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Hey math twitter, here's a thread about my favorite pass time, deriving things! You may (and should if you want to understand this thread) know about good old derived functors between abelian categories, well turns out they also make 'em nonabelian!
To making sense of this will require some basic infinity category stuff and simplicial methods and I will be quite sloppy with left/right bounded above/below stuff, so be warned.
First let me say what I mean by the derived category of an abelian category C. As you know, C embeds into the category of (non-positive/negative) chain complexes Ch(C) by sending an object A to the complex A[0], i.e. the one with A in degree zero and 0 elsewhere.
We can then get an infinity category D(C) by formally inverting all quasi isos which is called the derived category of C. This still comes with an induced map C -> D(C) but we no longer really understand its image sadly.
Now for any functor F: C-> E between abelian categories one can contemplate whether F refines to a functor DF :D(C) -> D(E) and if E is nice enough this can indeed be done by Kan extending along the inclusion C ->D(C).
The left resp. right Kan extension is referred to as the right resp. left derived functor of F. Now it may be a little surprising, that I'm able to state all this in such generality.
Commonly one has to worry about C having enough injectives/projectives, F being additive and left/right exact and I am deliberately not saying that here!
If we want to recover what's commonly known as the derived functors R^iF(A) or L^iF(A) for some A in C we need to look at the homotopy groups of DF(A[0]) but they will only give the right thing if we ask that F is say, additive and left/right exact.
The injective/projective stuff is a model structure on Ch(C) which gives a presentation of the derived category D(C) and allows us to actually compute stuff, but let me not get into that here.
In this generality now we've only really used that we have this category of chain complexes into which C embeds and where we know what our weak equivalences are. Now crucial to getting from here to nonabelian derived functors is the Dold-Kan correspondence.
Recall that there is an equivalence between the category of simplicial objects in C and the category of bounded chain complexes Ch(C) which sends a simplicial object A to the complex which is just A_i in degree i with differential the alternating sum of the face maps.
More is true, in fact we can take the geometric realization (in space) of any simplicial object and the homotopy groups of this space are precisely the homology groups of the associated chain complex!
Of course simplicial objects make sense in any category not just abelian ones, so we want to think of this as the correct direction for generalizing the derived category. Hence for any category C we can define its derived category D(C) to be the infinity category obtained from
the category of simplicial object by inverting the weak homotopy equivalences (defined via geometric realization). This is conceptually very clean but completely useless for actually computing the derived functor
, one can again work with model categories but another approach has been recently popularized by Scholze+Clausen, which they call the formalism of animated gadgets.
Recall that an object X in a category is called compact if Hom(X, -) commutes with filtered colimits and projective if Hom(X,-) detects isos. Denote the subcat of compact+projective objects of C as C^cp.
Now suppose C has all small colimits and is generated under small colimits by C^cp, then the animation Ani(C) is the infinity category freely generated under sifted colimits by C^cp.
As usual one can give a concrete model of this by setting Ani(C) be the subcat of Fun((C^cp)^op, Spaces) which commute with sifted colimits. The crucial point now is that in fact Ani(C) = D(C) as we constructed earlier!
This is a nontrivial fact, but just comes down to showing that Ani(Set) = Spaces. So why is this formalism useful? Well if we have some functor F :C-> D we can again define the derived functor DF:Ani(C) -> Ani(D) by Kan extending along C -> Ani(C),
but now by our concrete model Ani(C) = Fun((C^cp)^op, Spaces) we can describe DF! It suffices to give a functor which is just good old F on the compact projectives and commutes with sifted colimits and the better your functor plays with colimits the easier it is to write down.
In particular this shows that you can compute DF by simplicially resolving an object by compact projective ones and just applying F levelwise, since F commutes with geometric realizations!
By Dold Kan this also recovers the way you might ordinarily computed derived functors by resolving with honest finitely generated free/projective modules, without mentioning model categories at all. I highly encourage you to try and derive your favourite functor in this way,
maybe by convincing yourself first, that this works as intended in the abelian case. My favorite example thus far is actually the abelianization functor F:Grp -> Ab.
This is genuinely a great exercise but for those who want it I will spoil this much: The homotopy groups of DF(G) are the group homology of G! So you only need to give a functorial description of the chain complex.
