Introduction

Abstract

This is an introductory text in categorial proof theory. The aim is to explain fundamental notions of category theory, and in particular the notion of adjunction, from a proof-theoretical point of view. It will be shown that elementary notions, like the notions of category, functor, and natural transformation, and the more complex notion of adjunction, together with the related notion of comonad (we could as well deal with monads, also called triples), can all be formulated in such a manner that equalities between arrows tied to them are necessary and sufficient for eliminating composition in freely generated structures corresponding to these notions. This means, in particular, that for every arrow in categories involved in freely generated adjunctions and comonads we will have a term designating this arrow in which there is no composition. Moreover, such a term can be brought into a normal form unique for the arrow, so as to yield both syntactical and very simple model-theoretical, geometrical, decision procedures for the commuting of diagrams. This composition-free normal form serves also to demonstrate that strengthening the notions of adjunction and comonad with any new equality between arrows would trivialize these notions. The difference between these “composition-free” formulations of categorial notions and standard formulations is that natural transformations are not conceived as families of arrows, but as operations on arrows.