The simplicial set with a unique non-degenerate simplex in each dimension

Question

There is

a unique simplicial set with a unique non-degenerate simplex in each dimension, (updated) and such that all faces of the non-degenerate simplex are non-degenerate.

Does it have a name, and what can be said about it ? What is it geometrically ? Can one think of it as a "fat" point ?

I have seen it referred to as the dunce's hat but I cannot find a reference. An explicit construction is as follows: $ S_n $ is the set of equivalence relations on $\{0,1,\dotsc,n\}$ such that each equivalence class is an interval (i.e. of form $[a,a+1,\dotsc,b]$ for some $a$, $b$). In another notation, $S_n := \bigsqcup_{m\leq n} \operatorname{Hom}_\text{surj}(n,m)$ is the set of order-preserving surjections from $n$ to $m$, $m\leq n$ (i.e. $\operatorname{Hom}_\text{surj}(n,m)$ is the set of all order-preserving surjections from the linear order $n$ to linear order $m$).

(Update). By the answer of Dmitri Pavlov $S$ is weakly contractible. A comment by Reid Barton suggested $S$ is something like an "ordered" classifying space (BG) for the standard representation of the group of automorphisms of a dense linear order. Namely, $$S_n := Hom_{orders} (n^\leq, \Bbb Q^\leq)/Aut(\Bbb Q^\leq)$$ That is, (a representative of) an $n$-simplex is an ordered tuple $x_0\leq ... \leq x_n $, and two tuples $(x_0\leq ... \leq x_n)$ and $(y_0\leq...\leq y_n)$ are considered equal iff there is an order preserving map $g:\Bbb Q\to \Bbb Q$ such that $y_i=gx_i$,$0\leq i \leq n$. (Such a $g$ exists iff both tuples give rise to the same equivalence relation $i\approx j$ iff $x_i=x_j$.)

Does this construction have a name ? What is a reference and correct terminology for the classifying spaces of group representations (actions) ?

A related question:

let $\mathrm{Eq}_\bullet$ be the simplicial set where $\mathrm{Eq}_n$ is the set of equivalence relations on $\{0,1,\dotsc,n\}$. Equivalently, it is something like a classifying space for the standard representation of the symmetric group of an infinite set $X$: $$Eq_n := Hom_{Sets}(n, X)/Aut(X)$$

Where can I read about this simplicial set ?

Note it classifies equivalence relations in the following sense: to give an equivalence relation on a set $X$ is the same as to give a morphism to $\mathrm{Eq}_\bullet$ from the simplicial set represented by $X$.

Answer 1

As already pointed out in the comments, such a simplicial set is highly nonunique. For example, in addition to the simplicial set $S$ described in the second paragraph one could take the wedge of simplicial spheres, with one sphere for every dimension.

If we concentrate our attention on the simplicial set $S$ described in the second paragraph, it is easy to show that $S$ is weakly contractible. For example, it has a single vertex, therefore it is connected and by writing down an explicit presentation for the fundamental group we see that the fundamental group is trivial. Next, computing the chain complex of normalized simplicial chains on $S$ with coefficients in an abelian group $A$, we get the chain complex $$A←A←A←A←⋯,$$ where the differentials alternate between zero and identity on $A$. This chain complex is contractible, which proves that $S$ is weakly contractible.