nLab pushout-product axiom

Contents

Context

Model category theory

model category, model \infty -category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Contents

Idea

The pushout-product axiom for a two-variable adjunction between model categories is a condition on pushout-products of certain cofibrations which ensures (together with its equivalent dual pullback-power axiom) that the two-variable adjunction is homotopically meaningful in a way (see the remark here) analogous to how the axioms of a Quillen adjunction ensure that an ordinary adjunction between model categories is. Therefore one refers to such two-variable adjunctions also as Quillen bifunctors.

In particular, in the definition of enriched model categories the pushout-product axiom ensures that the enriched tensoring/cotensoring is homotopically meaningful.

Specialized to the case of simplicial model categories this is the origin of the notion of the pushout-product axiom, in its dual guise as the pullback-power axiom as “axiom SM7” in Quillen (1967).

Specialized, alternatively, to the case of self-enrichment the pushout-product axiom for monoidal model categories ensures that the tensor product in two-variable adjunction with its internal hom-functor is homotopically well-behaved.

This situation of monoidal model categories has come to be the case where the pushout-product axiom is most prominently discussed in the literature, and where it has received its name, see the references there.

Definition

Let CC be a closed symmetric monoidal category equipped with a model category structure.

Then CC satisfies the pushout-product axiom if for any pair of cofibrations f:XYf : X \to Y and f:XYf' \colon X' \to Y' their pushout-product, hence the induced morphism out of the coproduct

(XY)⨿XX(YX)YY, (X \otimes Y') \overset {X \otimes X'} {\amalg} (Y \otimes X') \longrightarrow Y \otimes Y' \,,

is itself a cofibration, which, furthermore, is acyclic if ff or ff' is.

This means that the tensor product

:C×CC \otimes \colon C \times C \to C

is a left Quillen bifunctor.

If the monoidal category underlying CC is a closed monoidal category, one can dually define the pullback-power axiom to mean that if f:ABf \colon A \to B is a cofibration and g:XYg \colon X \to Y is a fibration, their pullback power

[B,X][A,X]× [A,Y][B,Y], [B, X] \to [A, X] \times_{[A,Y]} [B, Y] \,,

is a fibration, which, furthermore, is acyclic if ff or gg is.

By Joyal-Tierney calculus, the pullback-power axiom holds if and only if the pushout-product axiom holds.

Remark

The pushout-product axiom implies in particular that tensoring with cofibrant objects preserves cofibrations and acyclic cofibrations.

However the plain tensor product of a pair of (acyclic) cofibrations is in general not an (acyclic) cofibration.

Remark

In a cofibrantly generated model category the pushout product axiom holds as soon as it holds for (acyclic) generating cofibration (see here).

Remark

The pushout-product axiom makes sense more generally in the context of a two-variable adjunction between model categories. This is important in enriched homotopy theory.

References

The pullback-power axiom in its role in enriched model categories, specifically in simplicial model categories originates in:

For more see the references at:

Discussion for model \infty -categories (such as with homotopy Kan fibrations):

Last revised on May 21, 2023 at 07:17:38. See the history of this page for a list of all contributions to it.