# nLab model structure on algebras over a monad

Contents

### Context

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for $\infty$-groupoids

for ∞-groupoids

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

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

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

## Definition

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

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

$(F \dashv U) \;\colon\; Alg_T(\mathcal{C}) \stackrel{\overset{F}{\longleftarrow}}{\underset{U}{\longrightarrow}} \mathcal{C} \,.$

## References

Last revised on October 5, 2016 at 18:34:31. See the history of this page for a list of all contributions to it.