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Stream: community: general

Topic: Is PresCat *-autonomous?

Alexander Gietelink Oldenziel (Aug 24 2020 at 10:36):

Hello fellow Category lovers!

The category $PresCat$ of (locally) presentable categories is symmetric monoidal closed. Given a presentable category $A$ I may define its dual $A^{\bot}:= Coc(A, Set)$ of cocontinuous functors from $A$ to $Set$. Does this make $PresCat$ a *-autonomous category?
To be precise, the $(_)^{\bot}$ functor goes from $PresCat$ to $PresCat^{op}$, and I believe it will be adjoint to itself.

The lower dimensional analogy of PresCat is SupLat, the category of suplattices. The nLab page on SupLat says that it is *-autonomous for the involution $A^{\bot}=Join(A, \Omega^{op})$ of join preserving maps to $\Omega^{op}$, the suplattice of truth values. This gives some evidence that $PresCat$ might also be *-autonomous. Unfortunately, the nLab page doesn't cite any literature.

Furthermore, for presheaf categories $A=Set^{\mathbb{C}^{op}}$, we have $A^{\bot}=Set^{\mathbb{C}}$, so $(A^{\bot})^{\bot}=A$ as required.

Attempt at a proof:
Any presentable category $A$ is a coinverter of presheaf categories.
$Set^{\mathbb{D}^{op}} \rightrightarrows Set^{\mathbb{C}^{op}} \to A$
Applying $(_)^{\bot}=Coc(_, Set)$ will send this diagram to an inverter diagram by the universal property of a colimit. An inverter diagram in the opposite category is a coinverter, so $A^{\bot}$ will be a coinverter of copresheaf categories in $PresCat^{op}$.
One would like to apply $(_)^{\bot}$ again and use the fact that it is involutive on (co)-presheaf categories to conclude it is involutive on $A$. Unfortunately, I don't think the argument quite goes through, since I can't be sure the colimit gets preserved.

Any information is appreciated!

Reid Barton (Aug 24 2020 at 10:50):

This functor kills lots of presentable categories, and definitely isn't an involution. For example, it sends any category C with a zero (= both initial and terminal) object to 0, because any cocontinuous functor F from C to Set has to send 0 to the empty set, but every object of C has a map to 0, and then F has to send it to an object with a map to the empty set, of which there's only one.

Alexander Gietelink Oldenziel (Aug 24 2020 at 12:39):

@Reid Barton
Right! Thank you, that is very helpful. I am actually mostly interested in presentable categories that are also topoi, which emphatically do not have a zero object. Would you by any chance have any idea what happens if I restrict to localizations of presheaf categories?

Todd Trimble (Aug 24 2020 at 13:09):

A better guess by way of the analogy with sup-lattices would have been $Set^{op}$ as the dualizer; unfortunately, this is not locally presentable. In fact the only categories for which both it and its opposite are locally presentable are (algebraic) sup-lattices.

The $\ast$-autonomy of sup-lattices is essentially in Joyal and Tierney's An Extension of the Galois Theory of Grothendieck, although perhaps only implicitly in the calculations (as for example in $A \otimes B \cong [A, B^{op}]^{op}$) -- they don't spell out a theorem using that language, though.

Todd Trimble (Aug 24 2020 at 13:13):

It is a consequence of Volpenka's principle that all reflective subcategories of presheaf categories are locally presentable, so I'm maybe not sure what the last question is asking.

Reid Barton (Aug 24 2020 at 13:13):

I think "localization" means "left-exact localization"

Reid Barton (Aug 24 2020 at 13:14):

(and "presentable categories that are also topoi" means "topoi" :upside_down:)

Todd Trimble (Aug 24 2020 at 13:14):

Yeah: Grothendieck toposes.

Reid Barton (Aug 24 2020 at 13:19):

I agree Set^op would have been the analogous dualizing object. One thing you could try to do is extend Pres somehow to a larger category which includes the opposites of presentable categories, and try to make that *-autonomous. @Mike Shulman and I had a related discussion here.

Reid Barton (Aug 24 2020 at 13:23):

It's not clear to me what happens for topoi, but I'm skeptical that you would get an involution.
For the category of sheaves on a topological space, for example, you'd get out the category of cosheaves, and while I know this has to be a locally presentable category, it's not at all obvious how to write down a suitable presentation (as a limit sketch, or as a coinverter of presheaf categories, say) that would let you compute the "dual" of that.
Probably the thing to do would be to write down the simplest concrete example of a topos that isn't a presheaf category and try to work out by hand a presentation of its "dual".

Mike Shulman (Aug 24 2020 at 14:56):

Reid Barton said:

One thing you could try to do is extend Pres somehow to a larger category which includes the opposites of presentable categories, and try to make that *-autonomous. Mike Shulman and I had a related discussion here.

Yes; such an extension is Chu(Cat,Set).

Peter Arndt (Aug 26 2020 at 09:13):

Hi Alexander!
Somewhere in the vicinity of your question is the Scott adjunction between (Grothendieck) topoi and accessible cats with filtered colimits. You go from topoi to accessible cats with filtered colims by taking points (i.e. by replacing the merely cocontinuous functors of your question by cocontinuous-and-finite-limit-preserving functors to Set). You go the other way round by taking filtered colimit preserving functors to Set.
If you are only interested in topoi, that might get you closer to what you want. Check out Ivan di Liberti's thesis for this: https://diliberti.github.io/Research.html
It is not clear on what exactly this duality restricts to an equivalence, but the question is discussed in Ivan's thesis (e.g. all presheaf toposes are fine, but not all toposes with enough points). I don't know anything about the 2-Chu construction but maybe you can use it to stuff both sides of this duality into a *-autonomous category.
Also Ivan's and Julia Ramos Gonzalez' Gabriel-Ulmer duality for topoi (see the same page) could be helpful with the same 2-Chu construction in mind.

Alexander Gietelink Oldenziel (Aug 29 2020 at 07:54):

Hi Peter! Good to hear from you.

The Scott adjunction is quite interesting. Like you, I am wondering about its true meaning.
Some how there are multiple notions of opposite/duals for categories. The actual opposite category and various categories of functors to Set.

(also recently I finally understood filtered colimits... compatible extensions of strategies as collections of possible interactions!)