Subject: categories: Re: Monoidal structure, take II

From: [email-masked] (Marco Grandis)
Date: Thu, 18 Mar 1999 18:27:30 +0100

On Francois Lamarche's question.

If I understand correctly, the tensor product  X tensor Y  has the obvious
objects  (x, y)  and arrows of three types

(a, y): (x, y) -->  (x', y),  for  a: x -> x'  in  X,  y  in  Y,

(x, b): (x, y) -->  (x, y'),  for  x  in  X,  b: y -> y'  in  Y,

(a, b): (x, y) -->  (x', y'),  for  a  and  b  as above.

***

Remark 1.
We are thus simulating "identities" of  X  and  Y  (which are not given).
In other words, we are considering the cartesian product  X'xY'  of the
free reflexive graphs over  X  and  Y,  and taking out its identities.

Would it not be simpler to work with  REFLEXIVE GRAPHS  and their cartesian
closed structure?
In my opinion, reflexive graphs are more natural than graphs:

reflexive graph = 1-truncated simplicial set = 1-truncated cubical set

It is the topos of presheaves over a FULL subcategory of non-empty ordinals
(or cardinals as well), actually the initial segment  1,  2.

Remark 2.
Roughly speaking, a category enriched over reflexive graphs (wrt the cc
structure) is a "2-category without vertical composition". It has cells  a:
f -> g: X -> Y,  with a categorical horizontal composition; it also has
trivial cells  f -> f: X -> Y  ("vertical identities").

All this is clearly related to homotopy and its abstract settings in
"2-dimensional categories" (in some sense).
And indeed topological spaces, with continuous maps and homotopies, form a
rather obvious example.
The horizontal composition of homotopies

   a: f -> g: X -> Y,    b: h -> k: Y -> Z

is     b(a(x, t), t)               t  in  [0, 1],

which is indeed categorical.

Remark 3.
[The sequel is relevant for homotopy; I do not know if it may be relevant
in CS, but I always had the impression that abstract homotopy should be of
use there, eg with respect to deformations of processes, in some sense.]

I do not think that the latter is the "right" 2-dimensional categorical
setting for abstract homotopy (even as a starting point).
The previous horizontal composition of homotopies is rather artificial; it
is what you get from the "double homotopy"

b(a(x, t), t')       (t, t')  in  [0, 1]^2

through the diagonal  t = t'  of the square.
(The "double homotopy" itself is quite natural, as produced by the cubical
enrichment due to the cylinder functor; it is also important in homotopy.)

When the diagonal of the "standard interval" is missing (eg for chain
complexes of abelian groups), there is no canonical horizontal composition
of homotopies (working with the vertical composition, you get two of them;
the middle four interchange does not hold). But there still are canonical
horizontal compositions of "maps with homotopies" and "homotopies with
maps".

This is why I think that the basic 2-dimensional categorical setting for
abstract homotopy should only treat such "reduced horizontal composition":
arrows with cells, cells with arrows, but NOT cells with cells.
Formally, it is again a category enriched over reflexive graphs, BUT wrt
the following monoidal closed structure:

X tensor Y:
 -  the subgraph of  XxY  whose arrows are pairs  (a, b),  where  a  or  b
is an identity;

[X, Y]:
-   vertices: the graph morhisms;
-   arrows: their transformations (without "diagonals").

References:

a) The last enrichment (with further developments) has been used for
abstract homotopy in:

M. Grandis, On the categorical foundations of homological and homotopical
algebra, Cahiers Top. Geom. Diff. Categ. 33 (1992), 135-175.   [sketch]

 - , Homotopical algebra in homotopical categories, Appl. Categ. Structures
2 (1994), 351-406.

b) A notion equivalent to a category enriched in the same sense had already
been studied in:

K.H. Kamps, Ueber einige formale Eigenschaften von Faserungen und
h-Faserungen, Manuscripta Math. 3 (1970), 237-255.

c) For homotopy in groupoid-enriched categories, see Gabriel-Zisman's text
(1967).

***

With best regards

Marco Grandis

Dipartimento di Matematica
Universita' di Genova
via Dodecaneso 35
16146 GENOVA, Italy

e-mail: [email-masked]
tel: +39.010.353 6805   fax: +39.010.353 6752

http://www.dima.unige.it/STAFF/GRANDIS/
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