Functors between shape categories

Abstract

Categorical shape theory was introduced in a series of articles by Deleanu, Hilton and Frei ([5], [6], [7] and (81). It provided not only a convenient language for handling purely categorical questions which arose in shape theory, but also fitted very neatly into several categorical areas, notably the study of Kan extensions (cf. Frei-Kleisli [9] and [lo]). In this note we give some methods for inducing functors between shape categories. A preliminary version of these constructions appeared as part of our 1978 Esquisses notes [3], however as there have been several advances since that time (notably arising from the use of distributors in [2]) and as those notes had an extremely limited distribution, it has seemed advisable to write an improved version. We include illustrative examples which show that many of the functors arising naturally in situations involving monads can be interpreted as shape induced functors. This is in line with the examples given by Deleanu, Hilton and Frei which calculate the shape category of a functor having a left adjoint.