model category

for ∞-groupoids

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

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