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I have been trying to write up an intelligible treatment of transfer in HQFTs and in the corresponding context of ’crossed algebras’. I have the feeling that there should be a categorification of this, but don’t want to ’reinvent the wheel’, so wonder if there is an already existing categorified version of finite index as in ’$G$ is a subgroup of $H$ of finite index’. I had the feeling that there should have been discussion of related topics in the café chats on n-geometries but cannot put my finger on the discussion.
There had been some discussion of sub-2-groups, but that was all about what the definiton might be in the first place and certainly did not get into question like what the analog of finite index might be.
The context is dual to that of finite sheeted covering spaces, so one thought I had was to look at higher connected coverings and to see if there was a finiteness condition on the fibre that looked reasonably well behaved.
Let me see: we do have a good notion of finite cover for 2-groups, cohesive 2-groups and generally cohesive $\infty$-groupoids. Would that help for what you are after?
There was never any summing up of the Klein 2-geometry thread. Plenty of ideas floated about. I don’t recall any finite index discussion. At one point Riemannian orbifolds were being considered as natural entities with symmetry 2-groups. Have people looked at finite sheeted covers of those?
It seemed to me to be a good question. With regard to what I have in mind, I’m not sure. The term ’transfer’ is used for various things in algebraic topology and some seem close to ideas that have been discussed a lot here and in the Lab. I will check up on finite covers, but I think the problem is more closely related to that of categorifying subobjects etc. as in the Klein 2-geometry thread. I am sort of ‘fishing’ for ideas that will suggest induction methods in TQFTs and HQFTs as they could be useful.
I guess the finest thoughts were contained in these threads:
The Stabilizer of a Subcategory
Then there was
Thanks, David :)
@Tim, I would say that a sub-2-group $i:H \hookrightarrow G$ (which I would define as a homomorphism with faithful underlying functor) is of finite index if the weak/homotopy quotient $G // H$ is equivalent to a finite groupoid. In particular, this would imply that the induced subgroups $H(a,a) \to G(ia,ia)$ are of finite index (equiv $\pi_1(H,a) \to \pi_1(G,ia)$ is a finite index subgroup for all $a\in Obj(H)$). It would also imply that the image of $\pi_0(H) \to \pi_0(G)$ is of finite index. Unless I’m completely mistaken, this should be an equivalent characterisation.
@everyone Thanks. My guess would have been what David R said (but in fact in more generality for a ’sub n-group’.) I was wanting reassurance that somewhere ’out there’, there was not a different notion with that name, and that that seemed a reasonable idea to examine (from the point of view of transfer at the algebraic level).
I was left unsure whether those threads quite delivered what I was after. The Klein geometry set up had a Lie group G acting on a space, which turned out to be the quotient of G by a subgroup H, which could be taken to fix a point in G/H. Other subgroups of G fixed other shapes, so that the quotient of G is the space of those shapes, with G acting on it. Double cosets could capture incidence relations, and so on.
So was I right to be left with the impression that an analogous story could be told with Lie 2-groups and spaces where points have internal symmetry, smooth groupoids such as homogeneous orbifolds or principle bundles?
I think you are right, but my approach is from a slightly different direction. Because a lot of the stuff I handle has a nice simplicial aspect, I tend to think of that as the combinatorics of geometry. I also like the multimodal logic idea as being partially about categorified combinatorial and geometric structures. (I also prefer the discrete to the continuous, but that is another point.)
I am currently trying to prove that transfer of HQFTs and crossed algebras (as described in Turaev’s papers and book) is left adjoint to pullback of backgrounds. It looks almost certainly the case, although a detailed proof is as yet lacking. Any ideas for generalisations, applications etc. would be very much appreciated. Also if someone knows of other similar instances on such a geometric adjoint (other than, of course, the geometric morphisms of toposes, etc.)
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