An old paper on 'cohesive categories'

An old paper on 'cohesive categories'

[Marco Grandis]

This is to announce a downloadable, slightly revised version of a  
1988 preprint:

       Marco Grandis, Cohesive categories and manifolds, Dip. Mat.  
Univ. Genova, Preprint 76 (1988),

published in 1990:

       –, Cohesive categories and manifolds, Ann. Mat. Pura Appl. 157  
(1990), 199-244.

The revised version of the preprint can be found at:

       http://www.dima.unige.it/~grandis/Chm.Pr1988(Rev.2013).pdf

In the present version the text has been slightly modified, to make  
it less concise and clearer. Moreover, in the proof of the "cohesive  
completion theorem" (Section 9.2) a correction has been inserted,  
that was published in:

       –, Cohesive categories and manifolds - Errata Corrige, Ann.  
Mat. Pura Appl. 179 (2001), 471-472.

With best regards

Marco Grandis

________________________

 	The purpose of this article is to treat structures, like manifolds,  
fibre bundles and foliations, that can be obtained by glueing  
together 'elementary spaces' of a given kind. These structures are  
here defined by a sort of glueing atlas, and - formally - as  
symmetric enriched categories over suitable 2-categories. Their  
morphisms are defined as 'compatible profunctors'.

 	The basis of the enrichment, called 'cohesive' and 'e-cohesive  
categories', are equipped with an order and a compatibility relation;  
they extend inverse categories and the categories of partial  
mappings. Two completion theorems, with respect to compatible joins  
and the glueing of 'manifolds', play a crucial role.
 	This matter was presented at the meeting 'Categorical Topology',  
Prague 1988 (see [G4]) and is developed in the present work.  
Applications to partially defined operators between Banach spaces  
were given in [G5], also published in 1990.
 	The links of this setting with Ehresmann's pseudogroups [E1, E2] and  
Lawvere's view of mathematical structures as enriched categories [La]  
are examined in the Introduction. Dominical categories and p- 
categories, that are also related with partial mappings and were  
previously introduced in the 1980's by Heller, Di Paola and Rosolini  
[He, Di, DH, Ro], have a natural e-cohesive structure (see Section  
3.8). Later, e-cohesive categories have also been used under the name  
of 'restriction categories' and equivalent axioms.

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