nLab
pushout-product

Contents

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.

Definition

Let : 1× 2 3 \otimes : \mathcal{E}_1 \times \mathcal{E}_2 \to \mathcal{E}_3 be a functor (e.g. a tensor product, tensoring). Let 3\mathcal{E}_3 have pushouts.

Definition

For f:ABf : A \to B in 1\mathcal{E}_1 and g:XYg : X \to Y in 2\mathcal{E}_2, the pushout product fgf \Box g morphism is the morphism

AY AXBXBY A \otimes Y \coprod_{A \otimes X} B \otimes X \to B \otimes Y

out of the coproduct, induced from the commuting diagram

AX BX AY BY. \array{ A \otimes X &\to& B \otimes X \\ \downarrow && \downarrow \\ A \otimes Y &\to& B \otimes Y } \,.

Properties

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

Proposition

Let 𝒞\mathcal{C} be a symmetric closed monoidal category with finite limits and finite colimits, and let I 1,I 2Mor(𝒞)I_1, I_2\subset Mor(\mathcal{C}) be two classes of its morphisms.

Then under pushout product \Box:

(I 1Cof)(I 2Cof)(I 1I 2)Cof. (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. 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.

Examples

Example

For nn \in \mathbb{N}, let

i n:S n1D n 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 1i n 2i n 1+n 2. i_{n_1} \Box i_{n_2} \simeq i_{n_1 + n_2} \,.

Let moreover

j n(id,δ 0):D nD n×I. j_n \coloneqq (id,\delta_0) \;\colon\; D^n \hookrightarrow D^n \times I \,.

Then

i n 1j n 2j n 1+n 2. 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 1i 1:(=||) i_1 \Box i_1 \;\colon\; (\; = \;\cup\; \vert\vert\;) \hookrightarrow \Box

and

i 1j 0:(=|). i_1 \Box j_0 \;\colon\; (\; = \;\cup\; \vert \; ) \hookrightarrow \Box \,.

Generally, D nD^n may be represented as the space of nn-tuples of elements in [0,1][0,1], and S nS^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×D n 2S^{n_1} \times D^{n_2} is the spaces of (n 1+n 2)(n_1+n_2)-tuples, such that one of the first n 1n_1 coordinates is equal to 0 or 1, and hence

S n 1×D n 2D n 1×S n 2S n 1+n 2. 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×D n 2D n 1+n 2D^{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×D n 2×IS^{n_1} \times D^{n_2} \times I is contractible to S n 1×D n 2S^{n_1} \times D^{n_2} in D n 1×D n 2×ID^{n_1} \times D^{n_2} \times I, and that S n 1×D n 2S^{n_1} \times D^{n_2} is a subspace of D n 1×D n 2D^{n_1} \times D^{n_2}.

Remark

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.

References

Revised on September 13, 2016 21:03:11 by Todd Trimble (67.81.95.215)