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I felt there should be an entry category of presheaves. So I started one.
I’ve wondered something for a long time. Why is it called a “free cocompletion” if Psh is not idempotent. Is it idempotent if we fix a universe U and look at U-Psh(C)? Or is it perhaps more subtle? For any subcanonical grothendieck topology, there is a natural assignment of a grothendieck topology on Sh(C) (cf. SGA 4. This follows by a theorem of Giraud) such that Sh(Sh(C)) is equivalent to Sh(C) (the theorem of Giraud specifically gives a natural assignment of a grothendieck topology on Psh(C) that restricts to the canonical topology on the full reflective subcategory Sh(C)). Can we say something similar about presheaf categories? In particular, the assignment gives the canonical topology on Psh(C) for the chaotic topology on C.
I say this simply because the definition of a closure/completion operator requires idempotence.
Good question, but no, even in contexts that permit it (such as small categories enriched in a quantale), taking presheaves can never be idempotent unless the base of enrichment is equivalent to a trivial category.
In this case “free cocompletion” comes from “free” as left adjoint to the forgetful functor going from “cocomplete” categories to categories, so “free cocompletion” is a linguistic back-formation which refuses to fit the general rule cited on that page. But there is plenty of relevant commentary on this point at completion: recall that a monadic functor gives an idempotent monad if and only if it is full and faithful. In the language of stuff, structure, and property, this is precisely the case where the monadic functor is a functor which forgets “only properties”. It is explained at completion that there is intermediate notion called “property-like structure”: where the structures are described by properties (e.g., small colimit structure is unique when it exists), but the forgetful functor is not full because not every morphism in the target category preserves that structure (e.g., not every functor between cocomplete categories preserves colimits). And if the monadic functor is not full, then the monad can’t be idempotent.
The usage is too well-established to do anything about it, so nLab will not play Bourbaki here; we’ll therefore be descriptive, not prescriptive.
The usage is too well-established to do anything about it, so nLab will not play Bourbaki here; we’ll therefore be descriptive, not prescriptive.
No arguments here. As I’ve said before, I’m against all attempts to Bourbaki definitions until we Bourbaki prestack to be a pseudofunctor into Cat.
Questions at category of presheaves
I think I answered your question. What I said is definitely true for subcanonical topologies, and I believe that for nonsubcanonical topologies, it’s rather trivial.
Well, you need to restrict to left-exact reflective subcategories, and I don’t think subcanonicality has much to do with it.
By the way, this is all described at category of sheaves.
What was that funny effect that all the “$\infty$“-signs in the last line of category of presheaves had been removed? I put them back in. What happened there?
I’m afraid I feel my question has not been answered yes or no, yet. Certainly there are topologies such that the associated categories of sheaves are sub-topoi, and left-exact reflective subcategories are sheaves on some topology, but what if I have some category $C'$ equipped with an adjoint equivalence to the subtopos? Is it wrong to think of $C'$ as a subtopos as well? If it is a subtopos, then is it a Grothendieck topos (i.e. actually a category of sheaves) or just equivalent to one? (this question, I am well aware, may be completely stupid, but please treat me as ignorant of finer points of topos theory - except you Harry; you can continue to revere me ;P )
a Grothendieck topos (i.e. actually a category of sheaves) or just equivalent to one?
What an evil question! We don’t distinguish between categories of sheaves and categories equivalent to them.
A Grothendieck topos is of course a category equivalent to one that is sheaves on something.
And any sub-(elementary-topos) of a Grothendieck topos is in fact itself a Grothendieck topos, if that’s what you’re worried about.
That’s fine, but what about a sub-(elementary topos) of a presheaf topos? Perhaps I should be asking whether a presheaf topos is a Grothendieck topos (this is standard knowledge I know, but I’m in the mood for extracting answers out of people today).
@Urs: thats pretty much what I thought.
whether a presheaf topos is a Grothendieck topos
Yes, because it is sheaves for the trivial topology. More elegantly: it is evidently an exact reflective subcategory of itself.
what about a sub-(elementary topos) of a presheaf topos?
Well, as we said, this is one of the equivalent definitions of Grothendieck toposes: geometric embeddings into presheaf toposes.
Well, as we said, this is one of the equivalent definitions of Grothendieck toposes: geometric embeddings into presheaf toposes.
Yes, and this is the property that is abstracted to the (∞,1) case.
because it is sheaves for the trivial topology.
ok - that clears thing up. Thanks for the help.
I added the page functoriality of categories of presheaves because I felt it should exist, even though we do have restriction and extension of sheaves. (Also added a link from category of presheaves.)
Ought to at least link to Kan extension
I copied over some elaborations on the fundamental theorem of presheaf slices from over-topos.
It looks like some work needs to be done to harmonize various relevant nLab articles. The notion of “atomic” used in atomic category is I think stronger than the notion used by Bunge: the former means there is a small dense (full) subcategory of atomic objects, whereas the latter seems to refer to the atomic objects forming merely a generating set. This difference might account for the more technical-looking statement that appears in category of presheaves.
The first thing I would do (not now, but later unless someone beats me to it) is to track down in the literature where the density formulation appears. Assuming I didn’t just hallucinate that formulation, we can take it from there.
added pointer to:
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