Endofunctor on the category of simplicial sets
Process of subdivision of the standard
2
{\displaystyle 2}
Δ
2
{\displaystyle \Delta ^{2}}
[
2
]
=
{
0
,
1
,
2
}
{\displaystyle [2]=\{0,1,2\}}
0
≤
1
{\displaystyle 0\leq 1}
1
≤
2
{\displaystyle 1\leq 2}
0
≤
2
{\displaystyle 0\leq 2}
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\}\}}
{
0
}
{\displaystyle \{0\}}
{
1
}
{\displaystyle \{1\}}
{
2
}
{\displaystyle \{2\}}
{
0
,
1
}
{\displaystyle \{0,1\}}
{
1
,
2
}
{\displaystyle \{1,2\}}
{
0
,
2
}
{\displaystyle \{0,2\}}
{
0
,
1
,
2
}
{\displaystyle \{0,1,2\}}
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.
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![{\displaystyle [2]=\{0,1,2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3f3d205b0da1924e67292bfd643ee205afe63d1)
![{\displaystyle s([2])=\{\{0\},\{1\},\{2\},\{0,1\},\{1,2\},\{0,2\},\{0,1,2\}\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81f09a7ccf486fabf691013d9bed4158742a3293)
