What is the name for the construction of this poset related to coherence of degeneracies of the simplex category?

Question

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.

Answer 1

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; 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.)