A note on the smash product and regular associativity

Abstract: This note is about the smash product of pointed topological spaces, without relying on some convenient subcategory. We deal with its partial associativity properties and their connection with the function spaces, introducing a property of regular associativity related to the colax monoidal structure of the product. We also study a large class of triples of pointed spaces 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.