Coequalizers and Free Triples

This paper is concerned with two problems which, although not apparently closely related, are solved in part by the same methods. The first problem is: given a bicomplete (=comple te and cocomplete) category X and a triple T on X, is X T also bicomplete? The second is: given a category X and a functor R: X---~X, does R generate a free triple? This paper began as an attempt to show that the category of contramodules over a coring is cocomplete (see (4.4)). Many people, too numerous to mention, have contributed materially to the results and their applications. All notation and terminology not explicitly defined below may be found in the introduction to [-2]. The first section of this paper gives the main definitions used and in section two we give the fundamental lemma on which the proofs are based. The next two sections prove and give applications of the cocompleteness theorem. Section five gives the construction of free triples and in section six we apply this to show that if 3--~ is a small theory, then under certain conditions the category of ~-~ algebras in X is tripleable over X. In the next section we apply these results to the category of sets and we show that for a certain large full subcategory of endofunctors on sets there is a "free triple triple". The last section gives another cocompteteness theorem, not related to that of section three. This latter is a generalization of the result that every category of algebras over sets is cocomptete.