model category, model $\infty$-category
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Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
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related by the Dold-Kan correspondence
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With braiding
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category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
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Morphisms
Internal monoids
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In higher category theory
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.
Let $C$ be a closed symmetric monoidal category equipped with a model category structure.
Then $C$ satisfies the pushout-product axiom if for any pair of cofibrations $f : X \to Y$ and $f' \colon X' \to Y'$ their pushout-product, hence the induced morphism out of the coproduct
is itself a cofibration, which, furthermore, is acyclic if $f$ or $f'$ is.
This means that the tensor product
is a left Quillen bifunctor.
If the monoidal category underlying $C$ is a closed monoidal category, one can dually define the pullback-power axiom to mean that if $f \colon A \to B$ is a cofibration and $g \colon X \to Y$ is a fibration, their pullback power
is a fibration, which, furthermore, is acyclic if $f$ or $g$ is.
By Joyal-Tierney calculus, the pullback-power axiom holds if and only if the pushout-product axiom holds.
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.
In a cofibrantly generated model category the pushout product axiom holds as soon as it holds for (acyclic) generating cofibration (see here).
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.
dually: pullback-power axiom in enriched model categories
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.