The most usual method of subdivision for a simplicial complex as used in elementary algebraic and geometric topology is the barycentric subdivision. There is however another very well structured subdivision construction encountered which can be useful. The basic geometric construction involves chopping up a geometric $n$-simplex by $n$-planes parallel to a face and halfway between that face and the opposite vertex.

Categorical descriptions

We first define the ordinal subdivision on the simplices, $\Delta[n]$. (Here $\oplus$ denotes the ordinal sum

The ordinal subdivision of $\Delta[n]$, the standard $n$-simplex in simplicial sets, is denoted by $Sd(\Delta[n])$, and is defined as follows:-