nLab
canonical model structure on Operad

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

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

Related concepts

• canonical model structure on Cat

• model structure on operads

• model structure on dendroidal sets

References

• Ittay Weiss, Dendroidal sets, PhD thesis (web)