I present you a family of posets here. I don't say the posets themselves have a conventional name. However I'm sure the general construction of this kind has received some terminology, related to Grothendieck construction, comma categories, category of elements, etc. I can't exactly nail it, so it would be very helpful if you could do it for me.


As usual, write $\Delta$ for the simplex category: the category of inhabited finite linearly ordered set and order-preserving sets. Let $\Delta\_-$ and $\Delta\_+$ denote the wide subcategory of degeneracy maps and face maps, respectively, of $\Delta$.


Let $\sigma$ be a simplex of the simplicial nerve $N(\Delta\_-)$:
$$
\sigma\colon [n\_0]\twoheadrightarrow [n\_1]
 \twoheadrightarrow \dotsb \twoheadrightarrow [n\_k],
$$
where each $\twoheadrightarrow$ lies in $\Delta\_-$.
I would like to define a poset $P(\sigma)$ as:
$$
P(\sigma) := \left\{ \tau \overset{d}{\rightarrowtail} \tau' \overset{u}{\subset} \sigma \right\}.
$$
Before we define an order on this set, we need to clearify the meaning of the symbols here. Firstly $\tau' \overset{u}{\subset} \sigma$ denotes a face $\tau'$ of $\sigma$ in the nerve $N(\Delta\_-)$. It is determined by a face map $u\colon [l] \rightarrowtail [k]$ in $\Delta\_+$, and we have
$$
\tau' = u^\*(\sigma)\colon [n\_{u(0)}]\twoheadrightarrow [n\_{u(1)}]
 \twoheadrightarrow \dotsb \twoheadrightarrow [n\_{u(l)}].
$$
Secondly, $\tau \overset{d}{\rightarrowtail} \tau'$ denotes the vertex-wise family of face maps, i.e.
$$
\tau\colon [m\_0]\twoheadrightarrow [m\_1]
 \twoheadrightarrow \dotsb \twoheadrightarrow [m\_l]
$$
is a diagram in $\Delta\_-$ and $d\_i\colon [m\_i] \rightarrowtail [n\_{u(i)}]$, for $i=0,1,\dotsc, l$, are face maps in $\Delta\_+$ which, in $\Delta$, commutes all the squares.


We need to define an order on $P(\sigma)$. Given $\tau \overset{d}{\rightarrowtail} \tau' \overset{u}{\subset} \sigma$ and $\omega \overset{e}{\rightarrowtail} \omega' \overset{v}{\subset} \sigma$,
we say
$$
\left(\omega \overset{e}{\rightarrowtail} \omega' \overset{v}{\subset} \sigma\right) \le \left(\tau \overset{d}{\rightarrowtail} \tau' \overset{u}{\subset} \sigma\right),
$$
iff we have $\omega \overset{\exists f}{\rightarrowtail} \exists\omega'' \overset{\exists w}{\subset} \tau$ in the commutative way, i.e. $v=u\circ w$ in $\Delta\_-$ and
$e\_i = d\_{w(i)}\circ f\_i$.


I feel a strong déjà-vu looking at this, but I can't write it down into a conventional categorical construction. This clearly looks like a slice category, so if we can name the category with its objects simplices of the simplicial nerve $N(\Delta\_-)$ and its morphisms $\bullet \rightarrowtail \bullet \subset \bullet$, we are done. However I can't go beyond that point, so your help would be very helpful.



What is the name for the construction of this poset related to coherence of degeneracies of the simplex category?
Here’s one way to see it, if I’m not misunderstanding your definition.


* For a small category $\newcommand{\C}{\mathbf{C}}\C$, take its *categorical nerve* $\newcommand{\N}{\mathbf{N}}\N\C$ to be the functor $\newcommand{\op}{\mathrm{op}} \Delta^{\op} \to \mathbf{Cat}$ defined by $(\N\C)\_k = \C^{[k]}$; and take its *semi-nerve* $\N\_{+}\C$ to be the restriction of this to to $\Delta\_{+}$.
* The Grothendieck construction $\int\_{\Delta\_{+}} \N\_{+}\C$ is a split fibration over $\Delta\_{+}$. Its objects are strings $\sigma\_0 \to \cdots \to \sigma\_n$ in $\C$; its morphisms are $(\tau \overset{g}{\to} \tau' \overset{u}{\subseteq} \sigma)$, where $u$ is a face map and $\tau'$ is the restriction of $\sigma$ along $u$, and $g$ is a ladder in $\C$ from $\tau$ to $\tau'$.


Taking $\C := \Delta$, this is very nearly the category you describe in your last para (and so its slices are nearly the category you want overall), but it’s a bit more general: its objects are arbitrary strings in $\Delta$ (not just degeneracies) and its vertical maps are arbitrary ladders, not necessarily of face maps.


So we cut down to a subcategory. This can be done already at the level of the functor $\N\_{+}\C : \Delta\_{+}^{\op} \to \mathbf{Cat}$, before taking the Grothendieck construction. Suppose $\C$ has two distinguished wide subcategories of maps; call them $a$, $b$. Then write $\N^{a,b}\C : \Delta^\op \to \mathbf{Cat}$ (and $\N\_{+}^{a,b}$ similarly) for the functor where $(\N^{a,b}\C)\_k$ is the subcategory of $\C^[k]$ whose objects are strings of $a$-maps, and whose arrows are ladders of $b$-maps.


Then $\int\_{\Delta\_{+}} \N\_{+}^{-,+}\Delta$ is the category you describe in the last paragraph; and its slices are the posets you want overall.


In particular, $\Delta$ is playing two different roles here, which can be generalised separately:


* the *base* of the categorical semi-nerve, $\Delta\_{+}$ along with its inclusion into $\mathbf{Cat}$;
* the *target* of the categorical semi-nerve, $\Delta$ with its two distinguished wide subcategories.


(This categorical (semi-)nerve is an instance of a well-established construction, the [generalised nerve/realisation](https://ncatlab.org/nlab/show/nerve+and+realization); and the Grothendieck construction is of course very standard. Cutting down to a subcategory of the nerve in this particular way is not something I’ve seen before.)


