Operads from the viewpoint of categorical algebra

We exhibit a parallel between Lawvere’s algebraic theories and operads; there is a common ancestor of both notions whose syntax is described by labeled trees, dummy variables corresponding to special labels. Operads, cyclic and braided operads alike arise as group objects in the category of compositional structures. Laplaza’s coherence theorem for “associativity not an isomorphism” is seen to correspond geometrically to the 1-skeleton of a variant of the Stasheff polytope. Several questions are raised concerning the extent of the above-mentioned parallel; applications to higher categories remain for future work. The purpose of this note is to locate operads within the realm of classical universal algebra, using a uniform language to describe both, and to sketch how they may be used to construct structures in category theory analogous to ones in Set – in the sense in which MacLane monoidal categories are analogous to monoids. There are many more pieces to be fitted within this picture, such as symmetric and braided monoidal categories, “lax” functor categories, tensor products and abelianizations of structures, that will not be touched upon here; the chosen application is to Laplaza’s coherence theorem. The thrust of the first section, mainly a review, is that what sets operads apart is not their describing structures via n-ary operations and identities between composites. (That is the very essence of universal algebra, and goes back at least to the 40’s, to the notion of clone or “closed set of operations”; see [12].) Rather, it is the fact that operads cannot identify or skip inputs, only permute them. Being of such limited syntax allows operadic theories to extend to enriched categories more readily than the rest of universal algebra, and makes features common to all algebraic theories – such as free models, coequalizers, tripleability and cohomology – explicitly constructible. It also lets them extend from Set-based structures to ones in Cat (small categories) in a way that non-operadic structures do not; an aspect of this is captured, perhaps, in our notion of “relaxation”. The ultimate goal of any formalism of its kind would be an analogous development for n-categories – an area where presenting algebraic structures in terms of generators and relations (commutative and “pasting” diagrams, that is) has been notoriously cumbersome. Correspondingly, the operads encountered in this article have diverse combinatorial structures (categories, graphs, posets) rather than topological or graded algebraic objects as coefficients. Finally, the essay is punctuated with (perhaps too many) questions. 1991 Mathematics Subject Classification. Primary 18C05; Secondary 08A02.