# nLab canonical model structure on Operad

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

# Contents

## Idea

A model category structure on the category Operad of Set-enriched coloured symmetric operads which generalizes the canonical model structure on Cat.

## Definition

###### Definition

Call a morphism of operads $f : P \to Q$ a weak equivalence if

1. its underlying functor of categories is an essentially surjective functor;

2. for every collection $(c_1, \cdots, c_n; c)$ of colours it induces an isomorphism

$P(c_1, \cdots, c_n; c) \to Q(f(c_1), \cdots, f(c_n); f(c))$

(the operadic analog of being full and faithful).

Call a morphism $f : P \to Q$ a fibration if for every isomorphism in $Q$ and a lift of its source object to $P$ there is an isomorphism in $P$ covering it under $f$.

Call a morphism a cofibration if it is an injection on objects (on colours)

###### Theorem

This defines a cofibrantly generated model category structure on Operad.

This is due to (Weiss 07).

## References

Last revised on February 29, 2012 at 01:56:50. See the history of this page for a list of all contributions to it.