Sections of Bundles and Question on Inner Homs in Comma Categories

Posted by Urs Schreiber

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

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

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

$\array{ Y^\bullet &&\stackrel{\sigma}{\to}&& V//_\rho G \\ & {}_g \searrow && \swarrow_r \\ && \mathbf{B} G }$

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

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

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

$\array{ X &&\stackrel{\sigma}{\to}&& V//_\rho G \\ & {}_g \searrow && \swarrow_r \\ && \mathbf{B} G } \,,$

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 $n$ -groupoids to the world of Lie $n$ -algebras. That’s described in $L_\infty$ -associated bundles, sections and covariant derivatives.

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

$hom_{n Cat}(\Sigma, --) \left( \array{ X &&&& V//_\rho G \\ & {}_g \searrow && \swarrow_r \\ && \mathbf{B} G } \right) \,,$

hence is the collection of completions $\sigma$

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

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

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

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

Question

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

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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March 4, 2008