4

The original motivation for extended TQFTs (as introduced by Freed, Lawrence, Baez-Dolan) is indeed giving a finer form of locality, as explained by Dmitri Pavlov. However I think there are two quicker, and arguably more physical, ways to see n-categorical structure in n-dimensional QFTs.
The first is not really about the states of a QFT (as axiomatized in ...

mp.mathematical-physics higher-category-theory quantum-field-theory cobordism topological-quantum-field-theory

3

It's just a Kan fibration with all fibres principal homogeneous $G$-spaces. Take an $\infty$-category $C$ and a functor $C\to BG,$ where BG is the classifying groupoid of an $\infty$-group (a grouplike $E_1$-space). Pulling back the overcategory projection $EG=BG_{/\ast}\to BG,$ where $\ast$ is the unique object of $BG$, gets you the Kan fibration you ...

3

The inclusion of groupoids into simplicial sets is fully faithful. Its left adjoint, $\Pi_1$ is given by left Kan extension of the functor $\Delta\to \mathcal{Gpd}$ sending the n-simplex to the contractible groupoid with objects $\{0,...,n\}$.
The entirety of the data of the homotopy type of the space $X$ is contained in its singular simplicial set, which ...

2

I have now (May 13) partitioned the answer into the blocks 1,2, as I think 2 is the simpler answer!
1 I hope the book Nonabelian Algebraic Topology will answer the question for you.
A groupoid is level one of a structure called a crossed complex which is a kind of nonabelian chain complex but also with the groupoid structure in dimensions $\leqslant 1$...

2

I'll have a go at answering your question (although higher category-theory is absolutely not my area of expertise).
The conditions for $(G,m,e,inv)$ to be a group object is stipulated by the following relations
$m\circ (e\times inv)\circ\Delta=m\circ (inv\times e)\circ\Delta=id_G$
$m\circ (m\times id_G)=m \circ (id_G\times m)$
$m\circ(e\times id_G)=m\circ(...

1

The physics motivation for extended QFTs (and not just TQFTs) comes from the locality principle (no spooky action at a distance).
The mathematical expression of locality is the descent property for extended QFTs.
See, for instance, Higher Algebraic Structures and Quantization by Daniel S. Freed.
Specifically, the assignment to X of the symmetric monoidal (∞...

mp.mathematical-physics higher-category-theory quantum-field-theory cobordism topological-quantum-field-theory

1

You cannot extend the Steenrod squares to integer coefficients (as in have a set of cohomology operations satisfying the same axioms).
Consider the tangent space $TS^2$ of the 2-sphere. With a little persistence, we can identify the Thom space of this vector bundle with the two cell complex $S^2 \cup_{2\eta} e^4$ (here $\eta$ is as usual the Hopf map ...

1

A very rough argument that can be (easily) formalized is as follows:
We have a notion of $\infty$-groupoids. These are like groupoids, but they have homotopies between morphisms, homotopies between homotopies, and so on. Every topological space presents an infinity groupoid by taking the objects to be points, morphisms to be paths, morphisms between ...

1

Fosco Loregian has surveyed the theory of co/lax co/ends in Section 7.1 of his new book, citing Bozapalides's PhD thesis and paper on the subject (see also [MO67083]).
Another reference is Chapter 2 of the Alexander Corner's PhD thesis or [arXiv:1709.01332]. These two references develop the theory of extrapseudonatural transformations, defining bicoends (...

1

A reference is Section 7.1 of Fosco Loregian's very nice book on coends, which treats co/lax co/ends. In particular, see Example 7.1.9 for a proof of the formula
$$\mathrm{Nat}_\mathrm{lax}(F,G)=\int_{A\in\mathcal{C}}\mkern-2.05em\square\mkern+1.0em\mathsf{Hom}_{\mathcal{D}}(F(A),G(A)).$$
Another reference for bicategorical coends is Chapter 2 of Alexander ...

1

The proof of the equivalence of 2-categories between the 2-category of "prestacks"(whose meaning is a pseudofunctor in the context of this question) and the 2-category of fibered categories is mentioned in the theorem 2.2.3. in the paper CATEGORICAL NOTIONS OF FIBRATION by FOSCO LOREGIAN AND EMILY RIEHL in https://arxiv.org/pdf/1806.06129.pdf.
Though my ...

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