Subject: preprints available

From: categories <[email-masked]>
Date: Thu, 4 Dec 1997 09:47:48 -0400 (AST)

Date: Thu, 4 Dec 1997 11:54:52 +0100
From: Marco Grandis <[email-masked]>

The following preprints are now accessible as ps-files, via web of ftp:

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

ftp://www.dima.unige.it/pub/STAFF/GRANDIS

(1). "Limits in double categories",  by Marco Grandis and Robert Pare
Dbl.Dec97.ps

(2). "Weak subobjects and weak limits in categories and homotopy
categories", by M.G.
Var1.Aug97.ps

(3). "Weak subobjects and the epi-monic completion of a category", by M.G.
Var2.Dec97.ps

***

The first was announced on this mailing list, on 13 Nov 1997.
(With respect to the printed preprint, this is a slightly revised version,
containing a more detailed comparison with Bastiani-Ehresmann's "limits
relative to double categories".)

The second and third form an expanded version of a printed preprint
("Variables and weak limits in categories and homotopy categories", Dec
1996), announced on this list on 13 Dec 1996.
Abstracts for (2) and (3) are given below.

***

(2). Abstract. We introduce the notion of "variation", or  "weak
subobject", in a category, as an extension of the notion of subobject. The
dual notion is called a covariation, or weak quotient.
    Variations are important in homotopy categories, where they are well
linked to weak limits, much in the same way as, in "ordinary" categories,
subobjects are linked to limits. Thus, "homotopy variations" for a space
S,  with respect to the homotopy category  HoTop,  form a lattice  Fib(S)
of "types of fibration" over  S.
    Nevertheless, the study of weak subobjects in ordinary categories, like
abelian groups or groups, is interesting in itself and relevant to classify
variations in homotopy categories of spaces, by means of homology and
homotopy functors.   (To appear in: Cahiers Top. Geom. Diff. Categ.)

(3). Abstract. Formal properties of weak subobjects are considered. The
variations in a category  X  can be identified with the (distinguished)
subobjects in the epi-monic completion of  X,  or Freyd completion  FrX,
the free category with epi-monic factorisation system over  X,  which
extends the Freyd embedding of the stable homotopy category of spaces in an
abelian category (P. Freyd, Stable homotopy, La Jolla 1965).
    If  X  has products and weak equalisers, as  HoTop  and various other
homotopy categories,  FrX  is complete. If  X  has zero-object, weak
kernels and weak cokernels, as the homotopy category of pointed spaces,
then  FrX  is a "homological" category. Finally, if  X  is triangulated,
FrX  is abelian and the embedding  X --> FrX  is the universal homological
functor on  X,  as in the original case. These facts have consequences on
the ordered sets of variations.

Marco Grandis

Dipartimento di Matematica
Universita' di Genova
via Dodecaneso 35
16146 GENOVA, Italy
e-mail: [email-masked]
tel: +39.10.353 6805   fax: +39.10.353 6752
http://www.dima.unige.it/STAFF/GRANDIS/