When equivariant homotopy theory meets combinatorics

Abstract

In many different ways, mathematics often amounts to finding and studying suitable \emph{algebraic structures} on various collections of objects. In any first class in algebra, one gets to know the bestiary of monoids, (abelian) groups, rings, modules, algebras, etc. Algebraic structures are omnipresent and can also be viewed in a broader sense. For instance, certain categories can be endowed with a commutative multiplicative structure, making them into a \emph{symmetric monoidal category}. The latter is a triple \((C,\otimes,\mathbb{1}_C)\), where \(C\) is a category, \(\otimes\) is a functor \(C\times C \to C\), viewed as a multiplication operation, and \(\mathbb{1}_C \in C\) is an object in \(C\), representing a unit for the multiplication. We also require several axioms to be satisfied, for example associativity and commutativity of \(\otimes\), and unitality of \(\mathbb{1}_C\) with respect to \(\otimes\). A concrete example is the category of vector spaces over a field \(k\), where \(\otimes\) is defined as the tensor product of vector spaces and \(\mathbb{1}_C\) as the field \(k\) itself.