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 I {\displaystyle I} , let s ( I ) {\displaystyle s(I)} 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 N : C a t → s S e t {\displaystyle N\colon \mathbf {Cat} \rightarrow \mathbf {sSet} } defines the subdivision functor Sd : Δ → s S e t {\displaystyle \operatorname {Sd} \colon \Delta \rightarrow \mathbf {sSet} } on the simplex category by:
Sd ( Δ n ) := N ( s ( [ n ] ) ) . {\displaystyle \operatorname {Sd} (\Delta ^{n}):=N(s([n])).}
On the full category of simplicial sets, the subdivision functor Sd : s S e t → s S e t {\displaystyle \operatorname {Sd} \colon \mathbf {sSet} \rightarrow \mathbf {sSet} } , similar to the geometric realization, is defined through an extension by colimits. For a simplicial set X {\displaystyle X} , one therefore has:
Sd ( X ) := lim → Δ n → X Sd ( Δ n ) . {\displaystyle \operatorname {Sd} (X):=\varinjlim _{\Delta ^{n}\rightarrow X}\operatorname {Sd} (\Delta ^{n}).}
With the maximum max : s ( I ) → I {\displaystyle \max \colon s(I)\rightarrow I} , 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 a : Sd ⇒ Id {\displaystyle a\colon \operatorname {Sd} \Rightarrow \operatorname {Id} } by extension. In particular there is a canonical morphism a X : Sd ( X ) → X {\displaystyle a_{X}\colon \operatorname {Sd} (X)\rightarrow X} for every simplicial set X {\displaystyle X} .
Sd∞ functor For a simplicial set X {\displaystyle X} , the canonical morphism a X : Sd ( X ) → X {\displaystyle a_{X}\colon \operatorname {Sd} (X)\rightarrow X} indudes an N {\displaystyle \mathbb {N} } -shaped cocone … → Sd 3 ( X ) → Sd 2 ( X ) → Sd ( X ) → X {\displaystyle \ldots \rightarrow \operatorname {Sd} ^{3}(X)\rightarrow \operatorname {Sd} ^{2}(X)\rightarrow \operatorname {Sd} (X)\rightarrow X} , whose colimit is denoted:
Sd ∞ ( X ) := lim ← n ∈ N Sd n ( X ) . {\displaystyle \operatorname {Sd} ^{\infty }(X):=\varprojlim _{n\in \mathbb {N} }\operatorname {Sd} ^{n}(X).}
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![Subdivision (simplicial set): Process of subdivision of the standard
2
{\displaystyle 2}
-simplex
Δ
2
{\displaystyle \Delta ^{2}}
: The partially ordered set
[
2
]
=
{
0
,
1
,
2
}
{\displaystyle [2]=\{0,1,2\}}
with
0
≤
1
{\displaystyle 0\leq 1}
,
1
≤
2
{\displaystyle 1\leq 2}
and
0
≤
2
{\displaystyle 0\leq 2}
forms a triangle, while the partially ordered set
s
(
[
2
]
)
=
{
{
0
}
,
{
1
}
,
{
2
}
,
{
0
,
1
}
,
{
1
,
2
}
,
{
0
,
2
}
,
{
0
,
1
,
2
}
}
{\displaystyle s([2])=\{\{0\},\{1\},\{2\},\{0,1\},\{1,2\},\{0,2\},\{0,1,2\}\}}
forms its subdivision with
{
0
}
{\displaystyle \{0\}}
,
{
1
}
{\displaystyle \{1\}}
and
{
2
}
{\displaystyle \{2\}}
being the original triangle,
{
0
,
1
}
{\displaystyle \{0,1\}}
,
{
1
,
2
}
{\displaystyle \{1,2\}}
and
{
0
,
2
}
{\displaystyle \{0,2\}}
subdividing the edges and
{
0
,
1
,
2
}
{\displaystyle \{0,1,2\}}
subdividing the face.](https://upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Barycentric_subdivision.svg/500px-Barycentric_subdivision.svg.png?utm_source=en.wikipedia.org&utm_campaign=parser&utm_content=thumbnail)
