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March 4, 2008

Sections of Bundles and Question on Inner Homs in Comma Categories

Posted by Urs Schreiber

In the spirit of groupoidification a section of an associated bundle can be conceived in the following way:

let GG be a group, BG\mathbf{B} G the corresponding one-object groupoid, XX a space, YXY \to X a “good” regular epimorphism, Y Y^\bullet the corresponding groupoid. Then GG-bundles [g]:PX[g] : P \to X on XX are equivalent to functors g:Y BG. g : Y^\bullet \to \mathbf{B} G \,.

Now let ρ:BGVect\rho : \mathbf{B} G \to Vect be a linear representation of GG (or ρ:BGC\rho : \mathbf{B} G \to C any other representation) and denote by V// ρGV//_\rho G the corresponding action groupoid, which sits canonically in the sequence VV// ρGrBG. V \to V//_\rho G \stackrel{r}{\to} \mathbf{B} G \,. Given these two morphisms, we are lead draw the cone Y V// ρG g r BG. \array{ Y^\bullet &&&& V//_\rho G \\ & {}_g \searrow && \swarrow_r \\ && \mathbf{B} G } \,. It is easy to convince oneself that the collection of completions

Y σ V// ρG g r BG \array{ Y^\bullet &&\stackrel{\sigma}{\to}&& V//_\rho G \\ & {}_g \searrow && \swarrow_r \\ && \mathbf{B} G }

of this diagram equals the collection of sections of the bundle associated to [g][g] via ρ\rho:

Hom BG(g,r)Γ([g] ρV). Hom_{\mathbf{B} G}(g,r) \simeq \Gamma( [g]\otimes_\rho V) \,.

If we allow the functors starting at XX to be anafunctors, we can simply write

X σ V// ρG g r BG, \array{ X &&\stackrel{\sigma}{\to}&& V//_\rho G \\ & {}_g \searrow && \swarrow_r \\ && \mathbf{B} G } \,,

This formulation of sections of bundles has the great advantage that its arrow theory remains valid when we pass this from the world of Lie nn-groupoids to the world of Lie nn-algebras. That’s described in L L_\infty-associated bundles, sections and covariant derivatives.

I am claiming that in extended QFTs (= extended cobordism representations) of Σ\Sigma-model type, those induced from an nn-bundle with connection on XX, the assignment of the representation to a kk-dimensional piece Σ\Sigma of cobordism is the “collection of sections” of the transgressed nn-bundle

hom nCat(Σ,)(X V// ρG g r BG), hom_{n Cat}(\Sigma, --) \left( \array{ X &&&& V//_\rho G \\ & {}_g \searrow && \swarrow_r \\ && \mathbf{B} G } \right) \,,

hence is the collection of completions σ\sigma

hom(Σ,X) σ hom(Σ,V// ρG) hom(Σ,g) hom(Σ,r) hom(Σ,BG) \array{ hom(\Sigma,X) &&\stackrel{\sigma}{\to}&& hom(\Sigma,V//_\rho G) \\ & {}_{hom(\Sigma,g)}\searrow && \swarrow_{hom(\Sigma,r)} \\ && hom(\Sigma,\mathbf{B} G) }

where this “collection” is to be read as an inner hom

hom hom(Σ,BG)(hom(Σ,X),hom(Σ,V// ρG)). hom_{hom(\Sigma,\mathbf{B}G)}( hom(\Sigma,X), hom(\Sigma, V//_\rho G)) \,.

(See maybe Bruce’s description of this situation).


Question

What I need to better understand are those inner homs in nn-categories over BG\mathbf{B} G.

What in general can be said about inner homs in over categories of closed categories? What’s a good way to handle them?

Posted at March 4, 2008 9:23 PM UTC

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