nLab pushout-product

Idea

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. Regarding arrows in a category as diagrams with domain the interval category, and giving the interval category the natural monoidal product, this is a kind of Day convolution tensor product with the natural copowering over Set replacing a tensor in the coend.

Definition

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.

Definition

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 pushout, induced from the commuting diagram

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

Properties

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.

Proposition

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 $\Box$:

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

(Hovey-Shipley-Smith 00)

Proof

By a little Joyal-Tierney calculus.

Remark

In the context of monoidal model category theory, prop. 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.

Examples

Example

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

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

Proof

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 $D^{n_1} \times D^{n_2}$.

Remark

The relations in example 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.

Related concepts

• pullback power– the dual concept

• pushout-product axiom

• Joyal-Tierney calculus

References

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