DK equivalences are Reedy equivalences for complete Segal spaces

Question

$\require{AMScd}$ Dear all,

I have a question concerning Charles Rezk's paper "A model for the homotopy theory of homotopy theory ", precisely Proposition 7.6 in this paper. It is proven there that if $U$ and $V$ are complete Segal spaces, then Dwyer-Kan equivalence $f\colon U\to V$ is a Reedy equivalence.

The proof (roughly) goes as follows. We consider the diagram $$ \begin{CD} U_0 @>s_0>> U_1 @>>>U_0\times U_0\\ @Vf_0VV @VVV @VVV\\ V_0 @>s_0>> V_1 @>>>V_0\times V_0. \end{CD} $$ The right-hand square is a homotopy pullback, because $f$ is a Dwyer-Kan equivalence, thus induces an equivalence on horizontal fibers in this square. The big rectangle is a homotopy pullback, because $U,V$ are complete Segal Spaces, so the horizontal fibers are spaces of paths between points, so also equivalent. Now from this we deduce that $f_0\colon U_0\to V_0$ is a weak equivalence.

I am not sure why the last statement holds. It sounds to me like a general fact from model categories saying that if $$ \begin{CD} X@>{\Delta}>>Y \\ @VfVV @V{f\times f}VV \\ Y@>\Delta>> Y\times Y \end{CD} $$ is a homotopy pullback, then $f$ is a weak equivalence. Is this true? And is my intuition here correct, or in the proof above we are using something else?

Answer 1

Let $U$ and $V$ be Segal spaces and let $f : U \to V$ be a Dwyer–Kan equivalence. That means two things:

1. The following diagram is a homotopy pullback square: $$\require{AMScd} \begin{CD} U_1 @>>> U_0 \times U_0 \\ @V{f_1}VV @VV{f_0 \times f_0}V \\ V_1 @>>> V_0 \times V_0 \end{CD}$$

2. $\operatorname{Ho} f : \operatorname{Ho} U \to \operatorname{Ho} V$ is essentially surjective on objects.

It seems to me you are already convinced of these two facts:

Lemma 1. If $V$ is a complete Segal space, then $\pi_0 (f_0) : \pi_0 (U_0) \to \pi_0 (V_0)$ is surjective.

Lemma 2. If $U$ and $V$ are complete Segal spaces, then the following diagram is a homotopy pullback square: $$\require{AMScd} \begin{CD} U_0 @>{\Delta}>> U_0 \times U_0 \\ @V{f_0}VV @VV{f_0 \times f_0}V \\ V_0 @>>{\Delta}> V_0 \times V_0 \end{CD}$$

Perhaps this is the fact you are missing:

Lemma 3. Let $h : X \to Y$ is a morphism of simplicial sets. The following are equivalent:

• $h : X \to Y$ is a weak homotopy equivalence.

• $\pi_0 (h) : \pi_0 (X) \to \pi_0 (Y)$ is surjective and the following is a homotopy pullback square: $$\begin{CD} X @>{\Delta}>> X \times X \\ @V{h}VV @VV{h \times h}V \\ Y @>>{\Delta}> Y \times Y \end{CD}$$

Morally, this is the homotopy-theoretic version of the familiar fact that a map of sets is bijective if and only if it is surjective and injective. Combining the three lemmas above yields the claim that $f_0 : U_0 \to V_0$ is a (weak) homotopy equivalence when $U$ and $V$ are complete Segal spaces.

Normally, I would advise to think about the discrete case first to separate the conceptual difficulties from the technical difficulties (of working homotopically), but the Rezk condition makes the discrete case rather uninteresting! Nonetheless, thinking about the discrete case would have alerted you to the fact that the diagram in lemma 3 being a pullback is not enough on its own – that is actually equivalent to $h : X \to Y$ being a monomorphism.