The simplicial category $\Delta$ is defined as the small category whose objects are the totally ordered finite sets

and whose morphisms are monotonic non-decreasing (order-preserving) maps. It is generated by two families of morphisms:

The $\delta^{n}_{i}$ morphisms are called face maps, and the $\sigma^{n}_{i}$ morphisms are called degeneracy maps. They satisfy the following relations,

All morphisms $[n]\to[0]$ factor through $\sigma^{0}_{0}$, so [0] is terminal.

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

where $f\colon[m]\to[m^{{\prime}}]$ and $g\colon[n]\to[n^{{\prime}}]$. Sometimes, the simplicial category is defined to include the empty set $[-1]=\emptyset$, which provides an initial object for the category. This makes $\Delta$ a strict monoidal category as $\emptyset$ is a unit for the bifunctor: $\emptyset+[n]=[n]=[n]+\emptyset$ and $\mathrm{id}_{\emptyset}+f=f=f+\mathrm{id}_{\emptyset}$. Further, $\Delta$ is then the free monoidal category on a monoid object (the monoid object being [0], with product $\sigma^{0}_{0}\colon[0]+[0]\to[0]$).

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