Manuscript

[categories <cat-dist@mta.ca>]

Date: Sun, 2 Feb 1997 17:11:05 GMT
From: MHEBERT@acs.auc.eun.eg

Here is an Abstract of a manuscript recently submitted, and available on
request. Part of it was presented at the last summer Sussex Category Meeting.

On generation and implicit partial operations in 
locally presentable categories

Michel Hebert (The American University in Cairo, Cairo, Egypt)

Abstract. In a locally a-presentable category C, seen as a category of a-ary 
S-sorted structures, we describe the subobjects (resp. the regular, strong
 subobjects) generated by a subset, first in terms of closure under specific 
types of implicit partial operations (IPO), and then in syntactic terms, 
using variations on the concept of dominion. This extends previous results 
from [Hebert, Can. J.Math 93]. The domain of definition of an IPO of arity
 s ->s  is a subfunctor V >--> U(s)  of the appropriate forgetful functor,
 and each limit-closed domain  V determines, in a natural way, a structure P(V)
 in C having as its elements of sort s the (s->s)-ary IPO's with domain V 
(This generalizes the fact that the elements of sort s of the free structure F(s) 
can be seen as the (s->s)-ary implicit total operations in C). The P(V)'s 
for subobject-closed V >--> Us with   | s | < a are precisely the 
a-generated objects (in the sense of Gabriel-Ulmer) of C. Finally we use 
IPO's to give a characterization of the so-called a-retractions, 
which parallels the known syntactic characterization of a-pure morphisms.
	The point of view adopted in this paper is the one of the algebraist or 
model-theorist wishing to use the tools of category theory without making
 radical changes in the concrete description of his/her favourite structures 
(in particular without modifying the type). A part of the paper deals with 
the translation problems which arise.