The concept of pushout product is a natural kind of pairing operation on morphisms in categories equipped with a pairing operation on objects (e.g. a tensor product) and having pushouts. It sends two morphisms to the universal morphism out of the pushout of the span-diagram they form by pairing their domain objects.
For in and in , the pushout product morphism is the morphism
Then under pushout product :
By a little Joyal-Tierney calculus.
In the context of monoidal model category theory, prop. 1 implies that for checking the pushout-product axiom in the case of cofibrantly generated model categories it is sufficient to check it on generating cofibrations.
For , let
be the canonical sphere inclusions in Top (the generating cofibrations of the classical model structure on topological spaces). Their pushout product (with respect to Cartesian product of topological spaces) is given by addition of indices:
To see the idea of the proof, consider the situation in low dimensions, where one readily sees that
Generally, may be represented as the space of -tuples of elements in , and as the suspace of tuples for which at least one of the coordinates is equal to 0 or to 1.
Accordingly is the spaces of -tuples, such that one of the first coordinates is equal to 0 or 1, and hence
And of course it is clear that . This shows the first case.
For the second, use that is contractible to in , and that is a subspace of .
The relations in example 1 are the key in proving that the classical model structure on topological spaces (on compactly generated topological spaces) is an enriched model category over itself. See there at topological enrichment for more.