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Stream: theory: algebraic topology

Topic: Generalized (non-abelian) Tor functors

Matteo Capucci (he/him) (Sep 02 2020 at 17:26):

Hello everyone, have people ever thought of defining Tor functors in a non-abelian context? The ideal thing would be on any nice enough monoidal category, but I can bargain.

David Michael Roberts (Sep 08 2020 at 05:26):

The only thing I can think of is that it would start from the nonabelian tensor product somehow (see eg http://web.math.unifi.it/users/fumagal/articles/McDermott.pdf). Then it seems it should be done in the general semiabelian context. I don't know about in general.

Morgan Rogers (he/him) (Sep 08 2020 at 10:32):

Strong caveat that I haven't read it, but I heard yesterday that Jean Giraud (student of Grothendieck, of Giraud axioms fame) wrote his thesis on non-abelian cohomology. There might be some relevant material in there if you can dig it up, although it's likely to have a topos-theoretic flavour to it.

Matteo Capucci (he/him) (Sep 08 2020 at 20:48):

Thanks! I will look into it.

Peter Arndt (Sep 09 2020 at 00:12):

Giraud's results have been much improved and rendered more conceptual (hence easier to digest); the nLab page gives a summary of the story and references: https://ncatlab.org/nlab/show/nonabelian+cohomology

Peter Arndt (Sep 09 2020 at 00:21):

But maybe you were asking for something different anyway. The Tor functor is the derived functor of tensoring with a fixed object in an abelian category. "Derived functor" can be understood as follows: You consider the induced functor on chain complexes in your given abelian category, given by tensoring levelwise. Chain complexes form a category with a class of weak equivalences (a "homotopical category"). Your induced functor does not respect these weak equivalences, and the derived functor is the closest one which does (formally: a certain Kan extension) -- seen like this the Tor functor is a functor on the category of chain complexes. You can get back the classical Tor functor by seeing an object of your original abelian category as a complex concentrated in degree 0, applying the derived functor, then taking cohomology.

Peter Arndt (Sep 09 2020 at 00:26):

Phrased like this you can repeat the story for a monoidal category as follows: Pass to the simplicial objects in your monoidal category (a homotopical category), look at the functor given by levelwise tensoring with an object, take the derived functor. If you want to go from objects in your original category to groups in the end, you can see an object as a constant simplicial object, apply the derived functor, then take homotopy groups of the simplicial hom-sets.

Peter Arndt (Sep 09 2020 at 00:27):

(Just a rough outline, and it doesn't tell you at all how to compute anything)

Peter Arndt (Sep 09 2020 at 00:28):

Here is an nLab page describing this kind of thing, with a comparison to classical homoogical algebra: https://ncatlab.org/nlab/show/derived+functor

Matteo Capucci (he/him) (Sep 12 2020 at 09:31):

Wow, that looks very interesting Peter. Thanks for your help.

Matteo Capucci (he/him) (Sep 12 2020 at 14:30):

Wow, I tought delving into that would have been easier lol. What I really struggle to understand now is the role of simplicial objects...? Do you have a good intuition?
I know that eventually I'll have to pick up a big textbook in homotopy theory (actually, I'm not even sure this is the right subject :joy: )

Peter Arndt (Sep 13 2020 at 12:54):

A simplicial set, at least in this context, can be seen as a blueprint for building a topological space: Roughly, For each element of the n-th set of your simplicial set you take a copy of the topological n-simplex $\{x \in \mathbb{R}^n \mid \Sigma x_i = 1, x_i \geq 0 \}$; you place all these simplices, for all n, side by side and glue two n-simplices together along their i-th face, whenever the i-th face map of the simplicial set sends them to the same (n-1)-simplex. The resulting space is called the geometric realization of the simplicial set.
Similar constructions work with many other shapes, not just simplices, e.g. one can take cubical sets, or generally any shape captured by a so-called "test category".

Peter Arndt (Sep 13 2020 at 13:00):

Homming into a simplicial object in any category gives you a simplicial Hom-Set. Now you can call a morphism in your category an equivalence if homming out of it produces a weak equivalence of simplicial set, or equivalently if the induced map of simplicial sets, after geometric realization, becomes a homotopy equivalence of topological spaces. So in the end the role of simplicial objects is to endow your category with a notion of when a morphism is a weak equivalence. Also you can embed your original category into the category of simplicial objects as the constant simplicial objects. Thus you can construct "resolutions" of your original objects by seeing them as constant simplicial object and replacing them with an equivalent (in the sense sketched above) simplicial object which is levelwise "good" (in whatever sense you want, e.g. such that tensoring with them behaves well).

Peter Arndt (Sep 13 2020 at 13:01):

An example of this general approach to resolutions is derived algebraic geometry, where for example you can take a singular scheme and replace it with a simplicial scheme that is levelwise smooth.

Peter Arndt (Sep 13 2020 at 13:03):

But this is a somewhat heavy machinery. If you have some concrete goals for your generalized Tor functors, something specific that they should do for you, maybe you can get by with less...

Ronald Brown (Sep 29 2020 at 17:27):

This is the first time I've looked at this site, so I have lot of different reactions. You can look at my web site www.groupoids.org.uk . My original vision in 1965 or so was higher dimensional versions of the fundamental group(oid) and so higher Van Kampen Theorems. The interesting aspect is that progress involved using an enhanced cubical theory (rather than simplicial) and this is expounded in the 2011 EMS Tract Nonabelian Algebraic Topology. The slogan is "algebraic inverses to subdivision". which is very convenient cubically (especially with the extra structure of connections, which allow one to discuss commutative cubes! ) . There is an expository paper in 2018 Indagationes Math.