Not signed in (Sign In)

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 book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma 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 finite foundation foundations functional-analysis functor 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-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 simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft 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.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 18th 2010
    • (edited Jun 18th 2010)

    It seems like we could somehow by means of black magic construct the localization functor γ:CW 1C\gamma: C\to W^{-1}C where WW satisfies the two-out of six axiom and contains all identities by messing around with the nerve in the following way:

    Let U:CatQuivU:Cat\to Quiv be the forgetful functor sending categories to quivers, let F:QuivCatF:Quiv\to Cat denote the free category functor, and let N:CatSet ΔN:Cat\to Set_\Delta denote the nerve functor. Then it seems like the first few steps of the localization procedure can be described as taking the nerve of the free category of the underlying quiver of C. This gives us the simplicial set where the n-simplices are zig-zags of morphisms. We can look at the simplicial subset of this guy given by the restricted zig-zags (I haven’t worked out an exact characterization of this, but it seems plausible that we could, but this is the weakest part of the idea). Then we construct the pre-hom-sets between vertices A and B to be the simplices whose initial vertex is A and terminal vertex is B. This seems like it should give a category, with composition given by concatenation. Now, the other weak part of the idea is how exactly to impose the equivalence relations in a functorial way (perhaps there’s a way to do this before we turn our simplicial set back into a category?).

    Is there anything to this, or will it not work (or even if it will work, will it be stupid and contrived)?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 18th 2010
    • (edited Jun 18th 2010)

    Then it seems like the first few steps of the localization procedure can be described as taking the nerve of the free category of the underlying quiver of C. This gives us the simplicial set where the n-simplices are zig-zags of morphisms.

    Hm, the n-simplices in the nerve are not zig-zags, but just sequences of morphisms. And for the simplicial localization it is important that every zag in a zig-zag is in WW. Where is that condition in your prescription?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 18th 2010

    It’s more like this:

    you form something like the Kan fibrant replacement of the nerve, but subject to the condition that certain 1-cells are in WW.

    • CommentRowNumber4.
    • CommentAuthorHarry Gindi
    • CommentTimeJun 18th 2010
    • (edited Jun 18th 2010)

    Ah, thanks. I meant to say take the underlying undirected graph, but you’re right, it doesn’t work anyway.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 18th 2010
    • (edited Jun 18th 2010)

    A useful exercise is this:

    consider the special case where WW contains all morphisms. So we have a category CC in which every morphisms is supposed to be a weak equivalence. This is suposed to present an (,1)(\infty,1)-category that is in fact an \infty-groupoid.

    So one can form the nerve N(C)N(C) and then its Kan fibrant replacement Ex N(C)Ex^\infty N(C) to get a Kan complex.

    Alternatively, one can form the Dwyer-Kan simplicial localization L WCL_W C to obtain a Kan-complex enriched category.

    Under homotopy coherent nerve, these two constructions ought to give equivalent results

    N hcL WCEx N(C). N_{hc} L_W C \simeq Ex^\infty N(C) \,.

    This should be a good exercise to do in order to warm up for understanding the question that you are getting at.

    (I need to do that exercise myself.)