model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
for $n$-groupoids
for $\infty$-groups
for $\infty$-algebras
general $\infty$-algebras
specific $\infty$-algebras
for stable/spectrum objects
for $(\infty,1)$-categories
for stable $(\infty,1)$-categories
for $(\infty,1)$-operads
for $(n,r)$-categories
for $(\infty,1)$-sheaves / $\infty$-stacks
With braiding
With duals for objects
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?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
symmetric monoidal (∞,1)-category of spectra
The monoid axiom is an extra condition on a monoidal model category that helps to make its model structure on monoids in a monoidal model category exist and be well behaved.
We say a monoidal model category satisfies the monoid axiom if every morphism that is obtained as a transfinite composition of pushouts of tensor products of acyclic cofibrations with any object is a weak equivalence.
(Schwede-Shipley 00, def. 3.3.).
In particular, the axiom in def. says that for every object $X$ the functor $X \otimes (-)$ sends acyclic cofibrations to weak equivalences.
Let $C$ be a
Then if the monoid axiom holds for the set of generating acyclic cofibrations it holds for all acyclic cofibrations.
(Schwede-Shipley 00, lemma 3.5).
If a monoidal model category satisfies the monoid axiom and
it is a cofibrantly generated model category;
all objects are small objects,
then the transferred model structure along the free-forgetful adjunction $(F \dashv U) : Mon(C) \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} C$ exists on its category of monoids and hence provides a model structure on monoids.
(Schwede-Shipley 00, theorem 4.1)
Monoidal model categories that satisfy the monoid axiom (as well as the other conditions sufficient for the above theorem on the existence of transferred model structures on categories of monoids) include
with respect to Cartesian product
with respect to tensor product of chain complexes:
and with respect to a symmetric monoidal smash product of spectra:
(Schwede-Shipley 00, section 5, MMSS 00, theorem 12.1 (iii))
Stefan Schwede, Brooke Shipley, Algebras and modules in monoidal model categories Proc. London Math. Soc. (2000) 80(2): 491-511 (pdf)
Michael Mandell, Peter May, Stefan Schwede, Brooke Shipley, part III of Model categories of diagram spectra, Proceedings London Mathematical Society Volume 82, Issue 2, 2000 (pdf, publisher)
Last revised on April 4, 2016 at 13:34:14. See the history of this page for a list of all contributions to it.