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: Presheaves of Presheaves

Fabrizio Genovese (Apr 11 2020 at 12:22):

I am in the weird situation where I have to consider the category $[[C^{op}, Set]^{op},Set]$, that is, a category of presheaves where its base category is itself a category of presheaves. Are there ways to link the internal logic of the base presheaf category $[C^{op},Set]$ to the internal logic of the presheaf category $[[C^{op},Set]^{op}]$? If yes, and more generally, what can be said if I have instead the category $[[C^{op},D]^{op},E]$? What properties $D$ and $E$ need to have to make this category nice enough to manipulate?
I have (a bit of) experience with the internal logic of elementary topoi but this is well above my skills at the moment, so everything, even just references, would be very much appreciated!

Fabrizio Genovese (Apr 11 2020 at 12:24):

Finally, is there an "obvious" way to generalize whatever technique we can apply here to towers of presheaves of arbitrary length, say $[\dots[[C^{op}, Set]^{op},Set]^{op} \dots Set]$?

Matteo Capucci (he/him) (Apr 11 2020 at 12:45):

Here Joyal and Anel outline a 'Stone duality' for topoi, i.e. an adjunction mediated in both ways by taking homs into a 'schizophrenic object'. The main example is that a frame $H$ is turned into a space $\rm{Spec}(H) := \rm{Hom}_{\bf{Frm}}(H, 2)$ and a topological space $X$ is turned into a frame $\mathcal{O}(X) = \rm{Hom}_{\bf{Top}}(X, 2)$. If you start from a sober topological space and do this twice, you get back from where you started.

Matteo Capucci (he/him) (Apr 11 2020 at 12:46):

The same happens between topoi and 'logoi', and the duality is mediates exactly by taking homs with $\bf{Set}$

Matteo Capucci (he/him) (Apr 11 2020 at 12:50):

So I guess taking presheaf twice should, at least for nice enough $\bf{C}$ s, be the same as doing nothing.

Fabrizio Genovese (Apr 11 2020 at 13:13):

This is the most "WAIT WAT?!" reaction I've had in a long time xD

Oscar Cunningham (Apr 11 2020 at 13:16):

I think what we get is that sheaves on the topos $[C^\mathrm{op},\mathbf{Set}]$ are precisely $[C^\mathrm{op},\mathbf{Set}]$. But there are presumably more presheaves.

Oscar Cunningham (Apr 11 2020 at 13:19):

Although it looks like Matteo is saying that the presheaves are $C$ itself. Surely that can't be right, $C$ needn't be complete.

Matteo Capucci (he/him) (Apr 11 2020 at 13:20):

Yeah I said if "$\bf{C}$ is nice enough"

Matteo Capucci (he/him) (Apr 11 2020 at 13:20):

See the analogy with topological spaces

Oscar Cunningham (Apr 11 2020 at 13:20):

Ah okay

Matteo Capucci (he/him) (Apr 11 2020 at 13:21):

Mmmh on the other hand it might have to be too nice... like "be a Grothendieck topos" nice os something like that XD

Matteo Capucci (he/him) (Apr 11 2020 at 13:24):

Another way to see the phenomenon: presheaves on $\bf{C}$ is the free cocompleton of $\bf{C}$. Then if you do this twice at least you get some canonical way to go back to $[\bf{C}^{\rm{op}}, \bf{Set}]$, by taking "formal colimits" in $[[\bf{C}^{\rm{op}}, \bf{Set}]^{\rm{op}}, \bf{Set}]$ to its true colimit.

Nathanael Arkor (Apr 11 2020 at 14:07):

Another way to see the phenomenon: presheaves on $\bf{C}$ is the free cocompleton of $\bf{C}$

this is generally true only when $\bf C$ is small

Nathanael Arkor (Apr 11 2020 at 14:07):

there's not a reasonable (naïve) way to iterate presheaves because of size issues

Nathanael Arkor (Apr 11 2020 at 14:07):

taking the free cocompletion takes you from $\bf{Cat} \to \bf{CAT}$

Nathanael Arkor (Apr 11 2020 at 14:08):

I would ask yourself whether this construction is really what you're after, and whether there's not some manipulation you can do to make sure you're not getting into these size problems

Fabrizio Genovese (Apr 11 2020 at 14:20):

Yes, all the categories involved in my question are small. I forgot to make this explicit, sorry :slight_smile:

Nathanael Arkor (Apr 11 2020 at 14:20):

the base category $\bf C$ might be small, but as soon as you take presheaves, you have a large category

Nathanael Arkor (Apr 11 2020 at 14:21):

so you run into size issues as soon as you try to take presheaves on this presheaf category

Fabrizio Genovese (Apr 11 2020 at 14:25):

Hmmmm. Yes, taking $Set$ may be too general, because in the end I am only interested in very specific functors from $C^{op}$ to $Set$

Fabrizio Genovese (Apr 11 2020 at 14:26):

But yes, this is a bit of a can of worms in any case. It may make sense to find a different way for me to say what I wanna say before diving straight into this

Morgan Rogers (he/him) (Apr 11 2020 at 14:29):

There are various ways to circumvent size issues here: one is to take the free cocompletion of $C$ under a restricted collection of colimits, another is to restrict yourself to functors to $Set$ with sufficiently nice properties. Without knowing the specifics of what you're looking at, it might be difficult to say which is preferable in your situation :wink:

Fabrizio Genovese (Apr 11 2020 at 14:31):

I'll try to think a bit more about what I am doing and I'll come back to you with more details then. I was basically hoping that this thing as I naively put it had already been studied, but this not being the case, it's probably wiser for me to reformulate my ideas before refining the question :slight_smile:

Joachim Kock (Apr 11 2020 at 14:34):

Fabrizio Genovese said:

I am in the weird situation where I have to consider the category $[[C^{op}, Set]^{op},Set]$, that is, a category of presheaves where its base category is itself a category of presheaves.

PUZZLE: Make $C \mapsto [C^{op}, Set]$ into a monad, with the Yoneda embedding as unit.

Morgan Rogers (he/him) (Apr 11 2020 at 14:49):

(spoiler: I have already pointed at one approach to the above problem in another thread, I can't remember exactly where. See:
Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures by Fiore, Gambino, Hyland and Winskel)

Nathanael Arkor (Apr 11 2020 at 15:00):

I think that gives away the answer a little too readily, @Morgan Rogers.

Fabrizio Genovese (Apr 11 2020 at 15:15):

I'm actually going through the paper right now and I think I'll need several weeks to digest it, I'm definitely not familiar enough with a lot of the concepts it covers. You can be assured the answer won't be readily given away for me! :slight_smile:

Fabrizio Genovese (Apr 11 2020 at 15:17):

(While skimming it I thought "Oh, so that's what the grown ups do...")

Morgan Rogers (he/him) (Apr 11 2020 at 15:24):

:information_desk_person: pointing out the existence of a solution without explaining what it is shouldn't be too much of a giveaway, although it does sound like Fabrizio went straight there without much delay :yum:

Morgan Rogers (he/him) (Apr 11 2020 at 15:25):

Also, it's not the only solution! cf my suggestion of restricting the class of colimits one is completing wrt, above.

Fabrizio Genovese (Apr 11 2020 at 15:39):

Well, the point is that I'm trying to model something, and that's my goal. So for now I'm just frantically vetting the possible ways to do this. Many things I stumble upon seem interesting, but in the end I have to make sure I don't deviate too much from my main goal, hence the reason why I went straight there without delay!

Matteo Capucci (he/him) (Apr 11 2020 at 16:07):

Joachim Kock said:

Fabrizio Genovese said:

I am in the weird situation where I have to consider the category $[[C^{op}, Set]^{op},Set]$, that is, a category of presheaves where its base category is itself a category of presheaves.

PUZZLE: Make $C \mapsto [C^{op}, Set]$ into a monad, with the Yoneda embedding as unit.

As pointed out by @Nathanael Arkor, I thought it didn't work for size issues, then I see this :laughing: indeed, this was what I had in mind when I wrote about iterating the free cocompletion. Moreover, Yoneda works for locally small cats, not just small. And taking presheaves doesn't change local smallness, right?
Therefore one should have $\rm{Psh}: \bf{CAT} \to \bf{CAT}$, with $y$ Yoneda embedding as monad and "turning formal colimits into real colimits" as multiplication. Did I guess right?

Thomas Read (Apr 11 2020 at 16:21):

If my understanding is correct then @Joachim Kock 's puzzle is a bit of a trick question - as mentioned the size issues mean that $C \mapsto [C^\text{op}, \text{Set}]$ is not actually an endofunctor, so not a monad in the usual sense. But it certainly "looks like it should be a monad" or something, and there are ways to make that idea much more formal.

John Baez (Apr 11 2020 at 16:39):

By the way, there is a monad on $\mathbf{CAT}$ that takes the cocompletion of a large category and produces another large category, but it's not taking the presheaf category. Steve Lack has explained how this works.

John Baez (Apr 11 2020 at 16:39):

(Here "cocomplete" means having all small colimits.)

Joachim Kock (Apr 11 2020 at 17:20):

Nathanael had already pointed out the size issue. The puzzle was to make it work. The solution (which is the one John refers to) is to consider only small presheaves: those that arise by left Kan extension from a small subcategory of C. The monad structure (as suggested by Matteo) consists in writing presheaves as colimits of representables and flatten colimits of colimits to colimits. For small presheaves, the colimits involved are small.

sarahzrf (Apr 11 2020 at 19:36):

Matteo Capucci said:

Joachim Kock said:

Fabrizio Genovese said:

I am in the weird situation where I have to consider the category $[[C^{op}, Set]^{op},Set]$, that is, a category of presheaves where its base category is itself a category of presheaves.

PUZZLE: Make $C \mapsto [C^{op}, Set]$ into a monad, with the Yoneda embedding as unit.

As pointed out by Nathanael Arkor, I thought it didn't work for size issues, then I see this :laughing: indeed, this was what I had in mind when I wrote about iterating the free cocompletion. Moreover, Yoneda works for locally small cats, not just small. And taking presheaves doesn't change local smallness, right?
Therefore one should have $\rm{Psh}: \bf{CAT} \to \bf{CAT}$, with $y$ Yoneda embedding as monad and "turning formal colimits into real colimits" as multiplication. Did I guess right?

read this and tell me what it says please https://arxiv.org/abs/1612.03678

sarahzrf (Apr 11 2020 at 19:36):

i started and never got very far

sarahzrf (Apr 11 2020 at 20:23):

one more reason why i like depleted category theory is that (0, 1)-PSh really is a legitimate monad (well, maybe a (1, 2)-monad) on (0, 1)-Cat

sarahzrf (Apr 11 2020 at 20:23):

this goes hand in hand with the full adjoint functor theorem; with complete<->cocomplete; with total<->cocomplete; etc

Fabrizio Genovese (Apr 11 2020 at 20:29):

sarahzrf said:

Matteo Capucci said:

Joachim Kock said:

Fabrizio Genovese said:

I am in the weird situation where I have to consider the category $[[C^{op}, Set]^{op},Set]$, that is, a category of presheaves where its base category is itself a category of presheaves.

PUZZLE: Make $C \mapsto [C^{op}, Set]$ into a monad, with the Yoneda embedding as unit.

As pointed out by Nathanael Arkor, I thought it didn't work for size issues, then I see this :laughing: indeed, this was what I had in mind when I wrote about iterating the free cocompletion. Moreover, Yoneda works for locally small cats, not just small. And taking presheaves doesn't change local smallness, right?
Therefore one should have $\rm{Psh}: \bf{CAT} \to \bf{CAT}$, with $y$ Yoneda embedding as monad and "turning formal colimits into real colimits" as multiplication. Did I guess right?

read this and tell me what it says please https://arxiv.org/abs/1612.03678

In general when I see Marcelo Fiore among the authors I know I'm dealing with some serious stuff. It's a guarantee! :slight_smile:

sarahzrf (Apr 11 2020 at 20:37):

btw, the yoneda lemma itself is nearly the statement of one of the pseudomonad laws :D

sarahzrf (Apr 11 2020 at 20:38):

...well, i worked that out once, but i think i had the wrong μ when i did, so maybe it isn't quite... hrm

Nathanael Arkor (Apr 11 2020 at 20:43):

@sarahzrf: one of the goals of the paper you linked is to show that the fact that free cocompletion is "morally" a 2-monad (and the Yoneda embedding is the unit) can actually be formalised: free cocompletion is a relative pseudomonad (i.e. a weak 2-monad that is not an endofunctor).

sarahzrf (Apr 11 2020 at 20:44):

oh yeah no i do know that much

sarahzrf (Apr 11 2020 at 20:44):

i was just questioning whether the yoneda lemma played the role in question

sarahzrf (Apr 11 2020 at 20:44):

also, is that lax 2-monad or weak 2-monad?

Nathanael Arkor (Apr 11 2020 at 20:45):

Weak, yes, sorry.

Matteo Capucci (he/him) (Apr 12 2020 at 10:12):

Joachim Kock said:

Nathanael had already pointed out the size issue. The puzzle was to make it work. The solution (which is the one John refers to) is to consider only small presheaves: those that arise by left Kan extension from a small subcategory of C. The monad structure (as suggested by Matteo) consists in writing presheaves as colimits of representables and flatten colimits of colimits to colimits. For small presheaves, the colimits involved are small.

Cool! This means that "presheaves are the free cocompletion of a category" is true only for small cats, right? Otherwise you need only these _small_ presheaves

Joe Moeller (Apr 12 2020 at 16:25):

Right, and you're talking about the small cocompletion here too.

Matteo Capucci (he/him) (Apr 12 2020 at 20:39):

yes

John Baez (Apr 12 2020 at 21:26):

Yes. The following paper explain this stuff rather briskly for enriched categories (a generalization of the ordinary case):

• Brian Day and Steve Lack, Limits of small functors.

I had trouble extracting all the information I wanted from this paper, so I asked Steve a question:

By CAT I will mean the category of locally small categories.

By CoComCAT I will mean the category of locally small cocomplete categories.

Is there a (pseudo)adjunction between CAT and CoComCAT, where the right adjoint is the obvious forgetful functor?

He answered:

Yes, that’s right. And the left biadjoint is precisely the thing we discuss. We may not have mentioned local smallness: in the enriched context this is automatic: a Set-category is a category whose homs lie in Set.