nLab
model structure on algebras over a monad

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

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

Higher algebra

Contents

Idea

For 𝒞\mathcal{C} a cofibrantly generated model category and T:𝒞𝒞T \colon \mathcal{C} \longrightarrow \mathcal{C} a monad on 𝒞\mathcal{C}, there is under mild conditions a natural model category structure on the category of algebras over a monad over TT.

Definition

Let 𝒞\mathcal{C} be a cofibrantly generated model category and T:𝒞𝒞T \colon \mathcal{C} \longrightarrow \mathcal{C} a monad on 𝒞\mathcal{C}.

Then under mild conditions there exists the transferred model structure on the category of algebras over a monad, transferred along the free functor/forgetful functor adjunction

(FU):Alg T(𝒞)UF𝒞. (F \dashv U) \;\colon\; Alg_T(\mathcal{C}) \stackrel{\overset{F}{\longleftarrow}}{\underset{U}{\longrightarrow}} \mathcal{C} \,.

See (Schwede-Shipley 00, lemma 2.3).

References

Revised on October 5, 2016 14:34:31 by Urs Schreiber (89.204.153.227)