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    • CommentRowNumber1.
    • CommentAuthorTim_Porter
    • CommentTimeDec 11th 2009
    I have made a comment on the groupoid cardinality page. It draws peoples attention to Quinn's notes on TQFTs which uses a notion of homotopy order to construct scaling factors in a simple TQFT. This does not seem to be mentioned in stuff that I have seen and googling gets very few hits for this. It is clearly the same as groupoid cardinality when both are defined.
  1. there’s something puzzling me about groupoid cardinality. namely, if XX is a tame groupoid and f:Xf:X\to\mathbb{C} a function, then we have a good prescription for the integral of ff over XX. since, as recalled at n-vector space, the set \mathbb{C} is identified wit te 0-category 0Vect 0Vect_\mathbb{C}, this means that we know how to integrate a functor X0Vect X\to 0Vect_\mathbb{C}.

    but what if we move from 0Vect 0Vect_\mathbb{C} to 1Vect 1Vect_\mathbb{C}? in this case we have a tentative answer given by sections of the vector bundle corresponding to f:X1Vect f:X\to 1Vect_\mathbb{C}, but this is a satisfactory answer only in case XX is a 1-type. in general, I feel the vector space of sections alone forgets about the higher homotopy of XX. this is upsetting, since the integral of a 0Vect 0Vect_\mathbb{C} remembers higher homotopy, so taking sections seems to break the pattern.

    however even in the 1-type case, things are not as neat as they could be. namely, assume XX to be a conected 1-type, so that X=BGX=\mathbf{B}G for some group GG. Then the datum of f:X1Vect f:X\to 1Vect_\mathbb{C} is the datum of a linear representation VV of GG, and taking sections corresponds to taking the subspace V GV^G of GG-invariants. but this is only the tip of the iceberg giving cohomology of GG with coefficients in the GG-module VV. so maybe one should think of the integral of a functor X1Vect X\to 1Vect_\mathbb{C} as the simplicial vector space whose associated complex computes the group cohomology of GG with coefficients in VV. have to think more on this.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 13th 2010
    • (edited May 13th 2010)

    I feel the vector space of sections alone forgets about the higher homotopy of XX. this is upsetting, since the integral of a 0Vect 0Vect_\mathbb{C} remembers higher homotopy, so taking sections seems to break the pattern.

    Yes, I know what you mean. Likely what we need is not the nn-category of nn-vector spaces, but the (,n)(\infty,n)-category of (,n)(\infty,n)-vector spaces.

    The extra \infty in there will make sure that everything always depends on all higher homotopies.

    It’s not too hard to make some guesses here:

    Set

    (,1)Vect k:=Ch (k) (\infty,1)Vect_k := Ch_\bullet(k)

    to be the (,1)(\infty,1)-category of chain complexes. This is still symmetric monoidal, so we can base the iterative definition

    (,n)Vect:=(,n1)VectMod (\infty,n)Vect := (\infty,n-1)Vect Mod

    on that.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 13th 2010
    • (edited May 13th 2010)

    And maybe better, what we should do is:

    take to be kk be a commutative simplicial ring. Then (,1)Vect:=kMod(\infty,1)Vect := k Mod its (,1)(\infty,1)-category of simplicial modules, and then iterate.

    This would also serve to “categorfy” the ground field itself. Which is now an \infty-groupoid with ring structure, i.e. really a special case of an E E_\infty-spectrum.

    Right, so probably in full generality we should go this way:

    • fix kk an E E_\infty-ring spectrum.

    • declare (,0)Vect k:=k(\infty,0)Vect_k := k – a symmetric monoidal \infty-groupoid.

    • then iterate: (,n)Vect k:=(,n1)Vect kMod(\infty,n)Vect_k := (\infty,n-1)Vect_k Mod.

  2. sounds good. that would fit what I was writing, the comment in section “Ch(Vect)-enriched categories” in 2-vector space and, most remarkably, the fact that the natural target in the cobordism hypothesis is a symmetric monoidal (,n)(\infty,n)-category. now, to have a pattern, we would like that in building the complex of sections on a (oo,1)-vector bundle, a weighted sum appears. let’s see what happens in the simplest case: a finite group representation, seen as a morphism f:BG1Vect k(,1)Vect kf:\mathbf{B}G\to 1Vect_k\hookrightarrow (\infty,1)Vect_k. let VV be the vector space associated with the unique object of BG\mathbf{B}G. then for every gg in GG we have an endomorphism ρ g:VV\rho_g:V\to V and V GV^G is the subspace of VV where the operator 1|G| gGρ g:VV\frac{1}{|G|}\sum_{g\in G}\rho_g:V\to V acts as the identity. but it’s still not clear to me which the general receipt should be.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 13th 2010
    • (edited May 13th 2010)

    Wait, once we embed all the way f:BG1Vect k(,1)Vect kf : \mathbf{B}G \to 1 Vect_k \hookrightarrow (\infty,1)Vect_k we also need to compute the limit/colimit of that functor there. This will no longer be just V GV^G, I think.

    Ahm, let me see, what will it actually be. Er. Have to think more about that.

    Notice the remarkable observation from FHLT, recorded at category algebra: if we compute the colimit of the trivial representation, but regarded as a representation on 2Vect k2 Vect_k

    BGconst 12Vect \mathbf{B}G \stackrel{const_{1}}{\to} 2 Vect

    the result, which by definition is an algebra, is the group alghebra of GG.

    • CommentRowNumber7.
    • CommentAuthordomenico_fiorenza
    • CommentTimeMay 13th 2010
    • (edited May 13th 2010)

    we also need to compute the limit/colimit of that functor there

    yes. we should obtain a complex whose H 0H^0 is V GV^G. what I was trying to obtain was a bit of this in which a weighted sum of the ρ g\rho_g’s appeared. namely, the general picture should say not only “closed top dimensional manifolds go to numbers given by weighted sums, closed codimension one manifolds go to vector spaces (or complexes) and top dimensional manifolds with boundary go to linear maps between these vector spaces”, but also something like “matrix entries of these linear maps are given by weighted sums”. stating this correctly (and proving it) should be the key to understand the interplay between sections and integrals with the groupoid measure.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 13th 2010

    Yes, right. I need to think…

  3. a good exercise could be computing the dimension of the space of sections of a 1-vector bundle over a connected 2-type (e.g. realized as a crossed module). I’ll try to work out the details of this later.

  4. still very confused about the above. need to think to something else for a few days before coming back to this.

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