categories

Christian Williams (May 08 2021 at 00:21):

Let $F:C\to D$ and suppose $P(F):[C^{op},Set]\to [D^{op},Set]$ is an equivalence. Then what do we know about $F$?

Christian Williams (May 08 2021 at 00:22):

motivating example: suppose $C$ is cartesian, and $F:C\to C^\ast$ freely adds exponentials. it seems like $P(F)$ might be an equivalence; but I'm not sure that's possible.

Morgan Rogers (he/him) (May 08 2021 at 08:19):

Nope, F extends to an equivalence between the idempotent completions of C and D.
While Yoneda factors through $C^*$ is your motivating example, presheaves on $C^*$ are much bigger than presheaves on $C$.

Morgan Rogers (he/him) (May 08 2021 at 08:21):

wait even what I said about Yoneda factoring through isn't accurate; if your F was the exponential completion of C (only adding ones which don't already exist) then Yoneda would factor through it. In any case, the same conclusion applies unless C is already cartesian closed.

Jade Master (May 08 2021 at 14:59):

Cool question. I think that if P(F) is an equivalence then F should be i.e. P reflects equivalences....this is my guess because I am pretty sure P is a 2-dimensional version of faithful.

John Baez (May 08 2021 at 15:10):

There are famous examples where P(F) is an equivalence even when F is not.

John Baez (May 08 2021 at 15:12):

For example, there's a category C with one object such that presheaves on it are reflexive graphs, and C is a subcategory of the usual category D with two objects such that presheaves on it are reflexive graphs.

John Baez (May 08 2021 at 15:13):

I think in this example i: C $\to$ D is fully faithful but not essentially surjective, so not an equivalence, yet P(i) is an equivalence.

John Baez (May 08 2021 at 15:16):

The point here is that D is the Cauchy completion of C - see the definition and also Theorem 2.2 at the link.

John Baez (May 08 2021 at 15:18):

Morgan's reply is a bit terse and cryptic:

Nope, F extends to an equivalence between the idempotent completions of C and D.

John Baez (May 08 2021 at 15:19):

I think he's trying to say: if F: C $\to$ D induces an equivalence of presheaf categories, F extends to an equivalence between the idempotent completions of C and D.

John Baez (May 08 2021 at 15:20):

(It's always nice to spell out details of what one is trying to say here, even if it takes a bit of extra time.)

John Baez (May 08 2021 at 15:20):

The idempotent completion is the same thing as the Cauchy completion.

John Baez (May 08 2021 at 15:21):

And this fact:

If F: C → D induces an equivalence of presheaf categories, F extends to an equivalence between the Cauchy completions of C and D.

John Baez (May 08 2021 at 15:21):

is easy to prove using the nLab's definition of Cauchy completion. I think.

John Baez (May 08 2021 at 15:25):

Heuristically, the Cauchy completion on C proceeds by splitting all idempotents in C. But the nLab describes the easy way to do this: form the category of presheaves on C, and then take the subcategory consisting of all retracts of representables.

John Baez (May 08 2021 at 15:27):

I think if F : C $\to$ D induces an equivalence of presheaf categories P(F) : P(C) $\to$ P(D), you can see P(F) restricts to an equivalence between the subcategories of the presheaf categories consisting of retracts of representables.

John Baez (May 08 2021 at 15:28):

But this is just a way of saying F extends to an equivalence between the Cauchy completion of C and the Cauchy completion of D!

John Baez (May 08 2021 at 15:28):

I say "I think if", because I haven't carefully checked.

Reid Barton (May 08 2021 at 15:30):

The main point is that the retracts of representables have an invariant (i.e. depending only on the presheaf category) description as the objects $A$ for which $\mathrm{Hom}(A, -)$ preserves all colimits.

John Baez (May 08 2021 at 15:31):

Thanks!

John Baez (May 08 2021 at 15:31):

That's great.

Reid Barton (May 08 2021 at 15:31):

BTW, I think nobody ever said it but I assume that this "P" thing takes a functor to left Kan extension?

John Baez (May 08 2021 at 15:31):

Yes.

John Baez (May 08 2021 at 15:32):

(I'd been talking to @Christian Williams about this stuff beforehand, so I can read his mind on this.)

Reid Barton (May 08 2021 at 15:34):

Right, that's how I use the notation too, but oddly it doesn't really matter that much in this situation, because the left adjoint in an adjunction is an equivalence if and only if the right half is.

John Baez (May 08 2021 at 15:36):

So based on Morgan's feedback, let me try to sharpen @Christian Williams's question:

Question: given a functor F: C $\to$ D between small categories, is it true that P(F): P(C) $\to$ P(D) is an equivalence iff K(F): K(C) $\to$ K(D) is an equivalence, where K is the functor sending any small category to its Cauchy completion?

John Baez (May 08 2021 at 15:37):

Less verbosely and precisely: is it true that a functor between categories induces an equivalence between their presheaf categories iff it induces an equivalence between their Cauchy completions?

John Baez (May 08 2021 at 15:37):

We've just seen the $\implies$ direction, but I have a feeling the converse is true too.

Nathanael Arkor (May 08 2021 at 15:40):

The other direction is trivial, because $PK \cong P$ and 2-functors preserve equivalences.

Tim Campion (May 08 2021 at 19:50):

Maybe the other thing to say in this setting is that this can all be done more generally in enriched category theory. There, you again define the Cauchy completion $K_V(C)$ of a V-category C to be the completion of C under absolute V-colimits, and you again have, for $F: C \to D$, that the induced functor $P_V(F): P_V(C) \to P_V(D)$ is an equivalence if and only if the induced functor $K_V(F): K_V(C) \to K_V(D)$ is an equivalence.

For example, taking V to be $([0,\infty],\geq,+)$, you conclude that a map of metric spaces induces an isomorphism of spaces of 1-Lipschitz functions if and only if it induces an isomorphism of Cauchy completions in the usual metric sense.

Taking $V = Ab$, you get the classical notion of Morita equivalence -- for example, modules over a ring $R$ form an equivalent category to modules over the matrix ring $Mat_n(R)$.

And if you go oo-categorical and enrich in spectra, you start talking about surprising equivalences of derived categories -- there are a lot of them because for spectral enrichment, all finite limits and colimits are absolute.

Tim Campion (May 08 2021 at 19:51):

Presumably there's a similar story for internal category theory, but I'm not familiar with it...

Mike Shulman (May 08 2021 at 23:27):

Internal categories are basically set-like, so Cauchy completion again coincides with idempotent-completion.

Tim Campion (May 09 2021 at 03:44):

Mike Shulman said:

Internal categories are basically set-like, so Cauchy completion again coincides with idempotent-completion.

Interesting! I'm having trouble guessing what adjectives are needed to make this even make sense... First we need to ensure that every internal category has a completion with respect to absolute colimits, as well as a completion with respect to idempotents. Then we need to show they coincide... I suppose for internal categories I'm usually happy to assume things like cartesian closure, though, so maybe I shouldn't be so surprised that once everything is set up, this identification results...

Tim Campion (May 09 2021 at 03:45):

I suppose my favorite "interesting" example of internal categories are internal categories in Cat. I suppose I find this statement plausible for double categories, but not obvious.

Emily (May 09 2021 at 09:20):

Joyal calls such a functor a Morita equivalence in https://ncatlab.org/joyalscatlab/published/Model+structures+on+Cat

Emily (May 09 2021 at 09:21):

Screenshot_20210509-062023.png

Morgan Rogers (he/him) (May 09 2021 at 09:31):

Funnily enough, on the nLab the Cauchy completion of a small category is taken to be the subcategory of the presheaf category on the retracts of representables, and then the Theorem that follows is that the Cauchy completion is a small category; it makes the same mistake that I was worrying about recently in another topic; the result would only be essentially small under that construction.

Fortunately there is a known construction of the Cauchy completion which doesn't rely on taking retracts in the category of presheaves.

Morgan Rogers (he/him) (May 09 2021 at 09:37):

Incidentally, I say "idempotent completion" in the ordinary category case because Cauchy has plenty of things named after him already, and the connection here is rather indirect. Karoubi is probably a much more reasonable name to associate with the concept.

Fawzi Hreiki (May 09 2021 at 09:49):

It makes sense to distinguish between the idempotent completion and the absolute-colimit completion. It's a non-trivial fact that they coincide for set-enriched categories.

Mike Shulman (May 09 2021 at 16:41):

I wouldn't call it a "mistake". Most category theorists don't bother worrying about the difference between small and essentially small, since we are generally happy to work with categories defined only up to equivalence.

Mike Shulman (May 09 2021 at 16:45):

Of course, the difference matters more when we start talking about things like internal categories, where the relevant axiom of choice can fail, but the nLab page is only about ordinary set-based categories.

Mike Shulman (May 09 2021 at 16:47):

Finite limits in the ambient category should suffice to construct the idempotent-completion of an internal category, having as objects the idempotents in the original internal category: the "object of idempotents" is a finite-limit construction, etc. I would expect it to then be possible to prove fairly straightforwardly that this completion is also the absolute-colimit completion, without needing to construct the latter separately.

Morgan Rogers (he/him) (May 09 2021 at 18:12):

Mike Shulman said:

I wouldn't call it a "mistake". Most category theorists don't bother worrying about the difference between small and essentially small, since we are generally happy to work with categories defined only up to equivalence.

Having seen @John Baez chide someone for describing the category of finite sets as small, the importance seems to vary between people. At any rate, I personally think it's a distinction easy enough to avoid and important in enough contexts that there isn't much excuse for conflating "small" with "essentially small".

Mike Shulman (May 09 2021 at 18:21):

Well, feel free to improve the nLab page!

Tim Campion (May 09 2021 at 18:26):

In this case, one doesn't need to really introduce the more diagrammatic construction to rectify the issue: if $C$ is small, then one can construct a small Karoubi envelope of $C$ as the full subcategory of $Psh(C)$ comprising those presheaves $F$ which are literally split subpresheaves of functors which are literally of the form $Hom_C(-,c)$. That is, each $F$ comes equipped with some $c \in C$ and natural subset inclusions $F(d) \subseteq Hom_C(d,c)$ such that yadda yadda yadda. Not that there's anything wrong with the more diagrammatic construction -- in fact, I kind of like it and think it clarifies what is going on.

Jens Hemelaer (May 09 2021 at 20:54):

Since internal categories are mentioned: in Bunge's paper "Stack completions and Morita equivalence for categories in a topos", it is shown that for internal categories $C$ and $D$ in a topos $\mathcal{S}$, the functor $[C^\mathrm{op},\mathcal{S}] \to [D^\mathrm{op},\mathcal{S}]$ is an equivalence if and only if $C \to D$ induces an equivalence $S(K(C)) \to S(K(D))$, where $K$ denotes the idempotent completion / Karoubi envelope and $S$ denotes stack completion (the stack completion construction is discussed in the paper).

Some examples are given that show (if I interpret them correctly) that only taking idempotent completions is not enough in this more general situation.

John Baez (May 09 2021 at 20:54):

Morgan Rogers (he/him) said:

Having seen John Baez chide someone for describing the category of finite sets as small, the importance seems to vary between people. At any rate, I personally think it's a distinction easy enough to avoid and important in enough contexts that there isn't much excuse for conflating "small" with "essentially small".

Chide? I thought I just told them it wasn't small. If someone doesn't know that some people distinguish between small and essentially small, they should learn about it. If they know and have decided to redefine small to mean essentially small, I wouldn't mind very much - as long as I know they're doing it!

Mike Shulman (May 10 2021 at 02:35):

Ah, yes, I forgot about stack completion. Thanks.

Mike Shulman (May 10 2021 at 02:37):

Although there's a sense in which stack completion is a different sort of operation. For instance, I believe that if by "Cauchy complete" you mean that for every absolutely weighted diagram there exists a colimit (in an unspecified sense), or equivalently you formulate some other notion of Cauchy completion in terms of anafunctors rather than functors, then I don't think the stack completion enters -- in the latter case that's because passing to anafunctors essentially has the effect of replacing everything by its stack completion anyway.

Jens Hemelaer (May 10 2021 at 07:10):

Hi @Mike Shulman , thanks for the explanation. I think in Bunge's paper, the idempotent completion/Karoubian envelope is constructed by splitting idempotents. I see now how the terminology "Cauchy completion" can mean something different in the enriched/internal case. Maybe I have to be careful with the meaning of idempotent completion or Karoubi envelope as well here.