inverse semigroups are closely related to pseudogroups and so are
Dear Ronnie, This is about your remarks: part of Charles Ehresmann local-to-global view
Is this a rash view that one should look more at inverse categories?
I worked on two subjects where inverse categories are important, homological algebra and a categorical approach to manifolds - this second point, of course, agrees with the first citation above, from your posting. 1. While studying universal models of spectral sequences, in the 70's-80s, I studied inverse categories, as well as a more general notion, orthodox categories (the analogue of orthodox semigroups, ie regular semigroups where idempotents are closed under multiplication). The link comes from the fact that the main theories which produce spectral sequences (the filtered complex, the double complex, Massey's exact couple) are 'distributive', in the sense that their classifying Puppe-exact category E has distributive lattices of subobjects (notice that this aspect would be lost if one uses the classifying *abelian* category, since the existence of products clashes with distributivity). Now, the distributivity of E is equivalent to the fact that the (involutive) category of relations of E is orthodox (i.e., idempotent endorelations are closed under composition), and also to the fact that canonical isomorphisms between subquotients in E are closed under composition (a crucial fact when working in homological algebra). Moreover, as an orthodox category, RelE has a canonical quotient which is an inverse category; in the latter, the study of subquotients of E is easier. (Notice that subquotients in E amount to subobjects in RelE, as well as in its inverse quotient. The first fact was already used in Mac Lane's 'Homology' to define induced homomorphisms; in an abelian context, of course.) All this is developed in various papers, but a synopsis can be found in [1]. Paper [2] shows that every small inverse category can be embedded in the standard example of inverse categories: sets and partial bijections (ie, bijections between subsets of domain and codomain, composed as relations). 2. Inverse categories are of interest when viewing manifolds as enriched categories, in the line of the well-known Lawvere's paper on metric spaces. More generally, the procedure works for 'glueing structures', defined by glueing elementary ones: manifolds, fibre bundles, foliations... Here, the basic inverse category for manifolds is the category C of open subspaces of all euclidean spaces, with partial homeomorphisms (or diffeomorphisms) between open subsets of domain and codomain. A manifold is then defined replacing the usual atlas of charts U_i --> R^n with the 'glueing atlas' of such partial bijections U_i --> U_j. This atlas is, in a sense to be made precise, a category enriched over C - as an involutive ordered category endowed with further structure. The underlying topological space is recovered as a lax colimit. See [3, 4]. References [1] M. Grandis, Sous-quotients et relations induites dans les categories exactes, Cahiers Topologie Geom. Differentielle 22 (1981), 231-238. [2] --, Concrete representations for inverse and distributive exact categories, Rend. Accad. Naz. Sci. XL, Mem. Mat. 8 (1984), 99-120. [3] --, Manifolds as enriched categories, in: Categorical Topology, Prague 1988, pp.358-368, World Scientific 1989. [4] --, Cohesive categories and manifolds, Ann. Mat. Pura Appl. 157 (1990), 199-244. With best regards Marco