ホモトピー型理論
[002A]

iiを階数とする。ReflGraph(i)\mathord {\textnormal {\textsf {ReflGraph}}}(i) ([008C])の同一視型を特徴付ける。A:ReflGraph(i)A:\mathord {\textnormal {\textsf {ReflGraph}}}(i)Z:ReflGraph(i)Z:\mathord {\textnormal {\textsf {ReflGraph}}}(i)に対し、E(Z):U(i)E(Z):\mathcal {U}(i)を次のレコード型と定義する。

  • -v:A.VertexZ.Vertexv:A.\mathord {\textnormal {\textsf {Vertex}}}\simeq Z.\mathord {\textnormal {\textsf {Vertex}}}
  • -e:{x1,x2:A.Vertex}A.Edge(x1,x2)Z.Edge(v(x1),v(x2))e:\prod _{\lbrace x_{1},x_{2}:A.\mathord {\textnormal {\textsf {Vertex}}}\rbrace }A.\mathord {\textnormal {\textsf {Edge}}}(x_{1},x_{2})\simeq Z.\mathord {\textnormal {\textsf {Edge}}}(v(x_{1}),v(x_{2}))
  • -r:x:A.Vertexe(A.refl(x))=Z.refl(v(x))r:\prod _{x:A.\mathord {\textnormal {\textsf {Vertex}}}}e(A.\mathord {\textnormal {\textsf {refl}}}(x))=Z.\mathord {\textnormal {\textsf {refl}}}(v(x))
要素record{ ⁣vid,eλ(x1,x2).id,rλx.refl ⁣}:E(A)\mathord {\textnormal {\textsf {record}}}\mathopen {\{ \negmedspace |}v\equiv \mathord {\textnormal {\textsf {id}}},e\equiv \lambda (x_{1},x_{2}).\mathord {\textnormal {\textsf {id}}},r\equiv \lambda x.\mathord {\textnormal {\textsf {refl}}}\mathclose {|\negmedspace \} }:E(A)を得る。レトラクトの列
Z:ReflGraph(i)E(Z)\sum _{Z:\mathord {\textnormal {\textsf {ReflGraph}}}(i)}E(Z)\triangleleft  {並び換え}X:U(i)v:A.VertexXYezx_\sum _{X:\mathcal {U}(i)}\sum _{v:A.\mathord {\textnormal {\textsf {Vertex}}}\simeq X}\sum _{Y}\sum _{e}\sum _{z}\prod _{x}\_\triangleleft  {一価性}Y:A.VertexA.VertexU(i)e:x1,x2:A.VertexA.Edge(x1,x2)Y(x1,x2)zx_\sum _{Y:A.\mathord {\textnormal {\textsf {Vertex}}}\to A.\mathord {\textnormal {\textsf {Vertex}}}\to \mathcal {U}(i)}\sum _{e:\prod _{x_{1},x_{2}:A.\mathord {\textnormal {\textsf {Vertex}}}}A.\mathord {\textnormal {\textsf {Edge}}}(x_{1},x_{2})\simeq Y(x_{1},x_{2})}\sum _{z}\prod _{x}\_\triangleleft  {関数外延性と一価性}z:x:A.VertexA.Edge(x,x)x:A.VertexA.refl(x)=z(x)\sum _{z:\prod _{x:A.\mathord {\textnormal {\textsf {Vertex}}}}A.\mathord {\textnormal {\textsf {Edge}}}(x,x)}\prod _{x:A.\mathord {\textnormal {\textsf {Vertex}}}}A.\mathord {\textnormal {\textsf {refl}}}(x)=z(x)
を得て、最後の型は関数外延性により可縮なので、Z:ReflGraph(i)E(Z)\sum _{Z:\mathord {\textnormal {\textsf {ReflGraph}}}(i)}E(Z)は可縮である。