Re: Composing modifications

[Richard Garner <rhgg2@hermes.cam.ac.uk>]

Dear Ronnie,

There are a number of "cubical" notions of bicategory or 
higher in the literature with which I am sure you are 
familiar; the earliest being Dominic Verity's double 
bicategories (which are really a particular kind of triple 
category), and later the weak (or pseudo) double categories 
studied in a series of papers by Bob Paré and Marco Grandis. 
However, none of these quite seem to fit the spirit of what 
you are asking for, and certainly do not describe the 
singular cubical complex of a space.

One approach which is indicated in the literature is based on 
Michael Batanin's theory of higher operads. In particular, 
Tom Leinster, in the refined presentation of this theory 
given in his book, discusses the possibility of using it to 
give a notion of weak cubical omega-category, though he does 
not go into any details. The basic idea is straightforward. 
On the category of cubical sets (without diagonals or 
symmetries) one has the monad T for strict cubical 
omega-categories; and one aims to define the monad for weak 
cubical omega-categories as a suitable deformation T' of this 
monad. In the language of homotopy theory, one would like T' 
to be a cofibrant replacement for T; this then ensures both 
that T' looks sufficiently like T, and also that its algebras 
are homotopy-invariant in a suitable sense.

To make this more formal, one defines a cubical operad to be 
a monad S on the category of cubical sets together with a 
cartesian monad morphism S => T. The existence of such a 
monad morphism---which I think will be unique if it 
exists---places some strong restrictions on the shapes of the 
operations out of which the monad S is built: a typical such 
operation will take an n_1 x n_2 x ... x n_k grid of 
k-dimensional hypercubes, and stick them together in some way 
to yield a single k-dimensional hypercube. This is precisely 
what one wants for a notion of (weak) cubical category.

To ensure that the monad S looks sufficiently like the monad 
T, one needs the notion of a contraction (= acyclic 
fibration). In fact, we only need to know how to say that the 
monad morphism S => T has a contraction, and this is quite 
simple. Suppose we are given a n_1 x ... x n_k grid of 
k-dimensional hypercubes as before. Suppose we are also given 
operations s_1, ..., s_2k in S which indicate how to compose 
up the (k-1)-dimensional grids of hypercubes obtained as the 
faces of the original grid. Then we require that there should 
be given an operation of S which acts on n_1 x ... x n_k 
grids, and whose actions on each of the (k-1)-dimensional 
faces are given by s_1, ..., s_2k.

There now arises a category whose objects are cubical operads 
S equipped with a contraction on S => T, and whose morphisms 
are monad maps over T which commute to the contractions in 
the obvious sense. This category is locally presentable, and 
hence has an initial object; which we declare to be the monad 
T' for weak cubical omega categories. One may in fact show 
that T' is a cofibrant replacement for T in a suitable model 
structure on cubical operads.

One may of course specialise this construction to low 
dimensions, and I have just sat down to work it out in the 
case where n=2. Rather surprisingly, it seems that there 
might not be a finite axiomatisation of the resultant 
structure; at least, one does not immediately leap off the 
page which is the usual situation. The best I have been able 
to do is the following (which is not exactly the result of 
applying the above construction but is a perfectly good 
surrogate).

Definition. A cubical bicategory is given by sets of objects, 
of vertical arrows, of horizontal arrows and of squares, 
satisfying the obvious source and target criteria, together 
with operations of identity and binary composition for 
vertical and horizontal arrows, satisfying no laws at all; 
and finally, for every n x m grid of squares (where possibly 
n or m are zero), and every way of composing up the 
horizontal and vertical boundaries using the nullary and 
binary compositions, a composite square with those 
boundaries. The coherence axioms which this structure must 
satisfy say that any two ways of composing up a diagram of 
squares must give the same answer.

I would be very interested to know if anyone can extract from 
this definition a finite collection of composition operations 
on squares, and a finite collection of equations between 
them, which together generate all the others. The key 
obstable seems to be problem that identity 1-cells are not 
strict in either direction.

Richard


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