Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 15th 2020

    I’ve had reason to think about locally internal categories/locally small fibrations over a base topos lately, and I was asked to what extent one can view these as categories of families of objects of a “locally small category” in a structural axiomatic set theory. To me it seems like one should take the fibration to be a stack, since given compatible families of objects on some cover, then one should definitely be able to glue them. Maybe I’m looking in the wrong places, but I don’t see any statements to this effect in the various papers on locally internal categories (in all their various guises and names), by Penon, Benabou, Paré–Schumacher the Baby Elephant, and The Elephant. I didn’t read them thoroughly, but I also didn’t see it in Mike’s Sets for category theory or Enriched indexed categories.

    Does anyone else concur, or know of a result in the literature close to this?

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 15th 2020

    Or maybe the more pointed question to ask is: why should I think of arbitrary (locally internal) fibred categories as encoding “families of objects of a large category” if the fibred category is not a stack? This is one of the motivations given for fibred/indexed categories, but they are much more general to that. I’m not saying the non-stack fibred categories are not important, just the intuition seems slightly mismatched.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJun 15th 2020

    My current opinion is that the best context in which to answer this question is HoTT. For any site CC we have a 2- (indeed an \infty-) topos of stacks (of groupoids) on CC, which therefore models HoTT (perhaps 2-truncated). Internal to that we can therefore define univalent categories, which interpret to stacks of categories on CC. If the self-indexing of CC is a stack over itself, say S CS_C, it provides an extra universe in that model of HoTT with respect to which we can define (still internally) a notion of “locally small category”, meaning a univalent category DD whose hom-sets lie in the universe S CS_C, which will interpret semantically as a locally internal category that’s a stack. The “II-indexed families” of an indexed category can then be identified with families IDI\to D in the usual sense (where II is identified with its representable sheaf).

    To your more pointed question, I would say that you can just choose the topology on CC as you like. Even if CC is a topos, you don’t have to equip it with its coherent or canonical topology; you can give it the trivial topology instead.

    One day I’m going to write all of this up…

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 15th 2020

    OK, thanks.

    To your more pointed question, I would say that you can just choose the topology on CC as you like.

    well, alright. I guess the more flexible way to say this would be to talk about the base category as a site, rather than as a bare category, and not presuppose the trivial topology.

    One day I’m going to write all of this up…

    I know the feeling (-:

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeJun 17th 2020

    It is different, but maybe it is relevant: Durov had in his thesis a chapter on homotopical algebra in sheaf setup. He is interested in cohomology of quasicoherent sheaves over generalized schemes, where rings are replaced by finitary monads. Because it is nonabelian he needs to resort to homotopical algebra. But regarding the local nature he needs to work not with model categories but with model stacks where he finds that he must weaken the analogues of axioms of Quillen for model categories when defining model stacks.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)