Associativity of Cosmash Products in Algebra

A. Carboni and G. Janelidze extend the definition of smash product from pointed topological spaces to pointed objects in a suitable category. Moreover, they study a condition they call smash associativity. In this talk, we interest ourselves in the dual notion: the associativity of cosmash products. An interesting fact about this categorical notion is that it characterizes the variety of commutative and associative K-algebras for an infinite field K. The promise of this talk is to convince you of the validity of the latter statement.

In order to present things in an understandable way, we will first introduce the notion of a binary cosmash product. We see how it naturally leads to a suitable definition of binary commutators by looking at some classical examples. We then try to extend these notions the ternary case, and even to the n-ary case for some natural number n. From here, what cosmash associativity means can be explained essentially without effort. In the end, we discuss the main result and, if time allows it, the techniques used to prove it.

Joint work with Ülo Reimaa and Tim Van der Linden.