#
nLab

model structure on differential graded-commutative superalgebras

### Context

#### Super-Algebra and Super-Geometry

#### Model category theory

**model category**, model $\infty$-category

## 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

#### specific

### 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

# Contents

## Idea

The category of differential graded-commutative superalgebras over a field of characteristic zero carries a *projective* model category structure whose weak equivalences are the underlying quasi-isomorphisms and whose fibrations are the degreewise surjections (all either in unbounded degree, in non-negative degree or in non-positive degree).

This is the transferred model structure of the projective model structure on chain complexes of super vector spaces, transferred along the forgetful functor to underlying chain complexes.

The model structure is hence the direct generalization of the projective model structure on differential graded-commutative algebras, to which it reduces on the objects concentrated in even super-degree.

## References

A unified treatmeant generalizing to arbitary super Fermat theories is in