Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

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).


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

TrackBack URL for this Entry:

0 Comments & 0 Trackbacks

Post a New Comment