Subject: categories: Re: more on monoidal structure of graphs

From: [email-masked] (Marco Grandis)
Date: Fri, 19 Mar 1999 16:03:40 +0100

Francois Lamarche wrote

>...But you cannot show that this
>operation is itself associative, it is only associative at a higher
>order. Ah! some kind of multiple category is involved! But so far all
>attempts to use higher-order categories to describe those weird algebras
>have failed. It only works at level one: when you collapse all proofs of
>equality between proofs of equality you get that the predicate J(x,y) is
>a groupoid structure. But when you try to go to level 2 the ordinary 2-
>or bi- or lax- or whatever- categorical machinery fails.

I wonder whether the notion of "h4-category" which I introduced for
abstract homotopy theory (see the references in my first reply) might be of
use for this.

It is a sort of "2-category vertically relaxed up to a 'shadow' of 3-cells"
(whereas bi-categories are horizontally relaxed up to invertible 2-cells).

To begin with, it is a category enriched over reflexive graphs in the sense
previously described: there is a reduced horizontal composition, of maps
with cells and cells with maps, which is strictly associative as far as
this makes sense.
There also are vertical identities, vertical involution and vertical
composition, which only satisfy the usual axioms up to an assigned
equivalence relation ("2-homotopy").
This approach is truncated at the level of usual (first-order) homotopies;
3-cells (homotopies of homotopies) only leave their 'shadow', as an
equivalence relation.

If you want to proceed further, my impression is that the relaxed
categorical machinery becomes too heavy (at least for my taste). However,
if your 2-cells (homotopies) can be represented

- by a cylinder functor  I,  as arrows  IX -> Y,
- or dually by a path functor  P,  as arrows  X -> PY,
     (or by both, forming an adjoint pair  I -| P)

as it happens for most "concrete" homotopy theories, then the powers of
such endofunctor and the clone of operations (natural transformations)
linking them, derived from the basic ones (faces, degeneracy,
concatenation, etc.), seem to give all what we need in any dimension.
This machinery of derived operations can be seen in a paper of mine

Categorically algebraic foundations for homotopical algebra, Appl. Categ.
Structures 5 (1997), 363-413,

but of course "abstract cylinder functors" go back, at least, to Kan; other
references can be found in the paper above.

Regards

Marco Grandis