Subsection 7.4. (0MTG) · ScalingStacks

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