model category, model -category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of -categories
Model structures
for -groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant -groupoids
for rational -groupoids
for rational equivariant -groupoids
for -groupoids
for -groups
for -algebras
general -algebras
specific -algebras
for stable/spectrum objects
for -categories
for stable -categories
for -operads
for -categories
for -sheaves / -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
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
symmetric monoidal (∞,1)-category of spectra
For a monoidal model category and a monoid in , there is under mild conditions a natural model category structure on its category of modules over . (Schwede & Shipley 2000, Thm. 4.1 (3.1 in the preprint)).
Let be a cofibrantly generated monoidal model category and let be a monoid object whose underlying object is cofibrant in . Then:
The category of internal -module objects carries a cofibrantly generated model structure whose weak equivalences and fibrations are those whose underlying maps are so in , hence which is right transferred along the forgetful functor :
If is in addition a commutative monoid object then the tensor product of modules makes itself into a monoidal model category.
(Schwede & Shipley 2000, Thm. 3.1 with Rem. 3.2)
General discussion:
Examples:
The special case of the model structure on modules in a functor category with values in a closed symmetric monoidal model category is (re-)derived (see the discussion here) in:
(there for the purpose of desribing representations of nets of observables in homotopical algebraic quantum field theory).
Last revised on November 8, 2023 at 08:52:46. See the history of this page for a list of all contributions to it.