## March 11, 2008

#### Posted by David Corfield

Guest post by Tim Porter

I have just been looking back over Todd’s guest posts (I and II) from last autumn, and a strange link has just occurred to me. In the paper (Me with A. Bak, R. Brown and G. Minian), Global Actions, Groupoid Atlases and Applications, Journal of Homotopy and Related Structures, 1(1), 2006, pp.101 - 167, we include some examples from group presentation theory (which has a tendency to be a good testing ground for ideas for presenting logics).

Take a group $G$ and a family of subgroups (not just one as in Jim and Todd’s discussion). This family can be just ‘discrete’ or may be completed under intersections, it may not make a lot of difference. You cannot form a direct quotient by the family to get a $G$-set because you have more than one (usually)!!! Try it with a nice finite group and two subgroups. The cosets of the subgroups in the family give a covering of the set of elements of $G$ and the nerve and Vietoris complexes of that covering give simplicial complexes with a $G$-action, and hence an orbi-hedron in Haefliger’s sense (see the big book by Bridson and Haefliger – Metric Spaces of Non-Positive Curvature).

Back to the nerve of the covering, its $\pi_0$ gives the set of cosets of the subgroup generated by the family, and its $\pi_1$ measures the extra relations that have to be added to presentations of the subgroups if you are to get a presentation of the group $G$. It is possible to have simple examples in which the situation that Jim envisages i.e. a simplex with a group action, occurs.

Also if you take $S_4$, with subgroups generated by (12), (23), (34), the resulting nerve is a 2-sphere with the (permutohedral?) decomposition. Again this is distantly related to Jim Stasheff’s polytopes.

My point is that a simple 2-type corresponding to a ‘theory’ is a group with three subgroups. I am sure that if I had thought a bit more here I could come up with a way of going from a group with three subgroups to a model of a 2-type directly without going through the space. The nice thing about that paper, that we wrote originally way back in 1998 or thereabouts, was that there are neat examples with simple structure. Tony Bak’s global actions were the original motivation together with some notions of Abels and Holst.

Posted at March 11, 2008 6:38 PM UTC

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Three messages from Tim Porter

I have been looking back at some of last autumn’s posts on Concrete Groups and Axiomatic Theories and there is another strange thought that occurs to me. About what seems a century ago I worked on categorical shape theory and there were lots of uses of Guitart’s exact squares which are closely linked to the Beck-Chevalley condition. I should backtrack a bit.

The setup is that one has a functor $K : A\to B$. The original one in shape theory was polyhedra $\to$ compact metric spaces, and the shape category is obtained by thinking of the comma categories as categories of approximations to an object $X$ of $B$ by objects of $A$. This leads under certain conditions on $K$ to the procategory on $A$. There are some formal similarities with the description that Todd gave of logics, and in fact, the shape category is related to the full clone of operations on $K$ in certain circumstances and I considered shape theory as a possible descriptive logic for objects in a fairly speculative paper I wrote in the 1980s. The functor $K$ was used to generate a ‘model induced triple’ or codensity monad and the shape category was the Kleisli category of algebras for this. (Details possibly incomplete or wrong as it is a long time since I looked at this!)

If we had a better understanding of the similarities of the two theories then there might be some chance of using STRONG CATEGORICAL SHAPE THEORY to attack the problem of forms of $n$-logical theories and a Galois correspondence with $n$-groupoids. The main work on categorical strong shape is based on simplicially enriched categories, but that would be fairly easily translated into other models of weak infinity categories. (The simplicial enrichment needs to be over Kan complexes to work well so that sort of fits.) That work was done by, guess who, Michael Batanin in some of the work he did before going down under. The theory was obtained by doing the obvious thing, replacing the limits by homotopy limits etc. I was convinced that this was closely linked with Grothendieck’s dream of generalised Poincaré-Galois correspondence and the Pursuit of Stacks.

Some discussion of this formed part of my correspondence with Grothendieck back in the 1980s. (Maltsiniotis will be publishing a transcript of Pursuing Stacks with the letters of Joyal, Baues, Ronnie and myself etc. soon. I am not sure of the dates. I was trying to understand higher order descent, models for $n$-types and how it all interacted with etale homotopy/shape theory.

Tim

Have you met the following ideas for modal logics? (The treatment in Kracht’s book is where I am getting this.) Let Bal be the category of Boolean algebras and ‘hemimorphisms’ (so $\phi : A\to B$ such that $\phi(1) = 1$ and $\phi(a\cap b) = \phi(a)\cap\phi(b)$ for all $a, b$ in $A$). Sambin and Vaccaro proved that the category of $k$-modal algebras is equivalent to a category of functors from a particular category (that seems to be the free monoid on k elements considered as a category) to Bal.

This is mystifyingly close to the functors from Fin to Bool. What do you think?

Tim

Somewhere along the way you were asking about a modal model and its automorphism 2-group (if that was possible). Here is a silly very simple example. Starting with a model for $S 5$. This has $X = \{1,2,3,4\}$, and as it is $S 5$ I need an equivalence relation and I will take 1~2, 3~4. It is possible to see that there are neat automorphisms of this, some operating inside an equivalence class others exchanging equivalence classes. (The latter works because I chose the two equvalence classes to be the same size) You can write down permutation descriptions of these ( I will just need (13)(24) and (14)(23) for my example.) but I claim the automorphisms form a 2-group since thinking of the $X$ as a groupoid, the automorphisms are autoequivalences and we can look for natural transformations between them. There is an obvious natural transformation from (13)(24) between (14)(23). (In fact $X = C_2\coprod C_2$ in groupoids.)

As I said this is a simple example, but it is easy to generate up others. It is noticeable that there are these two types of ways, one permuting within an orbit, the other permuting isomorphic orbits. This is just within propositional modal logic. Steve Awodey and Kishida look at a fibration of such Kripke frames and presumably this would allow permutation within a fibre as well. In other words, one has the possibility of examples like this and considering sheaves on them may allow additional certain locally defined automorphism and 2-automorphisms/homotopies in the fibres as well. One point is that the morphisms involved may need to be ‘$p$-morphisms’ which are interpretable as ‘fibrations’ in the groupoidal sense.

The above example lives inside not the symmetric group $S_4 = 4!$ but in some larger 2-group/crossed module. What is it? Well I just remembered a paper by Chris Wensley and Murat Alp in which exactly this sort of construction is examined, but I cannot recall the outcome!!! Life can get quite interesting in that approach. In my example there are no natural transformations in each component, but if you choose an arbitrary (finite) group $G$ and form the groupoid coproduct / disjoint union of 59 (why not) copies of $G$, then not you will have something like a wreath product of $S_59$ with the $AUT(G)$ crossed module/ automorphism 2-groupoid of $G$. Fun!

That is a Kripke frame for propositional $S 5$. I did not even try to look at $S 5_2$

The problem now is to determine what ‘concrete 2-group’ this is a subobject of. Perhaps the point of this example is that the component groups are in fact symmetric groups ($C_2 = S_2!$). I have a strange feeling that some sort of universal covering construction is needed at this point. (The construction is used in Modal logic according to the books I have.)

I will stop and do some more thinking.

I hope you enjoy playing with the example. Until I understand the ‘concreteness’ the other part is obscure to me.

Best wishes,

Tim

Posted by: David Corfield on March 17, 2008 11:13 AM | Permalink | Reply to this

We hit wreath products back here.

And we had a couple of nice smallish 2-groups and were going to see how they act.

Posted by: David Corfield on March 17, 2008 3:39 PM | Permalink | Reply to this

Checking back on those earlier entries, I was a bit surprised that the 2-group Cayley theorem did not seem to be known a year ago. At least for strict 2-groups, I thought it went something like this:

Take one crossed module / strict 2-group and look at the underlying groupoid (call it $G$) The 2-group $AUT(G)$ contains a copy of the original 2-group given by composition. (This is Yoneda of course but then cut back to size.)

If the 2-group is not strict, what replaces “isomorphic to subobject of”, surely it must be “equivalent to subobject of”, and that will be the case as you can rigidify a lack 2-group, then use the strict 2-Cayley. (Or have I missed some post where this was discussed.)

About the same time as John and Alissa were looking at 2-vector spaces, my then postgrad Magnus was looking at exactly 2-vector spaces for possible actions of a crossed module. They did not quite do the job as some of you know, but the basic argument was at the level of bases, i.e. permutation reps.

I think Chris Wensley may have something on this in that paper I mentioned earlier.

Posted by: Tim Porter on March 17, 2008 10:09 PM | Permalink | Reply to this

Could you provide a link to the automorphism group (general 2-linear group?) of a 2-vector space?

thanks

jim

Posted by: jim stasheff on March 18, 2008 1:24 AM | Permalink | Reply to this

One approach is to think 2-vector space $\simeq$ chain complex of length 1 (i.e. a linear transformation of vspaces). As a chain complex, it has an automorphism chain complex of groups and that object is one candidate for the Aut of a 2-vspace. That does leave some unanswered questions as it is not a 2-group/crossed module in general but that is fairly easy to handle as one is just working with collections of matrices! Magnus looked at this in some detail in

http://www.informatics.bangor.ac.uk/public/mathematics/research/preprints/04/algtop04.html#04.05 Representations of crossed modules and cat1-groups

I think there is some worth in exploring another variant (possibly equivalent but much older). A 2-vspace *is* a simplicial vector space (taking the nerve) and the theory of automorphisms of simplicial vector spaces is essentially ancient. All the stuff on simplicial groups, etc from the 1960s and 70s then is available (and reasonably readable) for analysing them. The structure of simplicial groups as hypercrossed complexes should then give a form of most of the pairings, associators, etc, that one needs from today’s outlook.

I should also point out an OOPS on my part. Yesterday evening I read David’s last post and reacted before reading back into the autumn’s 2-Klein discussion in this blog. There are several examples there and in the paper by Baez and Crans of the Cayley theorem for 2-groups.

Posted by: Tim Porter on March 18, 2008 8:10 AM | Permalink | Reply to this