In higher category theory in mathematics, the subdivision of simplicial sets (subdivision functor or Sd functor) is an endofunctor on the category of simplicial sets. It refines the structure of simplicial sets in a purely combinatorical way without changing constructions like the geometric realization. Furthermore, the subdivision of simplicial sets plays an important role in the extension of simplicial sets right adjoint to it.
Definition.
For a partially ordered set formula_1, let formula_2 be the set of non-empty finite totally ordered subsets, which itself is partially ordered by inclusion. Every partially ordered set can be considered as a category. Postcomposition with the nerve formula_3 defines the subdivision functor formula_4 on the simplex category by:
 formula_5
On the full category of simplicial sets, the subdivision functor formula_6, similar to the geometric realization, is defined through an extension by colimits. For a simplicial set formula_7, one therefore has:
 formula_8
With the maximum formula_9, which in partially ordered sets neither has to exist nor has to be unique, which both holds in totally ordered sets, there is a natural transformation formula_10 by extension. In particular there is a canonical morphism formula_11 for every simplicial set formula_7.
Sd∞ functor.
For a simplicial set formula_7, the canonical morphism formula_14 indudes an formula_15-shaped cocone formula_16, whose colimit is denoted:
 formula_17
Since limit and colimit are switched, there is no adjunction formula_18 with the Ex∞ functor.
The natural transformation formula_19 induces a natural transformation formula_20. In particular, there is a canonical morphism formula_21 for every simplicial set formula_7.
Examples.
Directly from the definition, one has:
 formula_23
 formula_24
Since formula_25, it is fixed under (infinite) subdivision:
 formula_26
 formula_27
 Using formula_35 with formula_36 results in the definition again.
 Since both functors are defined through extension by colimits, it is sufficient to show formula_46.