pushout-product

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.

Let $\otimes : \mathcal{E}_1 \times \mathcal{E}_2 \to \mathcal{E}_3$ be a functor (e.g. a tensor product, tensoring). Let $\mathcal{E}_3$ have pushouts.

For $f : A \to B$ in $\mathcal{E}_1$ and $g : X \to Y$ in $\mathcal{E}_2$, the **pushout product** $f \Box g$ morphism is the morphism

$A \otimes Y \coprod_{A \otimes X} B \otimes X
\to
B \otimes Y$

out of the coproduct, induced from the commuting diagram

$\array{
A \otimes X &\to& B \otimes X
\\
\downarrow && \downarrow
\\
A \otimes Y &\to& B \otimes Y
}
\,.$

For $\mathcal{C}$ any category and $K\subset Mor(\mathcal{C})$ any class of its morphisms, write $K Inj$ for the $K$-injective morphisms and $K Cof \coloneqq (K Inj)Proj$ for the $K Inj$-projective morphisms.

Let $\mathcal{C}$ be a symmetric closed monoidal category with finite limits and finite colimits, and let $I_1, I_2\subset Mor(\mathcal{C})$ be two classes of its morphisms.

Then under pushout product $\Box$:

$(I_1 Cof) \Box (I_2 Cof)
\subset
(I_1 \Box I_2) Cof
\,.$

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 $n \in \mathbb{N}$, let

$i_n \;\colon\; S^{n-1}\hookrightarrow D^n$

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:

$i_{n_1} \Box i_{n_2} \simeq i_{n_1 + n_2}
\,.$

Let moreover

$j_n \coloneqq (id,\delta_0) \;\colon\; D^n \hookrightarrow D^n \times I
\,.$

Then

$i_{n_1}\Box j_{n_2} \simeq j_{n_1 + n_2}
\,.$

To see this, it is profitable to model n-disks and n-spheres, up to homeomorphism, as n-cubes and their boundaries.

To see the idea of the proof, consider the situation in low dimensions, where one readily sees that

$i_1 \Box i_1
\;\colon\;
(\; = \;\cup\; \vert\vert\;)
\hookrightarrow
\Box$

and

$i_1 \Box j_0
\;\colon\;
(\; = \;\cup\; \vert \; )
\hookrightarrow
\Box
\,.$

Generally, $D^n$ may be represented as the space of $n$-tuples of elements in $[0,1]$, and $S^n$ as the suspace of tuples for which at least one of the coordinates is equal to 0 or to 1.

Accordingly $S^{n_1} \times D^{n_2}$ is the spaces of $(n_1+n_2)$-tuples, such that one of the first $n_1$ coordinates is equal to 0 or 1, and hence

$S^{n_1} \times D^{n_2} \cup D^{n_1} \times S^{n_2} \simeq S^{n_1 + n_2}
\,.$

And of course it is clear that $D^{n_1} \times D^{n_2} \simeq D^{n_1 + n_2}$. This shows the first case.

For the second, use that $S^{n_1} \times D^{n_2} \times I$ is contractible to $S^{n_1} \times D^{n_2}$ in $D^{n_1} \times D^{n_2} \times I$, and that $S^{n_1} \times D^{n_2}$ is a subspace of $D^{n_1} \times D^{n_2}$.

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.

- Mark Hovey, Brooke Shipley, Jeff Smith,
*Symmetric spectra*, J. Amer. Math. Soc. 13 (2000), 149-208 (arXiv:math/9801077)

Revised on May 2, 2016 05:05:12
by Urs Schreiber
(131.220.184.222)