Interpretation of HoTT in the Reedy model structure on bisimplicial sets

Question

I was trying to understand the interpretation of HoTT in the Reedy model structure on bisimplicial sets. While going through, it suggests to think of bisimplicial sets as having a "spatial" and a "categorical" direction, where simplicial sets can be embedded in the categorical direction as discrete simplical spaces or in the spatial direction as constant simplicial spaces. I exactly want to understand the meaning of the words, "spatial" and "categorical" in this context of horizontal and vertical directions?

Answer 1

Note that it has basically nothing to do with HoTT and it is simply a point of view on bisimplicial sets, stemming from two different points of view on simplicial sets, themselves emerging from two ways to embed the category $\Delta$.

Given a functor $i : \Delta \to \mathcal C$, one can consider its nerve $N_i : \mathcal C \to \hat\Delta$ given by $X \mapsto \mathcal{C}(i{-}, X)$. If both $i$ and $N_i$ are faithful, one can argue that simplical sets can be seen as a kind of "generalization" of objects of $\mathcal C$, especially when $\mathcal C$ has enough colimits to construct a left adjoint to $N_i$.

This happens, in particular, in the following situations:

1. when $i : \Delta \to \mathsf{Top}$ sends $n$ to the topological simplex of dimension $n$,
2. when $i : \Delta \to \mathsf{Cat}$ sends $n$ to the linear order with $n+1$ elements seen as a category.

So you can see a simplicial sets $X$ as a kind of generalized space: it is built from the points in $X_0$, by connecting them with paths according to the elements in $X_1$, themselves being the borders of triangles determined by elements in $X_2$, etc. But you can also see $X$ as a kind of generalized category where the "objects" are the elements of $X_0$, the "morphisms" are the elements in $X_1$ with source and target defined by the face maps of $X$, the elements of $X_2$ are witnesses of compositions of the "morphisms", etc.

Given a bisimplicial set $S : \Delta^\mathrm{op} \times \Delta^\mathrm{op} \to \mathsf{Set}$, consider all the simplicial sets $S(-,n)$ as generalized spaces, and all the simplicial sets $S(n,-)$ as generalized categories. Then if you have a simplicial set $X$, the bisimplicial set $(n,m) \mapsto X_n$ have all its generalized spaces being $X$ itself (this is the constant simplicial space you mention). On the contrary, the bisimplicial set $(n,m) \mapsto X_m$ have all its generalized category being $X$ itself but all its generalized spaces are just sets (discrete spaces).