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I’m not sure where to ask this. I thought the n-forum might have the right audience. I’m not sure if this is how the n-forum is supposed to be used, and I apologize if this question is inappropriate. Anyway I’m looking for a reference for some version of the following claim:
Claim: The forgetful functor from 2-groups to categories has a (weak) left adjoint and the corresponding weak adjunction is monadic, by which I mean there is a 2-monad on categories whose algebras are precisely the 2-groups.
It seems pretty clear to me that this should be the case and I am sure this has been thought about extensively already. I just don’t know where to look. I need references. Thanks!
Certainly is the right place to ask, but all I can do is add another question. What would the free 2-group on a category look like?
The forgetful functor from 2-groups to categories factors through the 2-category of groupoids and the left adjoint factors through groupoids as well. The left adjoint from categories to groupoids is the group-completion functor. It adds inverses for every morphism. (I think this is also still a left adjoint at the 1-categorical level, but that doesn’t matter for this discussion).
So it is sufficient to describe the free 2-group generated by a groupoid. This left adjoint will send coproducts (i.e. disjoint unions) of groupoids to coproducts of 2-groups. Hence, since (up to equivalence) every groupoid is a disjoint union of groups (viewed as one object categories BG), we just need to describe the free 2-group generated by BG. For this there is an explicit construction.
2-groups are classified by their group of objects, the automorphisms of the identity, and a certain 3-cocycle called the k-invariant. The 2-group L(BG) associated to BG has the integers as its group of objects and has the abelianization of G for the automorphisms of the identity. There is no choice for the cocycle since the 3-cocycles of Z for arbitrary coefficients vanish. A quick calculation shows that this 2-group has the correct universal property: 2-Grp(LBG, H) = Fun(BG, UH), where here the “=” means “equivalent”.
Thanks. I wonder if people have looked at generators and relations definitions of 2-groups. Is every 2-group a quotient of a free 2-group by a 2-group of relations?
Yes.
Any 2-group corresponds to a simplicial group, $G$ whose Moore complex is the crossed module. Resolve the $G_0$ with a free group, pull back to that. then take a set of generators for the Moore complex of the next level (like a step-by-step construction of Michel André) then truncate above to get a ‘free 2-group’. The kernel is a 2-group of relations as well. I had a student (Ali Mutlu) who wrote up that idea in his thesis (with a lot of other things that are interesting) and together we wrote some papers on this later. The combinatorial 2-group theory of this is probably related to the polygraph techniques used by Philippe Malbos and Yves Guiraud, with whom I was working in Lyon, but our explorations were not in that direction so I cannot be sure.
In the ordinary 1-categorical case, every algebra for a monad is a coequalizer of a pair of morphisms between free algebras, which we call a presentation. In the 2-categorical case, I would expect the notion of “presentation” for an algebra over a 2-monad (such as 2-groups are, correct?) to involve a codescent object of a 2-truncated simplicial object whose vertices are free algebras. Is that the same as the simplicial group you refer to, Tim?
@Mike It sounds like it. I am not too happy about codescent objects, but there is a slight eternal problem. Sloppily one speaks of something like a quotiet but in a higher categorical context and, very much as with homotopy colimits of course, this only defines an object up to ‘homotopy type’. I already have to pinch myself to use the correct definition of a presentation of groups, i.e., generators, relations plus the isomorphism from $F(X)/N(R)$ to $G$. When it comes to crossed modules it should be important to make the equivalence explicit, but I always forget.
Of some relevance may be the fact that crossed complexes form a variety in the category of simplicial groups (after a bit of Dold-Kan-nery). The structure is quite fun. That is algebraic not homotopic nor 2-categorical. That is the problem. My feeling is that here there may be a way of looking at simplicial groups homotopically/$\infty$-categorically first and then inducing the necessary things by some ’obvious’ approach, e.g. derived functors. That should give explicit forms for the presentation machinery. I should say that the
@David: Chris Wensley has code for working with crossed modules computationally using GAP. This exploits several methods of ’presenting’ a crossed module. I am not sure if the generators and relations approach is one.
Of course, there is the idea of identities among relations for a group presentation, so think of $G$ as a crossed modules $1\to G$ now ’present’ it as a crossed module. Any thoughts as to what you get?
generators, relations plus the isomorphism from $F(X)/N(R)$ to $G$
Part of what makes the last bit so easy to ignore is that it can be characterised (up to strict equality) with very little data: a set-morphism $X \to G$ (although not every set morphism qualifies). But higher up, we need more.
I should say that the
Something lost here?
:-(
I should say that structure such as Whitehead products, etc. can be almost explicitly seen in some versions of the idea of higher presentation. That structure is algebraic in nature, but homotopy invariant.
The case of crossed complexes that I mentioned above (and where the Whitehead products are trivial) was published with Phil Ehlers: Varieties of simplicial groupoids, I, Crossed complexes, Jour. Pure Applied Algebra, 120 (1997) 221 - 233.Erratum: Jour. Pure Applied Algebra, 134 (1999) 221-233. Some of the calculations were quite tricky (hence the erratum!).
There was a draft for a second paper, and some of that material was put into chapter 4 of the Menagerie. There were some problems that I was not sure how I wanted to tackle, hence I did not produce a separate paper, nor really get to a clear (to me) conclusion.
I wouldn’t think codescent objects should make any homotopy theorist unhappy; they’re just another word for geometric realization in a 2-truncated situation. Or have I misinterpreted the source of your unhappiness?
@ Mike It is the terminology that I am unfamiliar with. I looked up the n-Lab on it and did not get a feel for it although I understood the example.
Were there answers to Chris’s questions in #1? I wonder if there are opportunities to do more useful analogising from 1-groups to 2-groups. Is there, say, a (2-)functor sending a 2-group to a kind of abelianization? In other words, are there adjoints to the forgetful functors from (2)-categories of symmetric 2-groups and braided 2-groups to ordinary 2-groups?
There is at least a relative Abelianisation in some sense, but it depends on how much you want to Abelianise. This relates to the process of passing to chains on the universal cover and crops up in RB-PJH-RS’s new book (and also much earlier in RB and PJH’s papers). There are a chain of such relative Abelianisations in higher dimensions where the Abelianising kills the top dimensional non-trivial Whitehead product (that is the idea at least). If anyone is interested I can give more details but don’t want to burden this discussion with that as I am not sure that it answers David’s point.
Tim, you’re talking about the strict dimension to the cosmic cube, where I was looking in the stable direction? Is there an adjoint to forgetting strictness?
I do not really ’grok’ the cosmic cube! I think my viewpoint is perhaps costable! It is also a strictification process, thus from a 2-crossed module the process I am thinking of gives a crossed complex of length 2 (i.e. $C_2\to C_1\to C_0$, with groups or groupoids over a fixed base.) This is adjoint to the inclusion of the category of such beasties into that of 2-crossed modules. The 2-crossed modules have lax interchangers these are killed in the crossed complexes, which are strict beasties. The functors are easily constructed and are algebraically defined, my hesitancy is just that at a categorical level I would expect them to be 2-adjoints or similar and I think they are the simplicial analogues of such things but have not checked that or exactly what that should mean.
I do not quite see how this fits into the cosmic cube, and as I have been playing with my strictification for 15 years, and have a distrust of stable homotopy theory from the days when I was about the only homotopy postgrad in the UK not doing it, I have a feeling of wanting to keep the fundamental groups as intact as possible (I love non-Abelian phenomena so view Abelian things from their perspective , :-))
By the way the directedness direction of the cube seems to have similar structures to what I describe. :-) (At least I think it does.???)
I don’t think there has been an answer to my question in #1. I think what has been clarified is that there is at least a weak adjunction between categories and 2-groups. Depending on your model of 2-groups, eg crossed modules, you can probably make this a strict adjunction, but I’m asking something which is model independent, and so I don’t really care if it is strict or not.
This weak adjunction gives rise to a 2-monad on the 2-category of categories. I’m asking if the adjunction is monadic so that the algebras for this 2-monad are precisely the 2-groups. Actually I am sure this is the case. What I really want to know is if this has appeared anywhere in the literature before. If it has, I’d like to cite it. If not, then I’ll take the time to write it up. These sorts if calculations can become quite tedious, and this seems like the sort of result that should already be in the literature. I just don’t know where to look.
Also, there are certainly weak left adjoints (i.e. “abelianizations”) to the forgetful functors from symmetric and braided 2-groups. This follows from the infinity-categorical adjoint functor theorem, which you can read about as Cor 5.5.2.9 in Jacob Lurie’s Higher Topos Theory. Presumably these are also monadic.
I was wondering when you would notice that we weren’t answering your question. :-)
Isn’t there are free monoidal category monad? I do not have references,
then it should not be that hard to add in the quasi-inverses. Perhaps the work http://www.tac.mta.ca/tac/volumes/14/6/14-06.pdf on dagger categories is useful.
I cannot recall seeing an explicit construction of what you mention however.
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