Simplicial homotopy

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In algebraic topology, a simplicial homotopy is an analog of a homotopy between topological spaces for simplicial sets. Precisely,[1]pg 23 if

f , g : X → Y {\displaystyle f,g:X\to Y} {\displaystyle f,g:X\to Y}

are maps between simplicial sets, a simplicial homotopy from f to g is a map

h : X × Δ 1 → Y {\displaystyle h:X\times \Delta ^{1}\to Y} {\displaystyle h:X\times \Delta ^{1}\to Y}

such that the restriction of h {\displaystyle h} {\displaystyle h} along X ≃ X × Δ 0 ↪ 0 X × Δ 1 {\displaystyle X\simeq X\times \Delta ^{0}{\overset {0}{\hookrightarrow }}X\times \Delta ^{1}} {\displaystyle X\simeq X\times \Delta ^{0}{\overset {0}{\hookrightarrow }}X\times \Delta ^{1}} is f {\displaystyle f} {\displaystyle f} and the restriction along 1 {\displaystyle 1} {\displaystyle 1} is g {\displaystyle g} {\displaystyle g}; see . In particular, f ( x ) = h ( x , 0 ) {\displaystyle f(x)=h(x,0)} {\displaystyle f(x)=h(x,0)} and g ( x ) = h ( x , 1 ) {\displaystyle g(x)=h(x,1)} {\displaystyle g(x)=h(x,1)} for all x in X.

Using the adjunction

Hom ⁡ ( X × Δ 1 , Y ) = Hom ⁡ ( Δ 1 × X , Y ) = Hom ⁡ ( Δ 1 , Hom _ ( X , Y ) ) {\displaystyle \operatorname {Hom} (X\times \Delta ^{1},Y)=\operatorname {Hom} (\Delta ^{1}\times X,Y)=\operatorname {Hom} (\Delta ^{1},{\underline {\operatorname {Hom} }}(X,Y))} {\displaystyle \operatorname {Hom} (X\times \Delta ^{1},Y)=\operatorname {Hom} (\Delta ^{1}\times X,Y)=\operatorname {Hom} (\Delta ^{1},{\underline {\operatorname {Hom} }}(X,Y))},

the simplicial homotopy h {\displaystyle h} {\displaystyle h} can also be thought of as a path in the simplicial set Hom _ ( X , Y ) . {\displaystyle {\underline {\operatorname {Hom} }}(X,Y).} {\displaystyle {\underline {\operatorname {Hom} }}(X,Y).}

A simplicial homotopy is in general not an equivalence relation.[2] However, if Hom _ ( X , Y ) {\displaystyle {\underline {\operatorname {Hom} }}(X,Y)} {\displaystyle {\underline {\operatorname {Hom} }}(X,Y)} is a Kan complex (e.g., if Y {\displaystyle Y} {\displaystyle Y} is a Kan complex), then a homotopy from f : X → Y {\displaystyle f:X\to Y} {\displaystyle f:X\to Y} to g : X → Y {\displaystyle g:X\to Y} {\displaystyle g:X\to Y} is an equivalence relation.[3] Indeed, a Kan complex is an ∞-groupoid; i.e., every morphism (path) is invertible. Thus, if h is a homotopy from f to g, then the inverse of h is a homotopy from g to f, establishing that the relation is symmetric. The transitivity holds since a composition is possible.

Simplicial homotopy equivalence

If X {\displaystyle X} {\displaystyle X} is a simplicial set and K {\displaystyle K} {\displaystyle K} a Kan complex, then we form the quotient

[ X , K ] = Hom ⁡ ( X , K ) / ∼ {\displaystyle [X,K]=\operatorname {Hom} (X,K)/\sim } {\displaystyle [X,K]=\operatorname {Hom} (X,K)/\sim }

where f ∼ g {\displaystyle f\sim g} {\displaystyle f\sim g} means f , g {\displaystyle f,g} {\displaystyle f,g} are homotopic to each other. It is the set of the simplicial homotopy classes of maps from X {\displaystyle X} {\displaystyle X} to K {\displaystyle K} {\displaystyle K}. More generally, Quillen defines homotopy classes using the equivalence relation generated by the homotopy relation.

A map K → L {\displaystyle K\to L} {\displaystyle K\to L} between Kan complexes is then called a simplicial homotopy equivalence if the homotopy class [ f ] {\displaystyle [f]} {\displaystyle [f]} of it is bijective; i.e., there is some g {\displaystyle g} {\displaystyle g} such that f g ∼ id L {\displaystyle fg\sim \operatorname {id} _{L}} {\displaystyle fg\sim \operatorname {id} _{L}} and g f ∼ id K {\displaystyle gf\sim \operatorname {id} _{K}} {\displaystyle gf\sim \operatorname {id} _{K}}.[3]

An obvious pointed version of the above consideration also holds.

Simplicial homotopy group

Let S 1 {\displaystyle S^{1}} {\displaystyle S^{1}} be the pushout Δ 1 ⊔ ∂ Δ 1 1 {\displaystyle \Delta ^{1}\sqcup _{\partial \Delta ^{1}}1} {\displaystyle \Delta ^{1}\sqcup _{\partial \Delta ^{1}}1} along the boundary S 0 = ∂ Δ 1 {\displaystyle S^{0}=\partial \Delta ^{1}} {\displaystyle S^{0}=\partial \Delta ^{1}} and S n = S 1 ∧ ⋯ ∧ S 1 {\displaystyle S^{n}=S^{1}\wedge \cdots \wedge S^{1}} {\displaystyle S^{n}=S^{1}\wedge \cdots \wedge S^{1}} n-times. Then, as in usual algebraic topology, we define

π n X = [ S n , X ] {\displaystyle \pi _{n}X=[S^{n},X]} {\displaystyle \pi _{n}X=[S^{n},X]}

for each pointed Kan complex X and an integer n ≥ 0 {\displaystyle n\geq 0} {\displaystyle n\geq 0}.[4] It is the n-th simplicial homotopy group of X (or the set for n = 0 {\displaystyle n=0} {\displaystyle n=0}). For example, each class in π 0 X {\displaystyle \pi _{0}X} {\displaystyle \pi _{0}X} amounts to a path-connected component of X {\displaystyle X} {\displaystyle X}.[5]

If X {\displaystyle X} {\displaystyle X} is a pointed Kan complex, then the mapping space

Ω X = Map X ⁡ ( x 0 , x 0 ) {\displaystyle \Omega X=\operatorname {Map} _{X}(x_{0},x_{0})} {\displaystyle \Omega X=\operatorname {Map} _{X}(x_{0},x_{0})}

from the base point to itself is also a Kan complex called the loop space of X {\displaystyle X} {\displaystyle X}. It is also pointed with the base point the identity and so we can iterate: Ω n X {\displaystyle \Omega ^{n}X} {\displaystyle \Omega ^{n}X}. It can be shown[6]

Ω n X = Hom _ ( S n , X ) {\displaystyle \Omega ^{n}X={\underline {\operatorname {Hom} }}(S^{n},X)} {\displaystyle \Omega ^{n}X={\underline {\operatorname {Hom} }}(S^{n},X)}

as pointed Kan complexes. Thus,

π n X = π 0 Ω n X . {\displaystyle \pi _{n}X=\pi _{0}\Omega ^{n}X.} {\displaystyle \pi _{n}X=\pi _{0}\Omega ^{n}X.}

Now, we have the identification π 0 Map C ⁡ ( x , y ) = Hom τ ( C ) ⁡ ( x , y ) {\displaystyle \pi _{0}\operatorname {Map} _{C}(x,y)=\operatorname {Hom} _{\tau (C)}(x,y)} {\displaystyle \pi _{0}\operatorname {Map} _{C}(x,y)=\operatorname {Hom} _{\tau (C)}(x,y)} for the homotopy category τ ( C ) {\displaystyle \tau (C)} {\displaystyle \tau (C)} of an ∞-category C and an endomorphism group is a group. So, π n X {\displaystyle \pi _{n}X} {\displaystyle \pi _{n}X} is a group for n ≥ 1 {\displaystyle n\geq 1} {\displaystyle n\geq 1}. By the Eckmann-Hilton argument, π n X {\displaystyle \pi _{n}X} {\displaystyle \pi _{n}X} is abelian for n ≥ 2 {\displaystyle n\geq 2} {\displaystyle n\geq 2}.

An analog of Whitehead's theorem holds: a map f {\displaystyle f} {\displaystyle f} between Kan complexes is a homotopy equivalence if and only if for each choice of base points and each integer n ≥ 0 {\displaystyle n\geq 0} {\displaystyle n\geq 0}, π n ( f ) {\displaystyle \pi _{n}(f)} {\displaystyle \pi _{n}(f)} is bijective.[7]

See also

Notes

  1. Goerss, Paul G.; Jardin, John F. (2009). Simplicial Homotopy Theory. Birkhäuser Basel. ISBN 978-3-0346-0188-7. OCLC 837507571.
  2. Joyal & Tierney 2008, § 2.4.
  3. Joyal & Tierney 2008, § 3.2.
  4. Joyal & Tierney 2008, § 4.2.
  5. Cisinski 2023, Proposition 3.1.31.
  6. Cisinski 2023, (3.8.8.6)
  7. Joyal & Tierney 2008, Theorem 4.4.2.

References