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

## Re: A Strange Link

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 Stackswith 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