Monads on Composition Graphs

Abstract

Collections of objects and morphisms that fail to form categories, inasmuch as the expected composites of two morphisms need not always be defined, were introduced in [14,15] under the name composition graphs. In [14,16], notions of adjunction and weak adjunction for composition graphs are proposed. Building on these definitions, we now introduce a concept of monads for composition graphs and show that the usual correspondence between adjunctions and monads remains correct, i.e. that (weak) adjunctions give rise to monads and that all monads are induced by adjunctions. Monads are described in terms of natural transforms as well as in terms of Kleisli triples, which seem to be better suited in the absence of associativity. The realization of a monad by an adjunction relies on a generalization of the Kleisli construction to composition graphs; on the other hand, the Eilenberg–Moore construction produces only a weak adjunction and admits comparison functors from weak adjunctions. As a first nontrivial example, we determine the algebras of the word monad on the composition graph of full morphisms between sets with a distinguished subset.