ScalingStacks

Theorem 7.2. There exists a simplicial closed model category structure on the category $s{\operatorname{\mathcal{S}}}$ of simplicial spaces, called the complete Segal space model category structure, with the following properties.

1. (1)

   The cofibrations are precisely the monomorphisms.

2. (2)

   The fibrant objects are precisely the complete Segal spaces.

3. (3)

   The weak equivalences are precisely the maps $f$ such that $\Map_{s{\operatorname{\mathcal{S}}}}(f,W)$ is a weak equivalence of spaces for every complete Segal space $W$.

4. (4)

   A Reedy weak equivalence between any two objects is a weak equivalence in the complete Segal space model category structure, and if both objects are themselves complete Segal spaces then the converse holds.

Moreover, this model category structure is compatible with the cartesian closed structure on $s{\operatorname{\mathcal{S}et}}$ in the sense of Section 2.