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  1. (just a free note. I still don't know which is the right place for writing a post like this. but I promise I will learn...)

    a nice feature of extended TQFTs is that they (functorially) assign something to high codimension objects. so, for instance, a 2-extended d-dimensional TQFT with values in complex vector spaces assigns a complex number to d-dimensional closed manifolds, a vector space to (d-1)-dimensional closed manifolds, and a linear additive category to (d-2)-dimensional closed manifolds. restriction to up-to-(d-1)-dimensional manifolds produces a 1-extended (i.e. ordianry) (d-1)-dimensional TQFT, but not a complex vector space valued one. rather this (d-1)-dimensional TQFT takes its values in shifted-by-one-categorical degree vector spaces (that is, in zero degree are complex vector spaces rather than complex numbers, and so on).

    one could be more satisfied in obtaining a non-shifted TQFT, and this can nicely be obtained by shifting by one also on the cobordism side. namely, one takes the product of manifolds with the basic 1-dimensional closed manifold: S^1. this gives a (d-1) TQFT with values in the shifted d-dimensional 2-cobordism, and so by composing with the original complex vector spaces valued extended TQFT one gets a complex vector spaces valued (d-1)-dimensional TQFT.

    a way of thinking to this is to consider relative extended TQFTs. that is, all cobordisms are equipped with a map to a fixed manifold X. one clearly recovers the "absolute" situation by taking X={pt}. then the naive operation of multiplying by S^1 used above naturally leads to considering the loop space LX of X, and to saying that one goes from n-extended d-dimensional TQFT on X to (n-1)-extended (d-1)-dimensional TQFT on LX, and, composing maps to LX with the base-point-of-the-loop map LX --> X one gets an (n-1)-extended (d-1)-dimensional TQFT.

    a toy example of this is the n=1 d=1 case. 1-dimensional TQFTs on X with values in compex vector spaces encode complex K-theory on X, and 0-dimensional TQFTs on X with values in compex numbers encode ordinary cohomology of X with complex coefficients. the discussion above therefore suggests that one should naturally go from K^*(X) to H^*(X) passing through H^*(LX). this nicely fits the description of the Chern character as the trace of holonomy seen as a function on LX, as proposed e.g. by Toen and Vezzosi (arXiv:0804.1274 and subsequent papers)