nLab
canonical model structure on Operad

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

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:PQf : P \to Q a weak equivalence if

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

  2. for every collection (c 1,,c n;c)(c_1, \cdots, c_n; c) of colours it induces an isomorphism

    P(c 1,,c n;c)Q(f(c 1),,f(c n);f(c)) 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:PQf : P \to Q a fibration if for every isomorphism in QQ and a lift of its source object to PP there is an isomorphism in PP covering it under ff.

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

Revised on February 29, 2012 01:56:50 by Urs Schreiber (82.169.65.155)