Section 7. (0MTF) · ScalingStacks

7. Closed model category structures[0MTF]

Our main results deal with the existence of certain closed model category structures on $s{\operatorname{\mathcal{S}}}$ related to Segal spaces and complete Segal spaces.

[05UN]

Theorem 7.1. There exists a simplicial closed model category structure on the category $s{\operatorname{\mathcal{S}}}$ of simplicial spaces, called the Segal space model category structure, with the following properties.

(1)

The cofibrations are precisely the monomorphisms.
(2)

The fibrant objects are precisely the Segal spaces.
(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 Segal space $W$ .
(4)

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

We will prove (7.1) in Section 10.

[05UP]

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)

The cofibrations are precisely the monomorphisms.
(2)

The fibrant objects are precisely the complete Segal spaces.
(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)

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.

We will prove (7.2) in Section 12.

These theorems have the following important corollary.

[05UQ]

Corollary 7.3. If $W$ is a complete Segal space (resp. a Segal space) and $X$ is any simplicial space, then $W^{X}$ is a complete Segal space (resp. a Segal space).

[05UR]

Proof. This is a direct consequence of the compatibility of these model category structures with the cartesian closure, since any object is cofibrant in either of these model category structures. ∎

7.4. Dwyer-Kan equivalences[0MTG]

We would like to understand the relationship between the model category structures for Segal spaces and for complete Segal spaces.

We say a map $f\colon U\rightarrow V$ of Segal spaces is a Dwyer-Kan equivalence if

(1)

the induced map $\ho f\colon\ho U\rightarrow\ho V$ on homotopy categories is an equivalence of categories, and
(2)

for each pair of objects $x,x^{\prime}\in U$ the induced function on mapping spaces $\map_{U}(x,x^{\prime})\rightarrow\map_{V}(fx,fx^{\prime})$ is a weak equivalence.

For a Segal space $W$ let ${\operatorname{ob}}W/{\sim}$ denote the set of objects of $W$ modulo the equivalence relation of homotopy equivalence (or equivalently, the set of isomorphism classes in $\ho W$ ). If we define a condition 1’ by

1’.

the induced map ${\operatorname{ob}}U/{\sim}\rightarrow{\operatorname{ob}}V/{\sim}$ on equivalence classes of objects is a bijection,

then it is not hard to see that conditions 1’ and 2 together are equivalent to conditions 1 and 2.

[05US]

Lemma 7.5. If $U\xrightarrow{f}V\xrightarrow{g}W$ are maps of Segal spaces such that $f$ and $g$ are Dwyer-Kan equivalences, then $gf$ is a Dwyer-Kan equivalence.

If $U\xrightarrow{f}V\xrightarrow{g}W\xrightarrow{h}X$ are maps of Segal spaces such that $gf$ and $hg$ are Dwyer-Kan equivalences, then each of the maps $f$ , $g$ , and $h$ is a Dwyer-Kan equivalence.

[05UT]

Proof. Straightforward. ∎

[05UU]

Proposition 7.6. A map $f\colon U\rightarrow V$ of complete Segal spaces is a Dwyer-Kan equivalence if and only if it is a Reedy weak equivalence.

[05UV]

Proof. It is clear that a Reedy weak equivalence between any two Segal spaces is a Dwyer-Kan equivalence.

Conversely, suppose $f\colon U\rightarrow V$ is a Dwyer-Kan equivalence between complete Segal spaces. Then $\pi_{0}U_{0}\approx{\operatorname{ob}}U/{\sim}$ and $\pi_{0}V_{0}\approx{\operatorname{ob}}V/{\sim}$ by (6.5), so that $\pi_{0}U_{0}\rightarrow\pi_{0}V_{0}$ is a bijection. In the commutative diagram

the right-hand square is a homotopy pullback (since the induced maps of fibers are of the form $\map_{U}(x,y)\rightarrow\map_{V}(fx,fy)$ , which is assumed to be a weak equivalence), and the large rectangle is a homotopy pullback (since by (6.4) the induced maps of fibers are of the form $\hoequiv_{U}(x,y)\rightarrow\hoequiv_{V}(fx,fy)$ , which is also a weak equivalence). We conclude that $U_{0}\rightarrow V_{0}$ is a weak equivalence, and therefore that $U_{1}\rightarrow V_{1}$ is a weak equivalence. Since both $U$ and $V$ are Segal spaces, it follows that the map $f\colon U\rightarrow V$ is a Reedy weak equivalence as desired. ∎

[05UW]

Theorem 7.7. Let $f\colon U\rightarrow V$ be a map between Segal spaces. Then $f$ is a Dwyer-Kan equivalence if and only if it becomes a weak equivalence in the complete Segal space model category structure.

We will prove (7.7) in Section 14.

[05UX]

Remark 7.8. Note that if $C$ is a category, then the natural inclusion $\discnerve(C)\rightarrow N(C)$ of simplicial spaces is a Dwyer-Kan equivalence; thus by (7.7) this map is a weak equivalence in the complete Segal space model category structure.

[05UY]

Corollary 7.9. The homotopy category of complete Segal spaces may be obtained by formally inverting the Dwyer-Kan equivalences in the homotopy category of Segal spaces.

Charles Rezk

Original source: arXiv:math/9811037v3

Original source · math/9811037v3