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1. • CommentRowNumber1.
   • CommentAuthorSridharRamesh
   • CommentTimeOct 5th 2011
   • (edited Oct 5th 2011)
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   Author: SridharRamesh
   Format: MarkdownItexThis is a very introductory question, but I shall venture to ask it anyway: What is the relationship between the concept of a stack/2-sheaf on a site, and the concept of a category internal to the topos of (1-)sheaves on that same site? My naive thought, not yet knowing the first thing about what stacks are other than that they are something like Category-valued sheaves, is that these would be the same. Yet, surely, if they were the same, people would just say so. So that is indeed a naive thought, yet I suppose there must be some relation nonetheless?

   This is a very introductory question, but I shall venture to ask it anyway: What is the relationship between the concept of a stack/2-sheaf on a site, and the concept of a category internal to the topos of (1-)sheaves on that same site? My naive thought, not yet knowing the first thing about what stacks are other than that they are something like Category-valued sheaves, is that these would be the same. Yet, surely, if they were the same, people would just say so. So that is indeed a naive thought, yet I suppose there must be some relation nonetheless?

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeOct 5th 2011
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   Author: DavidRoberts
   Format: MarkdownItexAha, my own speciality! The following rough outline is from work in progress, so grains of salt need to be applied in strategic places. Consider not the topos of sheaves on the site, but the site itself. The easiest relation is between _geometric_ stacks and internal categories. Consider the following case study: Consider groupoids in algebraic spaces, such that source and target maps are smooth morphisms. There is a bicategory with these for objects and internal [[anafunctors]] for 1-arrows. This bicategory is equivalent to the 2-category of stacks on schemes which admit a representable smooth surjection from an algebraic space (aka algebraic stacks). This was proved by Pronk in 1996, as well as similar results for differentiable and topological stacks (replacing groupoids in algebraic spaces by Lie and topological groupoids resp.) Well, not the bit about anafunctors, but that the 2-category of algebraic stack is a localisation of the 2-category of internal categories. And I proved that the bicategory of anafunctors is also a localisation at the same 1-arrows. The same is true for other sites, and other sorts of internal categories. Essentially one considers stacks of groupoids with a representable surjection which arises 'from a pretopology on the site' (like the smooth surjection above), and these correspond to internal groupoids with source/target from that pretopology. Similar results are true for geometric stacks of categories, but these seem to be far and few between. I have a question on this on MathOverflow, if you care to glance at it (called something like 'What about stacks of categories')

   Aha, my own speciality!

   The following rough outline is from work in progress, so grains of salt need to be applied in strategic places. Consider not the topos of sheaves on the site, but the site itself.

   The easiest relation is between geometric stacks and internal categories. Consider the following case study:

   Consider groupoids in algebraic spaces, such that source and target maps are smooth morphisms. There is a bicategory with these for objects and internal anafunctors for 1-arrows. This bicategory is equivalent to the 2-category of stacks on schemes which admit a representable smooth surjection from an algebraic space (aka algebraic stacks). This was proved by Pronk in 1996, as well as similar results for differentiable and topological stacks (replacing groupoids in algebraic spaces by Lie and topological groupoids resp.) Well, not the bit about anafunctors, but that the 2-category of algebraic stack is a localisation of the 2-category of internal categories. And I proved that the bicategory of anafunctors is also a localisation at the same 1-arrows.

   The same is true for other sites, and other sorts of internal categories. Essentially one considers stacks of groupoids with a representable surjection which arises ’from a pretopology on the site’ (like the smooth surjection above), and these correspond to internal groupoids with source/target from that pretopology. Similar results are true for geometric stacks of categories, but these seem to be far and few between. I have a question on this on MathOverflow, if you care to glance at it (called something like ’What about stacks of categories’)

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeOct 5th 2011
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI would say that the difference between the two is in whether the gluing conditions have been "categorified". In a category internal to 1-sheaves, a family of objects over elements of a cover which agree *on the nose* on overlaps can be glued together to form a unique global object. But in a 2-sheaf, a family of objects over elements of a cover which agree *up to specified coherent isomorphisms* on overlaps and triple overlaps can be glued together to form a global object which is unique up to unique isomorphism. (Also, a 2-sheaf need only be a pesudo functor from the site to categories, whereas a category internal to 1-sheaves is necessarily a strict functor. There is also a gluing condition on morphisms, but this is the same in both cases.) In general, neither of these conditions implies the other. However, it is a fact that every 2-sheaf is equivalent to one which is also a category internal to 1-sheaves, and also that every category internal to 1-sheaves is "weakly equivalent" to one which is also a 2-sheaf. So up to equivalence, the two are interchangeable. (My understanding is that this fails to be true once we get up to ∞-sheaves: not every $(\infty,1)$-sheaf is equivalent to an $\infty$-groupoid internal to 1-sheaves, because not every $(\infty,1)$-topos is "hypercomplete".) I think the relationship to David's comment is that when you consider the topos of 1-sheaves as a site of definition for itself (which is always possible, and is the right thing to do if you are interested in internal categories in 1-sheaves rather than internal categories in some original site), then "geometricity" of a stack is basically automatic. The "localization" in question is at the class of "weak equivalences" between internal categories that I referred to above.

   I would say that the difference between the two is in whether the gluing conditions have been “categorified”. In a category internal to 1-sheaves, a family of objects over elements of a cover which agree on the nose on overlaps can be glued together to form a unique global object. But in a 2-sheaf, a family of objects over elements of a cover which agree up to specified coherent isomorphisms on overlaps and triple overlaps can be glued together to form a global object which is unique up to unique isomorphism. (Also, a 2-sheaf need only be a pesudo functor from the site to categories, whereas a category internal to 1-sheaves is necessarily a strict functor. There is also a gluing condition on morphisms, but this is the same in both cases.)

   In general, neither of these conditions implies the other. However, it is a fact that every 2-sheaf is equivalent to one which is also a category internal to 1-sheaves, and also that every category internal to 1-sheaves is “weakly equivalent” to one which is also a 2-sheaf. So up to equivalence, the two are interchangeable. (My understanding is that this fails to be true once we get up to ∞-sheaves: not every $(\infty,1)$-sheaf is equivalent to an $\infty$-groupoid internal to 1-sheaves, because not every $(\infty,1)$-topos is “hypercomplete”.)

   I think the relationship to David’s comment is that when you consider the topos of 1-sheaves as a site of definition for itself (which is always possible, and is the right thing to do if you are interested in internal categories in 1-sheaves rather than internal categories in some original site), then “geometricity” of a stack is basically automatic. The “localization” in question is at the class of “weak equivalences” between internal categories that I referred to above.