### Infinity-Groups with Specified Composition

#### Posted by Urs Schreiber

I have a certain desire to do the *one-two-three—-infinity* thing for $n$-groups while retaining *specified composition*.

What I mean is this: there is the

$\;\; \bullet$ bundle point of view

and the

$\;\; \bullet$ section point of view

on higher categories. The first one uses models where *the existence of compositions* of $n$-morphisms is guaranteed, but not specified, while the second one *explicitly specifies* for any two higher morphisms and all possible ways to attach them the resulting composite.

In the first approach it is easy to say $\infty$-group: “Kan complex with single 0-simplex”.

While that’s easy to *say*, it is in general hard to *do* anything with (at least for me). When we want to actually do something in concrete applications, we are often better off with having a model that has specified composites. (I discussed a concrete example for that recently in Construction of Cocycles for Chern-Simons 3-Bundles.)

Well, I might be just ignorant and prejudiced. But be that as it may, it should be an interesting question in its own right to see how far we can get with handling $\infty$-groups in the second approach, where composites are specified.

There is little chance, with present technology, to handle in the second case $\infty$-groups with full weakening allowed. On the other hand, entirely strict $\infty$-groups would be easy to handle, but a bit insufficient. I want something which is as strict as possible while still capturing a “sufficient” degree of weakening.

And here is my condition on what I will consider as sufficient weakening:

The model of $\infty$-groups must be

closedin that for $G$ an $\infty$-group also $AUT(G) := Aut(\mathbf{B} G)$ is an $\infty$-group.

Because that’s what is needed for doing differential nonabelian cohomology.

Here $\mathbf{B} G$ denotes the one-object $\infty$-groupoid given by $G$.

For instance, if $G$ is an ordinary group, then $AUT(G)$ is the 2-group whose objects are the ordinary automorphisms of $G$ and whose morphisms are the inner automorphisms of $G$.

Notice that if $G$ is a strict 2-group, then $AUT(G)$ is no longer a strict 3-group – but a Gray group, meaning that $\mathbf{B} AUT(G)$ is a Gray groupoid, a groupoid enriched over the category of 2-categories equipped with the Gray tensor product. In the language of crossed group structures, this amounts to passing from crossed complexes to crossed squares.

This is described in theorem 4.3 and 5.1 of

R. Brown, I. Icen
*Homotopies and automorphisms of crossed modules of groupoids*

(arXiv).

and David Roberts and myself talk about it in our article.

So, forming automorphism $(n+1)$-groups of $n$-groups takes one from the world of strict $n$-groups into the weakened realm. But how far? Do we need fully weakened $\infty$-groups to have that $AUT(G)$ is an $\infty$-group if $G$ is? Or is there some explicit “semistrict” notion of $\infty$-group in between, rather strict, but weak enough to allow for $AUT(G)$?

Here is my proposal for how to deal with that (following a similar remark I made in a comment here):

The (rather obvious, but still noteworthy) appearance of Gray-structure on the automorphism 3-group of a strict 2-group was what alerted me when I learned of the tensor product $\otimes$ on the category $\omega Cat$ of *strict* globular infinity-categories (from the immensely helpful Todd Trimble of course, here) as described in

Sjoerd Crans
*On combinatorial models for higher dimensional homotopies*

PdD thesis, chapter 3: *Pasting schemes for the monoidal biclosed structure on $\omega-Cat$*

(ps-files).

This tensor product generalizes the Gray tensor product on strict 2-categories. In particular, it makes $(\omega Cat, \otimes)$ a closed and biclosed category.

So I began to wonder if an $(\omega Cat,\otimes)$-enriched environment might be the right context in which to answer my question.

Being Gray, the automorphism 3-group of a strict 2-group corresponds in particular to a one object $(\omega Cat,\otimes)$-enriched groupoid.

So write $\omega Grp$ for the full subcategory of $(\omega Cat,\otimes)-Cat$ whose objects are one-object groupoids, and

$\mathbf{B} : \omega Grp \hookrightarrow (\omega Cat,\otimes)-Cat$

for the obvious inclusion.

Then: isn’t it true that $\omega Grp$ is closed in that for each $\omega$-group $G$ we have that

$AUT(G) := Aut_{(\omega Cat,\otimes)-Cat}(\mathbf{B} G)$

is itself such that

$\mathbf{B} AUT(G)$

is an $(\omega Cat,\otimes)$-enriched one-object groupoid?

From looking at section 2 of

G. M. Kelly
*Basic concepts of enriched category theory*

(pdf)

I get the impression that the answer is: yes, of course. But I feel I need to think more about enrichment in general, and enrichment over $(\omega Cat,\otimes)$ in particular.

## Re: Infinity-Groups with Specified Composition

Urs wrote:

It takes a while to get used to this Kan complex stuff, but it’s worth noting that homotopy theorists find Kan complexes incredibly nice to work with. So, they’ve been developing machinery for about 4 decades to do all sort of things with them. So, there’s a lot of stuff one can just look up.

For example, say you have an $\infty$-group described as a Kan complex with just one 0-simplex. You might prefer to turn it into some sort of $\infty$-group where multiplication is a specified operation. For this, homotopy theorists usually take the simplicial loop space of your Kan complex. This is an $A_\infty$ monoid — sort of like a simplicial monoid, but where multiplication is associative only up to coherent homotopy. But, there’s a way to massage this $A_\infty$ monoid to get an honest simplicial monoid, and then do ‘group completion’ to get an honest simplicial group. And, this simplicial group is another view of your original $\infty$-groupoid!

(So, weakening the group laws is not really essential in this context, if you are willing to work with a big ‘puffed-up’ version of your $\infty$-group: every $\infty$-group is equivalent to one where associativity, the left and right unit laws and inverses hold strictly! Someday people will prove this in a globular setting; I don’t know if they have yet.)

It takes practice to use all this simplicial machinery, and it seems hard to find it all explained in one place. Our guardian angels, Peter May and James Stasheff, were responsible for a lot of the early work on this topic. May’s online book

The Geometry of Iterated Loop Spaceshas a lot of good stuff in it. HisSimplicial Objects in Algebraic Topologyis also useful. Jardine and Goerss’Simplicial Homotopy Theorygives a good modern overview.But, I’ve been studying this stuff for years (when not wasting my time on other things) and I still feel like I’m just a beginner. That’s why I’m perfectly willing to do a lot of stuff with 2-groups, just to test out certain ideas, and leave the $\infty$-group generalizations for energetic youngsters like you.

I guess my main point is that if you want to study weak $\infty$-groups, most homotopy theorists would consider it insane to use anything other than simplicial methods, since they’re so well-developed.