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.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 16th 2018

    This is a bit speculative; some of it might be wrong, silly, or both.

    Suppose GG is a groupoid all of whose connected components are finite, and kk is a field of characteristic zero. Consider the category SuperDG k(G)SuperDG_k(G) consisting of superspace representations (V 0,V 1)(V_0, V_1) consisting of functors from GG into finite-dimensional superspaces, equipped with GG-equivariant differentials 0:V 0V 1, 1:V 1V 0\partial_0: V_0 \to V_1, \partial_1: V_1 \to V_0. I propose to define the category of virtual representations of GG to be the localization of SuperDG k(G)SuperDG_k(G) with respect to quasi-isomorphisms (morphisms that induce isomorphisms in homology).

    This might very well dissolve into something much more trivial than it sounds, but the rough idea anyway is that an object represented by (V 0,V 1, *)(V_0, V_1, \partial_\ast) plays the role of V 0V 1V_0 - V_1. Ordinary representations embed fully and faithfully as V(V,0)V \mapsto (V, 0) with zero differentials. There is a supersymmetry functor given by degree shift modulo 22 (if I’m using the word correctly, something that exchanges bosons and fermions), playing the role of additive inversion.

    The category has a symmetric monoidal structure in the expected way. I’m hoping it decategorifies to the representation ring.

    Some old notes of mine on the Lie operad (housed at Baez’s website) seemed to use this idea implicitly, but I had never bothered to formalize it. Perhaps something like this appears in the literature?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 16th 2018

    Gomi has a model of K-theory that works just like this. See at vectorial bundle

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 16th 2018

    Ah, great; thanks! So maybe it’s not such a silly idea. :-)

    For what it’s worth, the methodological lesson I took away way back when was: when doing these categorified calculations that pertain to “virtual species”, remain for as long as you can up in SuperDG k()SuperDG_k(\mathbb{P}) before descending down to the homotopy category, taking advantage of any/all benefits such as model category structure.

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 16th 2018

    Could one look at the 2-category instead, incorporating homotopies? It’s not a priori (to me) obvious that quasi-isomorphisms and equivalences in this 2-category coincide.

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 16th 2018

    Sorry, by equivalences you mean what? By “homotopy category” I simply meant the localization.

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 17th 2018

    Equivalences as in equivalence in a 2-category.

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 17th 2018

    Well, I haven’t thought about a 2-category here, yet.

    • CommentRowNumber8.
    • CommentAuthorDylan Wilson
    • CommentTimeSep 17th 2018

    Have you seen Construction 6.2.1 in Lurie’s “Rotation invariance in algebraic K theory” paper? (The construction is much older- I think due to Quillen?- but that was just the quickest reference I could find). If you do the same thing with representations then you categorify the representation ring. I’m not sure how to relate it to your super-construction…

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 17th 2018

    Dylan, it looks related, but I don’t think it’s doing what I quite want. For example, it seems to be the core groupoid of VectVect that embeds fully and faithfully into the category, whereas (for this toy case of the terminal groupoid GG) I would want all of VectVect to embed.

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)