Abstract: A multiplication on an object $X$ is a map $m:X\times X\to X$ with certain properties which capture the axioms of associativity, unit and inverses of a group. An action of a group $G$ on an object $X$ is given in terms of a map $a:G\times X\to X$. These notions are interesting in many different contexts, e.g., when the objects are topological spaces, groups, monoids, etc. The purpose of this talk is to answer the question: When can these notions be dualized, i.e., when can all the arrows be reversed in these definitions, and what results are obtained in the different settings? There is a strong motivation from algebraic topology where the duals in the case of topological spaces are well known constructions such as suspensions. To define a "comultiplication" we need the notion of a coproduct which is dual to that of a product. This notion is available for topological spaces, groups (the free product "$*$") and monoids. Thus a comultiplication on a group $G$ is a homomorphism $m:G\to G*G$ whose composition with the projections $G*G\to G$ is the identity. A coaction of $G$ on a group $K$ is similarly defined by a homomorphism $G\to G*K$. In this talk we will describe the structure of groups (and monoids) which admit a comultiplication or a coaction.\par Although the motivation comes from topology, the talk will be purely algebraic. Only a little group theory will be assumed.
This talk will be accessible to graduate students.