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I meant it the other way. That if Z --> X has local sections, then Sing(Z) --> Sing(X) is surjective. But maybe this depends on what you mean by surjective. I'm guessing from your Serre fibration comment that you mean surjective on each level of the simplicial set. I was thinking surjective on pi_0 (which is a good notion when you view (Kan) simplicial sets as infinity-groupoids, but probably not what you want).
Actually, I decided to leave it as it was. So does T(-z) then make sense on the lower half plane? or is the convergence issue worse then that. What if z is real? Can we make sense of T(z)/T(-z) then?
I see your point. Yes, I meant for N(v) to be the quadratic form that defines the inner product on the lattice; I forgot the factor of one-half. I think there was another mistake. I'll go fix this a bit.
As far as I can tell, just looking the monoid ring associated to a monoid doesn't cut it. For example let N be the natural numbers and consider the set E= N x N with multiplication (a,b) (x,y) = (a + bx, by) This is a monoid and is an extension of (N,x) by (N, +). The monoid ring of (N, x) is a polynomial ring on infinitely many generators (one for each prime). I don't see any way to extract this extension from the homological algebra of this ring.
I took a look at this book, and it did seem to me that they form a sort of TQFT (with "exotic anomaly") out of non-semisimple MTCs. The anomaly they use is "exotic" (my word) and corresponds to a central extension of the bordism 2-category, not by a group but by a monoid (I think it was the natural numbers). This way they can get a weird TQFT where the value of, say, S<sup>1</sup> x S<sup>2</sup> is zero. (This can't happen in a usual TQFT without this anomaly since the value of S<sup>1</sup> x S<sup>2</sup> is the trace of the identity. )
Ahh. No. If I had thought about this a little longer, I wouldn't have been worried about your assumption. As you can see the exact same argument shows that dp: T<sub>c</sub>C --> T<sub>p(c)</sub>B must be injective. So the only question is if it can have dimension less than B. Again, by looking at H, you can see that the tangent space to C must have dimension at least n. btw: Your question is completely local, so the fact that you have a fiber bundle is a red herring. As long as your notion of "bundle" includes local trivializations, you can reduce to the case that E is a trivial bundle.
I think this can be found in the standard texts "Model Categories" by Mark Hovey or "Model Categories and Their Localizations" by Philip Hirschhorn. It is also probably in Quillen's original papers. I would look at Quillen's stuff first.
