# nLab model structure on algebras over a monad

model category

## Model structures

for ∞-groupoids

### for $\left(\infty ,1\right)$-sheaves / $\infty$-stacks

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

For $C$ a monoidal model category and $T:C\to C$ a monad on $C$, there is under mild conditions a natural model category structure on the category of algebras over a monad over $T$.

## Definition

Let $C$ be a cofibrantly generated model category and $T:C\to C$ a monad on $C$.

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

$\left(F⊣U\right):\mathrm{Alg}T\stackrel{\stackrel{F}{←}}{\underset{U}{\to }}C\phantom{\rule{thinmathspace}{0ex}}.$(F \dashv U) : Alg T \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} C \,.

See (SchwedeShipley, lemma 2.3).

## References

Revised on November 18, 2010 15:56:02 by Urs Schreiber (131.211.232.149)