Given an abstract simplicial complex $K$, one can make a simplicial set $X(K)$ with $n$-simplices given by sequences $(x_0, \ldots, x_n)$ such that $\{x_0, x_1, \ldots, x_n\}$ is a simplex of $K$. The face maps delete entries and the degeneracy maps repeat entries. I'd like a reference for the fact that the geometric realization of $X(K)$ is homotopy equivalent to the geometric realization of $K$ itself. (Note that $|X(K)|$ is typically very big: for $K$ a single edge, $|X(K)|$ is the infinite-dimensional sphere $S^\infty$.)

I've sketched a proof of this fact here, but hope there is a reference I can just cite since, as I expected, every algebraic topologist I've asked in person already knew the fact. :)

Also, does this $X(K)$ have a standard name or notation? Or if not, can someone think of a catchy name or nice notation?

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    $\begingroup$ As for the notation, I think the simplicial set $K(X)$ is strictly linked to symmetric simplicial sets. If $\Upsilon$ denotes the category of symmetric simplices, there is a canonical funtor $v \colon \Delta \to \Upsilon$ which induces a Quillen equivalence pair $(v_!,v^*)$ between the category of presheaves (see §8.3). It seems to me that $K(X)$ is precisely $v_!(K_{\leq}(X))$ which in turn is precisely your $E \otimes_{\Delta} K_{\leq}(X)$. Moreover, you show that the unit $1 \to v^*v_!$ is always a weak homotopy equivalence. $\endgroup$ – Andrea Gagna Mar 26 '19 at 22:39
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    $\begingroup$ If one only reads the question title, one foolish way that comes to mind is this: allow only neighboring repetitions. That is, if $x_i=x_{i+j}$ then $x_i=x_{i+k}$ for all $0<k<j$ too. I wonder if this is also equivalent to $K$... $\endgroup$ – მამუკა ჯიბლაძე Apr 8 '19 at 17:45
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    $\begingroup$ @მამუკაჯიბლაძე I don't think the simplicial set you describe is homotopy equivalent to the complex you start with. For example if $K$ is a single simplex of dimension $n$, then I believe your simplicial set is the simplicial set obtained by freely adjoining degenerate simplicies to the semi-simplicial set known as the "complex of injective words on $n+1$ letters", which is not contractible. Specifically, for $K$ a single interval your simplicial set has the homotopy type of $S^1$. $\endgroup$ – Omar Antolín-Camarena Apr 9 '19 at 18:19
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    $\begingroup$ Concerning terminology and notation - it might be natural to call elements of your $X(K)$ singular simplices of $K$ and, accordingly, denote $X(K)$ by $\operatorname{Sing}(K)$. $\endgroup$ – მამუკა ჯიბლაძე Apr 9 '19 at 21:38
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    $\begingroup$ @მამუკაჯიბლაძე Yes, very appropriate! After all $X(K)_n$ is precisely the set of morphisms of simplicial complexes from the n-simplex to $K$. $\endgroup$ – Omar Antolín-Camarena Apr 9 '19 at 22:38

Let $K$ be a simplicial complex with vertex set $V$. Let $S_\bullet (K)$ be the simplicial set whose p-simplices are the maps $f:[p]\to V$ such that $f([p])$ is a simplex of $K$, or alternatively the set of maps $\Delta^p \to K$ of simplicial complexes. There is an obvious map $$\pi_K:|S_\bullet (K)| \to |K|$$ which you ask to be a homotopy equivalence. Here is an argument. I find it easier to work with the fat geometric realization $||S_\bullet (K)||$ instead, but the difference is minimal, since the quotient map to the ordinary geometric realization is a homotopy equivalence.

Step 1. Consider first the case $K=\Delta^n$ (rather, it is the full simplicial complex with vertex set $[n]$). I claim that $||S_\bullet \Delta^n||$ is contractible. For sake of notational clarity, let me write $\nabla^p$ for the topological $p$-simplex. Consider the map $$H_p: S_p (\Delta^n) \times \nabla^p \times [0,1] \to S_{p+1}(\Delta^n) \times \nabla^{p+1} $$ which is given by the formula $$H(f,v,t):= (f \ast n,((1-t)v,t)). $$ Explanation: $f \ast n: [p+1] \to [n]$ is the map whose restriction to $[p]$ is $f$ and which has $f(p+1)=n$. Furthermore $((1-t)v,t) \in \mathbb{R}^{p+1} \times \mathbb{R}$ is a point of $\nabla^{p+1}$. It is easily checked that the different $H_p$ glue together to a map $H:|| S_\bullet (\Delta^n)|| \times [0,1] \to ||S_{\bullet}(\Delta^n) ||$ (use that products and quotients commute in this setting, as the interval is compact, or work in the context of compactly generated spaces). It is clear that $H(0,\_)$ is the identity, and $H(1,\_)$ is the constant map to the vertex $n$. So we are done in this case.

Step 2. Now we prove the claim for finite complexes, by induction over both, the dimension and the number of top-dimensional simplices. The induction beginning $K=\emptyset$ is trivial. For the induction step, let $K$ be $n$-dimensional and let $L$ be obtained from $K$ by deleting one $n$-simplex. Then $|K| \cong |L| \cup_{|\partial \Delta^n|} |\Delta^n|$ and $||S_\bullet (K)|| \cong ||S_\bullet (L)|| \cup_{||S_\bullet (\partial \Delta^n)||} ||S_\bullet (\Delta^n)||$. The map $\pi_K$ is the pushout of the maps $\pi_{\Delta^n}$ and $\pi_L$, along $\pi_{\partial \Delta^n}$. These maps are homotopy equivalences, by step 1 and by induction hypothesis, respectively. The maps $|\partial \Delta^n| \to |\Delta^n|$ and $||S_\bullet (\partial \Delta^n)|| \to ||S_\bullet (\Delta^n)||$ are cofibrations, and so the gluing lemma implies that $\pi_K$ is a homotopy equivalence.

Step 3. Having shown the claim for finite complexes, it follows by a colimit argument that $\pi_K$ is a weak homotopy equivalence for arbitrary $K$, and hence a homotopy equivalence, by Whiteheads theorem.

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  • $\begingroup$ Thanks Johannes! I think for Step 1 I prefer the argument that goes "$S_\bullet \Delta^n$ is the nerve of the indiscrete category on $n+1$ vertices, and this category is equivalent to the terminal category", but I really like the directness of the rest of the proof. $\endgroup$ – Omar Antolín-Camarena Apr 18 '19 at 17:07
  • $\begingroup$ I'm not sure how surprised I should be that we now have 3 proofs and 0 references for this fact I always thought was "standard". $\endgroup$ – Omar Antolín-Camarena Apr 18 '19 at 17:10

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