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Is there an established term for a presheaf on a site whose descent morphisms are not monos as for a separated presheaf, but epis ? And is there any theory for such a situation?
Is there any theory for these? Monopresheaves form a quasitopos. What do epipresheaves form? Has this ever been considered?
Where in the literature is the term “epipresheaf” used?
I learned the term from Rosenberg who generalized these notions to Q-categories in 1987. In his MPI2004-35 preprint with Kontsevich "Noncommutative spaces", they take the Q-category of nilpotent thickenings on CRing and then sheaves, monopresheaves (=separated presheaves) and epipresheaves are respectively, formally etale, formally unramified and formally smooth functors -- see page 2 in dvi, or ps. I do not know if somebody looked at the formal properties of the category of ALL epipresheaves of sets in the special case of the ordinary Grothendieck topologies.
Thanks, Zoran. Do these links to dvi and ps-files work for you? For me they don’t, but maybe it’s my machine. I’l try again later.
Yes, they work.
Cf. also Tomasz Brzeziński, Notes on formal smoothness, http://arxiv.org/abs/0710.5527
He studies some interesting case of that epi condition for Q-categories and his generalization "S-categories".
Do they consider this epi-condition for ordinary Grothendieck topologies? Presheaves whose map into their ordinary sheafification is an epi?
By the way, now that I look at this definition of Q-sheaves again:
Every locally connected and connected topos is an example of what they call an a “supplemented Q-category” over the category of sets. In fact a topos becomes a supplemented Q-category over sets precisely if it is locally connected and connected.
Then an object in the topos being “formally cosmooth” according to def. 2.3 in Brzeziński’s article says precisely that this object thinks that “pieces have points” in the sense discussed at cohesive topos.
For ordinary Grothendieck topology, there is a canonical way that the category of sieves turns into a -category (or rather -category); a light generalization is to consider more general categories of (co)cones; the idea is that while Yoneda is left exact it is not right exact, so one forces just some cones to be preserved.
While it is possible that some topoi are Q-categories (as subcategories of a presheaf category) by accident this is not intended. Q-categories are not generalizations of categories of sheaves, but rather the categories of arrows, sieves or cones…
On the other hand, the presheaves on a Q-category make a Q-category.
I still can’t open these dvi and ps files. At best I get “file corrupted”. Do you have a pdf version?
No pdf version, I can send you just ps or dvi, and they display fine on my computer. I have sent it in gmail.
While it is possible that some topoi are Q-categories (as subcategories of a presheaf category)
Just for the record: this is not what I was saying. I was saying that for a connected and locally connected topos we have a Q-category “over ”
Or what would you call this? Maybe you would say this exhibits as a Q-category under .
It might be that you are tracing something interesting. On the other hand, the sense of my warning is still the same: the adjunctions in the originally intended examples of Q-categories involve adjunctions between arrow categories, cone categories or sieve categories or alike (or presheaves on such, in which case there are two different notions of sheaves: on and in). I do not know what are the sheaves, epipresheaves or monopresheaves ON your Q-category, but they are not the object of the original topos. For sheaves IN the category, which are less fundamental, it might be.
Thanks for sending the files. Now it works. Strange.
One useful idea with Q-categories is I think the one implied by 3.7.1 in the paper: inverting tau local isomorphisms for Grothendieck topology is subsumed into this (and this is a way of thinking of sheafification). But one can take some category of cones over a nondiscrete diagram, as generalized "covers| and the localization still works...yielding sheafification in the corresponding Q-category.
Now it works
It may be that you got corrupted the first time and somehow the connection did not refresh the file later but kept sending the same from the cache.
Maybe you would say this exhibits $Set$ as a Q-category under $\mathcal{T}$.
The whole adjunction with iso unit or counit is a $Q$- or $Q^\circ$, provided it's content is about distinguishing a subcategory of objects (usually cones) which have to have a distinguished behaviour with respect to taking some categories of presheaves...
Not sure about the “in” and “on”. But lok at it this way:
if you regard a locally connected and connected topos
as a supplemented Q-category, you regard the objects of as presheaves and the sets in as sheaves in with respect to the toplogy corresponding to the geometric embedding
Equivalently this says that sets are identified with those objects of for which
is an isomorphism. This map is to be thought of projecting a cohesive space onto its set of connected components. So if instead you demand this map to be an epimorphism (an epi-presheaf) then this means that is a “thickening” of the set : it is something that fattens up the set of its connected components.
In and on is a difference between a site and a topos, or in another corner, between the category and the category of presheaves on it. Q-categories were invented to generalize SITES, not topoi. It happens of course, that in very good cases one can consider the Q-categories of presheaves and do something with them and look for sheaves inside them, but the adjunction in the definition is only used to define sheaves, it is not that one of the two categories involved in the adjunctions is intended to be the category of sheaves. Of course one can formally find such examples of Q-categories but the Q-category is not only an adjoint pair with fully faithful left adjoint but such a pair with a specific class of interpretations to do sheaf theory. The adjunction is NOT the one between sheaves and presheaves in the intended interpretation.
This does not say that your insight above is not interesting, it is just not what Q-categories are for, I think, but rather to study nontrivial mechanisms of sheafification. For example, the Gabriel localization functor (endofunctor on the category of modules over a ring) for a Gabriel filter can be obtained as a square of a more primitive endofunctor . This reminds of sheafification on sites being a square of the Artin-Grothendieck plus construction. And indeed, the two are the two special cases of the same procedure defined in terms of Q-categories.
Edit: Probably Brzeziński’s motivation is closet to your setup (at least he considers mainly the sheaf conditions IN).
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