A Note on the Smash Product and Regular Associativity

We want to study the smash product of pointed topological spaces, in an organic way and full generality, without relying on some ‘convenient subcategory’. The n-ary smash product has a ‘colax’ form of associativity, which supplies a categorical framework for the properties of this operation and its connection with the function spaces. Various concrete computations of smash products are given, including a large class of cases where associativity fails. Lax and colax monoidal structures are unusual and interesting, in category theory. Some parts of this note will be obvious to a topologist and others to a categorist, in order to take into account both backgrounds.