In algebraic topology, a simplicial homotopy is an analog of a homotopy between topological spaces for simplicial sets. Precisely,pg 23 if
f , g : X → 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}
such that the restriction of 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}} is f {\displaystyle f} and the restriction along 1 {\displaystyle 1} is g {\displaystyle g} ; see [1]. In particular, f ( x ) = h ( x , 0 ) {\displaystyle f(x)=h(x,0)} and 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))} , the simplicial homotopy h {\displaystyle h} can also be thought of as a path in the simplicial set Hom _ ( X , Y ) . {\displaystyle {\underline {\operatorname {Hom} }}(X,Y).}
A simplicial homotopy is in general not an equivalence relation. However, if Hom _ ( X , Y ) {\displaystyle {\underline {\operatorname {Hom} }}(X,Y)} is a Kan complex (e.g., if Y {\displaystyle Y} is a Kan complex), then a homotopy from f : X → Y {\displaystyle f:X\to Y} to g : X → Y {\displaystyle g:X\to Y} is an equivalence relation. 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} is a simplicial set and K {\displaystyle K} a Kan complex, then we form the quotient
[ X , K ] = Hom ( X , K ) / ∼ {\displaystyle [X,K]=\operatorname {Hom} (X,K)/\sim }
where f ∼ g {\displaystyle f\sim g} means 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} to 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} between Kan complexes is then called a simplicial homotopy equivalence if the homotopy class [ f ] {\displaystyle [f]} of it is bijective; i.e., there is some g {\displaystyle g} such that f g ∼ id L {\displaystyle fg\sim \operatorname {id} _{L}} and g f ∼ id K {\displaystyle gf\sim \operatorname {id} _{K}} . An obvious pointed version of the above consideration also holds.
Simplicial homotopy group Let S 1 {\displaystyle S^{1}} be the pushout Δ 1 ⊔ ∂ Δ 1 1 {\displaystyle \Delta ^{1}\sqcup _{\partial \Delta ^{1}}1} along the boundary S 0 = ∂ Δ 1 {\displaystyle S^{0}=\partial \Delta ^{1}} and S n = S 1 ∧ ⋯ ∧ 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]}
for each pointed Kan complex X and an integer n ≥ 0 {\displaystyle n\geq 0} . It is the n-th simplicial homotopy group of X (or the set for n = 0 {\displaystyle n=0} ). For example, each class in π 0 X {\displaystyle \pi _{0}X} amounts to a path-connected component of X {\displaystyle X} . If 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})}
from the base point to itself is also a Kan complex called the loop space of 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} . It can be shown
Ω n X = 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.}
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)} for the homotopy category τ ( C ) {\displaystyle \tau (C)} of an ∞-category C and an endomorphism group is a group. So, π n X {\displaystyle \pi _{n}X} is a group for n ≥ 1 {\displaystyle n\geq 1} . By the Eckmann-Hilton argument, π n X {\displaystyle \pi _{n}X} is abelian for n ≥ 2 {\displaystyle n\geq 2} . An analog of Whitehead's theorem holds: a map 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} , π n ( f ) {\displaystyle \pi _{n}(f)} is bijective.
See also Kan complex Dold–Kan correspondence (under which a chain homotopy corresponds to a simplicial homotopy) Simplicial homology Homotopy category of an ∞-category
Notes
References Joyal, André; Tierney, Myles (2008). "Notes on simplicial homotopy theory" (PDF). Quillen, Daniel G. (1967), Homotopical algebra, Lecture Notes in Mathematics, No. 43, vol. 43, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0097438, ISBN 978-3-540-03914-3, MR 0223432 Cisinski, Denis-Charles (2023). Higher Categories and Homotopical Algebra (PDF). Cambridge University Press. ISBN 978-1108473200.
External links Simplicial homotopy