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.
    • CommentAuthorUrs
    • CommentTimeJan 4th 2010

    started entry on stabilization (in the sense of sending an (oo,1)-category to its free stable (infinity,1)-category)

    I want to eventually state more properties of the effect of stabilization on objects here.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 2nd 2013

    added in the References of stabilization:

    Discussion of the relation between stabilization of (∞,1)-categories (to stable (∞,1)-categories) and of model categories (to stable model categories) is in section 4 of

    • Marco Robalo, Noncommutative Motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes, June 2012 (arxiv:1206.3645)
    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 2nd 2013
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 30th 2018
    • (edited Dec 30th 2018)

    we had pointer to

    • Marco Robalo, section 4 of Noncommutative Motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes, June 2012 (arxiv:1206.3645)

    I have added pointer to the published version, which is:

    diff, v15, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJan 2nd 2019
    • (edited Jan 2nd 2019)

    added also pointer to

    • Marc Hoyois, section 6.1 of The six operations in equivariant motivic homotopy theory, Adv. Math. 305 (2017), 197-279 (arXiv:1509.02145)

    for making explicit stabilization with respect to a set of objects.

    diff, v16, current

    • CommentRowNumber6.
    • CommentAuthorn.mertes
    • CommentTimeMay 14th 2020
    What is the stabilization of the nerve of the category of sets? Is it the nerve of the category of abelian groups? I would like to compute this explicitly in terms of reduced excisive functors, but I'm having trouble.
    • CommentRowNumber7.
    • CommentAuthorRichard Williamson
    • CommentTimeMay 14th 2020
    • (edited May 14th 2020)

    Nice question! I haven’t thought about it quasi-categorically, but (,1)(\infty, 1)-limits in the nerve of a category should just be ordinary limits, and then the loop space functor is trivial (takes everything to the point).

    One can also observe that the nerve of the category of abelian groups cannot really be stable, since its homotopy category would then be triangulated, and any such structure (if there even exists one?) will be very artificial/trivial.

    I would say that to make the analogy precise, one can proceed as follows. One observes that the definition of a stable (,1)(\infty, 1)-category is a categorification of the notion of an abelian category. The category of abelian groups is the ’free abelian category on the category of sets’, and the (,1)(\infty, 1)-category of spectra is the ’free stable (,1)(\infty, 1)-category on the (,1)(\infty, 1)-category of spaces’. But the actual construction of these things is different, and only exists under circumstances which are more or less mutually exclusive (localisation of category of presheaves of sets vs (,1)(\infty, 1)-localisation of (,1)(\infty, 1)-category of presheaves of spaces).

    • CommentRowNumber8.
    • CommentAuthorGuest
    • CommentTimeMay 15th 2020

    The stabilization of the category of sets is zero. An object in Stab(Set) is a sequence of pointed sets (E 0,E 1,)(E_0, E_1, \ldots) together with isomorphisms Ω(E i+1)E i\Omega(E_{i+1}) \simeq E_i. But Ω(X)=*× X**\Omega(X) = \ast \times_X \ast \simeq \ast for every pointed set XX. So every object is isomorphic to (*,*,)(\ast, \ast, \ldots). The space of endomorphisms of this object is a limit of the endomorphism spaces Map Set *(*,*)*Map_{Set_*}(\ast, \ast) \simeq \ast, which is again contractible.

  1. Yes, this is what I had in mind when I wrote that the loop space functor is trivial.

    • CommentRowNumber10.
    • CommentAuthorn.mertes
    • CommentTimeMay 15th 2020

    Thank you for the answers. This clears up the fact that the existence of a nontrivial loop functor on the \infty-category of pointed spaces is something which significantly distinguishes \infty-category theory from ordinary category theory (where the category of pointed sets has trivial loop functor).

    • CommentRowNumber11.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 15th 2020

    Added an example: stabilization of Set.

    diff, v17, current

    • CommentRowNumber12.
    • CommentAuthorDELETED_USER_2018
    • CommentTimeAug 13th 2021
    • (edited Apr 11th 2023)

    [deleted]

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeNov 11th 2022

    added pointer to formalization in dependent linear homotopy type theory:

    diff, v20, current