Subject: categories: Re: query

From: [email-masked] (Marco Grandis)
Date: Thu, 19 Nov 1998 10:31:09 +0100

I have not seen Fukaya's paper.

If the problem is to formalise higher homotopies for, say, topological
spaces or chain complexes,  together with their operations, there is a
simple solution based on the path endofunctor (or, dually, the cylinder
endofunctor; or their adjunction) and its powers.

The path endofunctor  P  is constructed so that a homotopy  a: f -> g: X ->
Y  amounts to a map  a: X -> PY;  the maps  f, g  are recovered by means of
the two faces  d-, d+: PY -> Y.
(For  Top,  PY  is obviously the space  Y^[0, 1]  of paths in  Y,  with the
compact-open topology; for chain complexes  (PY)_n  =  Y_n + Y_(n+1) + Y_n,
with suitable differential.)

P  comes equipped with various natural transformations (faces, degeneracy,
connections, symmetries, concatenation...), satisfying "algebraic"
coherence axioms (a sort of "cubical comonad" with additional structure).
These transformations represent the basic structure of lower order
homotopies.

But now you have, practically for free:

- n-tuple homotopies, represented by the power endofunctor  P^n,
- n-homotopies, represented by a subfunctor  P_n  of  P^n,
- their operations, via the usual algebra of natural transformations,
- the deduced coherence relations of the latter.

Eg, a double homotopy, represented by a map  X -> P^2(Y),  has four faces
given by the four natural transformations  d-P, d+P, Pd-, Pd+: P^2 -> P;
if  k  is the concatenation of homotopies,  kP  and  Pk  are the vertical
and horizontal concatenation of double homotopies, and so on; if  k  is
associative, as is the case for chain complexes but not for spaces, so are
all higher concatenations. A double homotopy is said to be a 2-homotopy
when its "vertical" faces (say) are trivial, i.e. factor through the
degeneracy  1 -> P.
(For  Top,  P^2(Y)  is the space of maps from the standard square  [0, 1]^2
to Y;  P_2(Y)  is the subspace of those maps which are constant on the
vertical faces of the square.)

This way of deducing higher homotopies and their operations from the lower
ones can be found in:

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

M. Grandis, On the homotopy structure of strongly homotopy associative
algebras, J. Pure Appl. Algebra, 134 / 1 (1999 ?), 15-81.

Regards,   Marco Grandis

Dipartimento di Matematica
Universita' di Genova
via Dodecaneso 35
16146 GENOVA, Italy
e-mail: [email-masked]

http://pitagora.dima.unige.it/webdima/STAFF/GRANDIS/
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