The simplicial category $\Delta$ is defined as the small category whose objects are the totally ordered finite sets \begin{equation} [n] = \{0<1<2<\ldots<n\}, \quad n\geq0, \end{equation}and whose morphisms are monotonic non-decreasing (order-preserving) maps. It is generated by two families of morphisms: \begin{eqnarray*} \delta^n_i & \colon & [n-1] \to [n] \quad\mbox{is the injection missing\ } i\in[n], \\ \sigma^n_i & \colon & [n+1] \to [n] \quad\mbox{is the surjection such that\ } \sigma^n_i(i)=\sigma^n_i(i+1)=i\in[n]. \end{eqnarray*}The $\delta^n_i$ morphisms are called face maps, and the $\sigma^n_i$ morphisms are called degeneracy maps. They satisfy the following relations,

		for	(1)
		for	(2)
			(3)

There is a bifunctor $+\colon \Delta\times\Delta \to \Delta$ defined by

			(4)
			(5)

There is a fully faithful functor from $\Delta$ to $\mathcat{Top}$ , which sends each object $[n]$ to an oriented $n$ -simplex. The face maps then embed an $(n-1)$ -simplex in an $n$ -simplex, and the degeneracy maps collapse an $(n+1)$ -simplex to an $n$ -simplex. The bifunctor forms a simplex from the disjoint union of two simplicies by joining their vertices together in a way compatible with their orientations.

There is also a fully faithful functor from $\Delta$ to $\mathcat{Cat}$ , which sends each object $[n]$ to a pre-order $\mathcat{n+1}$ . The pre-order $\mathcat{n}$ is the category consisting of $n$ partially-ordered objects, with one morphism $a \to b$ if and only if $a \leq b$ .