Base-change for simplicial spaces

Question

Base-change for simplicial spaces

Let us say that a map of simplicial spaces $X_* \to Y_*$ is a base-change if for all $n$ the canonical map $$ X_n \to (X_0)^{n+1} \times_{(Y_0)^{n+1}}^h Y_n $$ is an equivalence. (Here I use the map $X_n \to (X_0)^{n+1}$ that sends an $n$-simplex to its $(n+1)$ vertices.)

Then one has the following base-change theorem:

If $f_*: X_* \to Y_*$ is a base-change and $\pi_0(f_0): \pi_0(X_0) \to \pi_0(Y_0)$ is surjective, then $\|f\|: \|X_*\| \to \|Y_*\|$ is a weak equivalence.

Here $\|-\|$ denotes the fat geometric realisation, but one could also use the usual geometric realisation if one assumes that the simplicial spaces are "good". In general, I have tried to state everything in a way that is invariant under weak equivalences.

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Reference request:

Is this result well-known and if so, is it written up somewhere?

This seems to be a rather fundamental tool for computing geometric realisations. For example if $Y_*$ is the nerve of a topological category then this implies that Dwyer-Kan equivalences induces equivalences on classifying spaces.

I think this is closely related to this question, but unlike there I do not want to restrict to nerves of topological categories. Another difference is that I am only interested in the homotopy invariant statement and not in the point-set topological conditions.

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A proof sketch

I know how to piece together the proof from various sources, so let me give the idea here for completeness sake.

Let $f:A \to B$ be some map of spaces that is surjective on connected components and let $T(A)_*$ and $T(B)_*$ be the simplicial spaces defined by $T(A)_n := A^{n+1}$ where the face and degeneracy maps delete and repeat entries. Then one can prove the more general statement that for any simplicial space $Y_*$ with a map $p:Y_* \to T(B)_*$ the following map is an equivalence: $$ \alpha_Y: \|T(A)_* \times_{T(B)_*} Y_* \| \longrightarrow \|Y_*\|. $$ Following the ideas of Rezk in When are homotopy colimits compatible with homotopy base change? (see Lanari's thesis for more details) one can reduce this to the case that $Y_*$ is the simplicial set $\Delta^n$ thought of as a discrete simplicial space. (In an $\infty$-categorical setting this a formal consequence of the fact that colimits in spaces are preserved by base-change and that every simplicial space is the colimit of its simplices.)

For the case of $Y_* = \Delta^n$ this statement follows from the base-change theorem of Ebert and Randal-Williams in [Semi-simplicial spaces][4]. (Replace $A \to B$ by a fibration, let $A_i := A \times_B \{i\}$, and then base-change the unital category $[n]$ along the map $\coprod_{i=0}^n A_i \to \{0,\dots,n\}$.)

The case of $Z_* = \Delta^n \xrightarrow{p} T(B)_*$ can be dealt with by hand. Replace $A \to B$ by a fibration; in particular this map will be surjective. The simplicial space $T(A)_* \times_{T(B)_*} \Delta^n$ is the nerve of the topological category $t(A) \times_{t(B)} [n]$ where $t(A)$ is the category with objects $A$ and morphisms $A \times A$. (Exactly one morphism between any two objects.) The category $t(A) \times_{t(B)} [n]$ has a terminal object given by $(a, n)$ for any $a \in A$ with $f(a) = p(n)$ and hence its classifying space is contractible. Therefore: $$ \|T(A)_* \times_{T(B)_*} \Delta^n\| \simeq * \simeq \|\Delta^n\|. $$

While this seems to be a complete proof, I would still much prefer to have a reference that states the base-change theorem in full generality.