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 -simplex by -planes parallel to a face and halfway between that face and the opposite vertex.
We first define the ordinal subdivision on the simplices, . (Here denotes the ordinal sum
The ordinal subdivision of , the standard -simplex in simplicial sets, is denoted by , and is defined as follows:-
For a general simplicial set we can then define:
the ordinal subdivision of a simplicial set, , is denoted by and is defined by
This expands to
This is related to the total décalage of by
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