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

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.

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.)