*-Autonomous Categories

Abstract

The category of finite dimensional vector spaces over a fieldK is an autonomous category (F. E. J. Linton's name for asymmetric closed monoidal category [see J. Algebra 2 (1965), 315349; MR 31#4821]) with an object K and having the propertythat the internal hom functor ( ,K) establishes an equivalencewith its opposite category. The author calls such a category *autonomous.\par The main goal in these notes is to construct a *autonomous category from a lesser structure. More precisely, if{\bf V} is both an autonomous category and a semivariety (i.e.a full subcategory of a variety closed under projective limitsand containing all free algebras ) and if {\bf A} is a{\bf V} category equipped with subcategories {\bf C} and{\bf D} which determine what the author calls an enriched pre *autonomous situation, the aim is to find a full subcategory {\bf G}\subset{\bf A} which contains {\bf C} and {\bf D} andcan be equipped with a * autonomous structure extending the givenstructure on {\bf C} and {\bf D}. The hypotheses under whichthis construction is achieved are introduced as needed but areneatly summarised at the end of Chapter III.\par In order to makethe notes largely self contained, basic concepts are introducedin Chapter I. The main results on the extensions of the dualityand the internal hom functor and the construction of the desired* autonomous category {\bf G} are contained in Chapters II andIII, respectively. Apart from the main motivating example oftopological vector spaces (over a discrete field), which isdiscussed parallel to the development of the theory, considerabletrouble has been taken to apply the theory to the followingexamples (Chapter IV): topological vector spaces (over {\bf R}or {\bf C} with the usual topology ); dualizing modules; Banachspaces; modules over a Hopf algebra; topological abelian groups;semilattices. The hypotheses under which the main constructionapplies are verified in each case, but it would have been moreinteresting had the * autonomous extension category {\bf G}been described more explicitly in the various examples.\par In anappendix Po Hsiang Chu, a student of the author, describes aconstruction which yields an embedding (not full) of anyautonomous category. The theory is applied to the double envelopeof a symmetric monoidal category