We finished some elementary discussion of monad, in the usual category.

$\gdef\bDelta{\mathbf{\Delta}}$ $\gdef\colim{\text{colim}}$

We start with simplex category $\Delta$, with objects $[0], [1], \cdots$, and order preserving morphisms.

A simplicial set is a functor $X: \Delta^{op} \to Set$. The set $X([0])$, or denoted as $X_0$, is the set of vertices, $X_1$ the set of edges. Given $[0] \to [1]$ mapping to the first vertices, we have the corresponding $X_1 \to X_0$, extracting the vertex in the edge labelled by $0$. These are called 'pull-back' maps, or face maps.

• $\Delta^n$ is the sSet given by $Map_{\Delta}(-, [n])$.
• Horn: $\Lambda^n_i$ is the sub sSet of $\Delta^n$, that avoid the facet opposite to vertex $i$.
• mapping sSet: given two sSets $X,Y$, we denote $Map(X,Y)_n = Map_{sSet}(\Delta^n \times X, Y)$. The usual mapping set is $Map(X,Y)_0$. This sSet remembers more information.
• $\bDelta^n$ is the geometrical $n$-dim simplex.

We have geometric realization of a simplicial set, it is a functor $|-|: sSet \to Top$, that $$ |X| = \colim_{\Delta^n \to X} \bDelta^n $$

Actually, I am not sure, what's going on here. What is the indexing category? It is like, we have $Map(pt, X)_n$ the mapping sSet from a point to $X$. It is a category with objects $[0], [1], \cdots$