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The result sought by the question is a consequence of the well-known analogous property for simplicial sets: the geometric realization of the singular complex of a space is weakly equivalent to the space itself. This will follow once one finds a weak equivalence between the multi-simplicial realization in the question and the realization of that singular complex. If you are comfortable with Tom Goodwillie's comment above on the diagonal of a $k$-fold multi-simplicial set, and just want to see the final argument, skip to section 3 below. Sections 1 and 2 just set up the details of the problem (mostly for my own benefit) and, for convenience, are quickly summarized in section 3.
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### 1. Background and terminology ###
First of all, let me fix some notation and background. Feel free to skip to the next section. Let $\SimplSet$ be the category of *simplicial sets*, i.e. the category of presheaves on the simplicial category $\Delta$. Further, let $\MultiSimplSet{k}$ be the category of *$k$-fold multi-simplicial sets*, also abbreviated as just *$k$-fold simplicial sets*. These are presheaves on the $k$-fold power of the simplicial category $\MultiDeltaCat{k}$.
Consider the functor
$$ \DeltaDot:\DeltaCat\To\SimplSet $$
which is just the Yoneda embedding for the simplicial category $\Delta$. Next, consider the composition
$$ \MultiDeltaDot{k} : \MultiDeltaCat{k}\overset{(\DeltaDot)^{\times k}}{\To}\SimplSet^{\times k}\overset{\times}{\To}\SimplSet $$
where the second functor is given by the product of simplicial sets. In particular, $\MultiDeltaDot{1}=\DeltaDot$. We now define the *simplicial realization* of a multi-simplicial set $X\in\MultiDeltaCat{k}$ to be the simplicial set given by the coend
$$ \SimplReal{X}=\coend{\MultiDeltaDot{k}}{\MultiDeltaCat{k}}{X} $$
Observe the Yoneda lemma implies the simplicial realization of a simplicial set $X$ is simply $X$ itself: $\SimplReal{X}=X$.
Another Yoneda lemma argument — not specific to $\DeltaCat$ — shows that $\MultiDeltaDot{k}$ is just the left Kan extension of $\DeltaDot$ along the diagonal functor $\diag{\op{\DeltaCat}}^k:\op{\DeltaCat}\to\op{(\MultiDeltaCat{k})}$
$$ \LKE{\diag{\op{\DeltaCat}}^k}{\DeltaDot} = \MultiDeltaDot{k} \label{(Ia)} $$
This isomorphism is induced (by adjunction) by the diagonal map
$$ \DeltaDot \To \MultiDeltaDot{k}\circ\diag{\DeltaCat}^k \label{(Ib)} $$
Objectwise, this map is the diagonal $\simplex{n}\to\simplex{n}^{\times k}$ of the n-simplex $\simplex{n}$. As a consequence of (Ia), we get the well-known natural isomorphism
$$ \SimplReal{X} = (\diag{\op{\DeltaCat}}^k)^\ast X \label{(Ic)} $$
between the simplicial realization of a multi-simplicial set and its diagonal.
Let us now consider the *singular complex* functor
$$ \Sing:\Top\To\SimplSet $$
and its left adjoint, the *geometric realization* functor
$$ \GeomReal{-}:\SimplSet\To\Top $$
To simplify the notation, abbreviate $\GeomReal{\MultiDeltaDot{k}}=\GeomReal{-}\circ\MultiDeltaDot{k}$ and $\GeomReal{\DeltaDot}=\GeomReal{-}\circ\DeltaDot$. It is well-known that geometric realization preserves finite products. In particular, $\GeomReal{\MultiDeltaDot{k}}$ is naturally isomorphic to the following composition
$$ \MultiDeltaCat{k}\overset{\GeomReal{\DeltaDot}^{\times k}}{\To}\Top^{\times k}\overset{\times}{\To}\Top \label{(II)} $$
where the last functor is given by the product of topological spaces.
Define the *multi-singular complex* functor
$$ \MultiSing{k}:\Top \To \MultiSimplSet{k}$$
on a space $X$ as
$$ \MultiSing{k}(X) = \Top(\GeomReal{\MultiDeltaDot{k}},X) \label{(IIIa)} $$
This coincides with the usual singular complex functor $\Sing$ in the case of simplicial sets. Also, given the expression (II) for $\GeomReal{\MultiDeltaDot{k}}$, it follows readily that $\MultiSing{k}(X)$ is naturally isomorphic to the multi-simplicial set defined in the question.
On the other hand, the *multi-geometric realization* of a $k$-fold simplicial set $X$ is defined by
$$ \MultiGeomReal{k}{X} = \coend{\GeomReal{\MultiDeltaDot{k}}}{\MultiDeltaCat{k}}{X} = \GeomReal{\SimplReal{X}} \label{(IIIb)} $$
where the last isomorphism is a consequence of geometric realization being a left adjoint. In view of (Ic), we have
$$ \MultiGeomReal{k}{X} = \GeomReal{(\diag{\op{\Delta}}^k)^\ast X} \label{(IIIc)} $$
as Tom Goodwillie states in a comment above. Importantly, note that this multi-simplicial geometric realization coincides with the usual geometric realization in the case of simplicial sets. Observe also that multi-geometric realization of $k$-fold multi-simplicial sets is left adjoint to the multi-singular complex functor $\MultiSing{k}$:
$$ \MultiGeomReal{k}{-} \leftadjointto \MultiSing{k}{} \label{(IIId)} $$
This is an immediate consequence of the usual Kan extension formalism applied to the definitions above. This adjunction also reduces in the case $k=1$ to the usual adjunction between geometric realization and singular complexes for simplicial sets.
### 2. Reducing the problem ###
Finally, we can rephrase the question. Is the counit of the adjunction (IIId):
$$ \counit{k}{X}:\MultiGeomReal{k}{\MultiSing{k}{X}} \to X \label{(IVa)} $$
a weak equivalence for any topological space $X$? We reduce this a bit further. Recall the diagonal map from (Ib):
$$ \DeltaDot\To\MultiDeltaDot{k}\circ\diag{\DeltaCat}^k $$
This induces upon applying geometric realization, a diagonal map
$$ \GeomReal{\DeltaDot} \To \GeomReal{\MultiDeltaDot{k}}\circ\diag{\DeltaCat}^k \label{(IVb)} $$
By the definition of multi-singular complexes (and the particular case of singular complexes) in (IIIa), we obtain
$$ F_X:(\diag{\op{\DeltaCat}}^k)^\ast(\MultiSing{k}{X}) \To \Sing{X} \label{(IVc)} $$
by composing the preceding map (IVb) with $\Top(-,X)$. This map is natural on topological spaces $X$. Applying geometric realization to this map of simplicial sets, we obtain
$$ \GeomReal{F_X}:\MultiGeomReal{k}{\MultiSing{k}{X}} \To \GeomReal{\Sing{X}} $$
in view of the natural isomorphism (IIIc). It is straightforward to check that the composite
$$ \MultiGeomReal{k}{\MultiSing{k}{X}} \overset{\GeomReal{F_X}}{\To} \GeomReal{\Sing{X}} \overset{\counit{1}{X}}{\To} X \label{(IVd)} $$
is actually just the counit from (IVa). Recall that the second map in (IVd) is a weak equivalence, by the familiar case $k=1$ of the present question. If we now show that the first map in (IVd) is also a weak equivalence, we obtain the desired conclusion that the counit in (IVa) is a weak equivalence. So it remains to show that the above map $\GeomReal{F_X}$ is a weak equivalence, or equivalently, that the map $F_X$ from (IVc) is a weak equivalence for any topological space $X$.
### 3. Summary of previous sections ###
Recall that the $k$-fold multi-simplicial set $\MultiSing{k}{X}$ is the multi-singular complex of a topological space $X$, as described in the question. The diagonal of a $k$-fold multi-simplicial set $Y$ is the simplicial set $(\diag{\op{\DeltaCat}}^k)^\ast Y$. We have seen — and Tom Goodwillie comments above — that the geometric realization of the diagonal of $Y$ is isomorphic to the realization of $Y$ itself:
$$ \MultiGeomReal{k}{Y} = \GeomReal{(\diag{\op{\Delta}}^k)^\ast Y} $$
Applying this to $Y=\MultiSing{k}{X}$, and using the well-known case $k=1$ of the question — i.e. the map $\GeomReal{\Sing{X}}\to X$ is a weak equivalence — we have reduced our task to analysing the map (V) immediately below.
### 4. The argument ###
In conclusion, we are left with showing that the map
$$ F_X:(\diag{\op{\DeltaCat}}^k)^\ast(\MultiSing{k}{X}) \To \Sing{X} \label{(V)} $$
is a weak equivalence for any space $X$.
[[Comment: I got this far before Peter May posted his answer. The reformulation/correction of Peter's argument below finishes the argument by proving that $F_X$ in (V) is a weak equivalence. One just has to use the simplicial realization instead of the geometric realization in the answer below.]]
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[This answer is mostly a long comment to Peter May's answer.]
**Edit:** I have corrected some arrows which were pointing the wrong way.
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I was unable to prove the map $X\to\real{Y}$ that Peter May uses in his answer is always a weak equivalence. Unfortunately, the answer Peter links to gives no details. The result would hold if $Y$ were Reedy cofibrant, but it does not seem to be.
So here is an alternative method which proves directly that the map $\real{SY}\to X$ is a weak equivalence, thus answering the question. It avoids any use of realizations of (multi) simplicial spaces, which are problematic by not always being homotopy invariant. Perhaps there is an easier method. In any case, I want to make clear that Peter's idea to use the multi-simplicial space $Y$ is *absolutely fulcral*. I am essentially dealing with some technical details.
Let $X$ be a space. Recall that the object $Y:\Delta^{\times(k-1)}\to\Top$ from Peter's answer is a multi-simplicial space whose space of $(n_2,\cdots,n_k)$-simplices is the space of maps $\Map(\Delta^{n_2}\times\ldots\times\Delta^{n_k},X)$. Let $cX$ be the constant $(k-1)$-fold multi-simplicial space with value $X$. Then we have an obvious map $cX\to Y$ which is objectwise a homotopy equivalence — as Peter remarks — due to the contractibility of products of simplices.
Let $S:\Top\to\sSet$ be the singular complex functor taking a space $Z$ to the simplicial set whose set of $n$-simplices is the set of maps $\Top(\Delta^n,Z)$. Applying $S$ objectwise to $cX$ and $Y$, we get a map $S(cX)\to SY$ (where $SY$ actually stands for $S\circ Y$, and similarly for $cX$). Since $S$ preserves weak equivalences, then $S(cX)\to SY$ is objectwise a weak equivalence. Now we conclude the map on realizations $\real{S(cX)}\to\real{SY}$ is an equivalence. This follows from the fact that any functor $\Delta^{\times(k-1)}\to\sSet$ is Reedy cofibrant: this follows from a straightforward generalization of proposition 15.8.7 of Hirschhorn; alternatively, it is a particular case of the main theorem (proposition 3.15) of the article <a href="http://arxiv.org/abs/1110.1066">*"Reedy categories and the $\Theta$-construction"*</a> by Charles Rezk and Julia Bergner.
Let us now view functors $\Delta^{\times(k-1)}\to\sSet$ as $k$-fold multisimplicial sets instead. As Peter observes, $SY$ is then just the $k$-fold multisimplicial set described in the question. So it suffices to prove that the natural map $\real{SY}\to X$ is a weak equivalence. But we already saw above that $\real{S(cX)}\to\real{SY}$ is a weak equivalence. Moreover, $\real{S(cX)}=\real{\diag(S(cX))}=\real{SX}$ as Tom Goodwillie observes in a comment above (together with the simple fact that $\diag(S(cX))=SX$). But $\real{SX}$ is well-known to be weakly equivalent to $X$ (the case $k=1$ of the question). Finally, this weak equivalence $\real{SX}\to X$ is the composite of the maps
$$ \real{SX}=\real{S(cX)}\To\real{SY}\To X $$
and the first map is a weak equivalence. In conclusion, the second map $\real{SY}\To X$ is also a weak equivalence, as desired.