nLab
model structure on modules over an algebra over an operad

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Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (∞,1)-categories

Model structures

for ∞-groupoids

for ∞-groupoids

for n-groupoids

for ∞-groups

for ∞-algebras

general

specific

for stable/spectrum objects

for (∞,1)-categories

for stable (∞,1)-categories

for (∞,1)-operads

for (n,r)-categories

for (∞,1)-sheaves / ∞-stacks

Higher algebra

higher algebra

universal algebra

Algebraic theories

Algebras and modules

Higher algebras

Model category presentations

Geometry on formal duals of algebras

Theorems

Contents

Definition

Theorem

Let

Then then category Mod P(A) of modules over an algebra over an operad carries the transferred model structure along the forgetful functor U:Mod P(A)→ℰ.

Every morphism of cofibrant P-algebras f:A→B induced a Quillen adjunction

(f !⊣f *):Mod P(B)→f *←f !Mod P(A)(f_! \dashv f^*) : Mod_P(B) \stackrel{\overset{f_!}{\leftarrow}}{\underset{f^*}{\to}} Mod_P(A)

which is a Quillen equivalence if f is a weak equivalence.

This is (BergerMoerdijk, theorem 2.6).

Examples

References

  • Benoit Fresse, Modules over operads and functors Springer Lecture Notes in Mathematics, (2009) (pdf)
Revised on February 11, 2013 01:36:37 by Urs Schreiber (89.204.137.65)
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