Re: KT Chen's smooth CCC, a correction

["R Brown" <ronnie.profbrown@btinternet.com>]

Dear Bill and Colleagues, 

I would like to explain my own interest in function spaces and function objects since it has a different origin to what Bill explains and a different direction which could be of interest for comment and investigation. 

Michael Barratt suggested to me in 1960 the problem of calculating the homotopy type of the space X^Y by induction on the Postnikov system of X, in contrast to Michael's own work on Track Groups, where he used a homology decomposition of Y, and using Whitney's tube systems gave explicit description of some group extensions in examples of the Barratt(-Puppe) exact sequence. (amazing!!?) 

Now the first Postnikov invariant in its simplest form is a Sq^2 but the extension is described by a Sq^1. How did the one transform into the other? Clue: the Cartan formula for Sq^2 on a product. How did a product get into the act? Answer: the evaluation map! 

Trying to write all this down led to using a number of `exponential laws' in spaces, spaces with base point, simplicial sets, pointed simplicial sets, chain complexes, simplicial abelian groups, etc. So it was dinned into me that an exponential law depended on the product as well as the function object. So why not try the known weak product for topological spaces? Surprise, surprise, it all worked, and was part 1 of my thesis, submitted 1961, with a sketchy account of what we now call monoidal closed categories, exemplified,  but not developed in general terms. 

Subsequent work with Philip Higgins has continued to use monoidal closed categories in algebraic topology. Indeed the category of crossed complexes is cartesian closed, but the homotopy theory one wants is given by the (different) monoidal closed structure. So the category of filtered spaces is usefully enriched  over this monoidal closed category. 

My question is then: what is the potential influence of this need for monoidal closed? It clearly does not lead to topos theory as such. It does lead to the possibility of some not previously available calculations, even of nonabelian homotopical invariants, and is relevant to the study of local-to-global problems. (I first heard these words from Dick Swan in connection with sheaf theory.)  But in this work cubical sets became essential, for ease of discussing subdivision, multiple compositions,  and homotopies, and here the monoidal closed structure is crucial. Kan's initial cubical work was neglected in favour of the (convenient in many ways) cartesian closed category of simplicial sets. 

One specific problem for me was a general notion of symmetry (naively, and using buzz words (!), higher order groupoids should yield higher order notions of symmetry!). In a cartesian closed category C we have for a specific object x not only Aut(x), the isomorphisms of x, but also AUT(x), the internal group object of automorphisms of x. This has been developed for the topos of directed graphs, in John Shrimpton's thesis, but actually the unaccomplished aim was to understand Grothendieck's Teichmuller Groupoid, and his envisaged computations of this by gluing or clutching procedures, but which needed topos theory, he claimed! When I asked for any notes on this he just said nothing was written down, it was all in his mind. Baffling!  

In the monoidal case we can get only that END(x) is an internal monoid wrt tensor. But in some cases  we have a candidate for AUT(x), even if `internal group' wrt tensor makes no sense.  One example was worked out with Nick Gilbert (published 1989). In this case there is a forgetful functor U to a cartesian closed category, in this case Set, and you can make up the rest. It worked in this dimension, relevant to homotopy 3-types, but still did not lead by induction to even higher order notions of symmetry. Pity! 

My question is now: given this background, how should  we  match the beautiful ideas and insights of Bill with what seem to be some monoidal closed realities? Could this be important for geometry, and, better still, even for analysis, and dynamics? 

Ronnie


From: Bill Lawvere <wlawvere@buffalo.edu>
To: categories@mta.ca
Sent: Tuesday, 26 August, 2008 9:07:58 PM
Subject: categories: Re: KT Chen's smooth CCC, a correction

Dear Jim and colleagues,

By urging the study of the good geometrical ideas and constructions of
Chen and Froelicher, as well as of Bott, Brown, Hurewicz, Mostow, Spanier,
Steenrod, I am of course not advocating the preferential resurrection of
the particular categories they tentatively devised to contain the
constructions.

...