Simplicial homotopy

In algebraic topology, a simplicial homotopy is an analog of a homotopy between topological spaces for simplicial sets. Precisely,{{Cite book|last1=Goerss|first1=Paul G.|url=http://worldcat.org/oclc/837507571|title=Simplicial Homotopy Theory|last2=Jardin|first2=John F.|date=2009|publisher=Birkhäuser Basel|isbn=978-3-0346-0188-7|oclc=837507571}}^{pg 23} if

: $f, g: X \to Y$

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

: $h: X \times \Delta^{1} \to Y$

such that the restriction of $h$ along $X \simeq X \times \Delta^{0} \overset{0}\hookrightarrow X \times \Delta^{1}$ is $f$ and the restriction along $1$ is $g$ ; see [https://books.google.com/books?id=iFv2BwAAQBAJ&dq=%22simplicial+homotopy%22&pg=PA23]. In particular, $f(x) = h(x, 0)$ and $g(x) = h(x, 1)$ for all x in X.

Using the adjunction

: $\operatorname{Hom}(X \times \Delta^1, Y) = \operatorname{Hom}(\Delta^1 \times X, Y) = \operatorname{Hom}(\Delta^1, \underline{\operatorname{Hom}}(X, Y))$ ,

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

A simplicial homotopy is in general not an equivalence relation.{{harvnb|Joyal|Tierney|2008|loc=§ 2.4.}} However, if $\underline{\operatorname{Hom}}(X, Y)$ is a Kan complex (e.g., if $Y$ is a Kan complex), then a homotopy from $f : X \to Y$ to $g : X \to Y$ is an equivalence relation.{{harvnb|Joyal|Tierney|2008|loc=§ 3.2.}} 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 $X$ is a simplicial set and $K$ a Kan complex, then we form the quotient

: $[X, K] = \operatorname{Hom}(X, K)/\sim$

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

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

An obvious pointed version of the above consideration also holds.

Simplicial homotopy group

Let $S^1$ be the pushout $\Delta^1 \sqcup_{\partial \Delta^1} 1$ along the boundary $S^0 = \partial \Delta^1$ and $S^n = S^1 \wedge \cdots \wedge S^1$ n-times. Then, as in usual algebraic topology, we define

: $\pi_n X = [S^n, X]$

for each pointed Kan complex X and an integer $n \ge 0$ .{{harvnb|Joyal|Tierney|2008|loc=§ 4.2.}} It is the n-th simplicial homotopy group of X (or the set for $n = 0$ ). For example, each class in $\pi_0 X$ amounts to a path-connected component of $X$ .{{harvnb|Cisinski|2023|loc=Proposition 3.1.31.}}

If $X$ is a pointed Kan complex, then the mapping space

: $\Omega X = \operatorname{Map}_X(x_0, x_0)$

from the base point to itself is also a Kan complex called the loop space of $X$ . It is also pointed with the base point the identity and so we can iterate: $\Omega^n X$ . It can be shown{{harvnb|Cisinski|2023|loc=(3.8.8.6)}}

: $\Omega^n X = \underline{\operatorname{Hom}}(S^n, X)$

as pointed Kan complexes. Thus,

: $\pi_n X = \pi_0 \Omega^n X.$

Now, we have the identification $\pi_0 \operatorname{Map}_C(x, y) = \operatorname{Hom}_{\tau(C)}(x, y)$ for the homotopy category $\tau(C)$ of an ∞-category C and an endomorphism group is a group. So, $\pi_n X$ is a group for $n \ge 1$ . By the Eckmann-Hilton argument, $\pi_n X$ is abelian for $n \ge 2$ .

An analog of Whitehead's theorem holds: a map $f$ between Kan complexes is a homotopy equivalence if and only if for each choice of base points and each integer $n \ge 0$ , $\pi_n(f)$ is bijective.{{harvnb|Joyal|Tierney|2008|loc=Theorem 4.4.2.}}

Notes

References

{{cite web|first1=André|last1=Joyal|author1-link=André Joyal|first2= Myles|last2= Tierney|author2-link=Myles Tierney|title= Notes on simplicial homotopy theory|url=https://ncatlab.org/nlab/files/JoyalTierneyNotesOnSimplicialHomotopyTheory.pdf|year=2008}}
{{Citation | last=Quillen | first=Daniel G. | authorlink=Daniel Quillen| title=Homotopical algebra | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Mathematics, No. 43 | doi=10.1007/BFb0097438 | year=1967 | mr=0223432 | volume=43| isbn=978-3-540-03914-3 }}
{{cite book |last=Cisinski |first=Denis-Charles |author-link=Denis-Charles Cisinski |url=https://cisinski.app.uni-regensburg.de/CatLR.pdf |title=Higher Categories and Homotopical Algebra |date=2023|publisher=Cambridge University Press |isbn=978-1108473200 |location= |language=en |authorlink=}}

External links

[http://ncatlab.org/nlab/show/simplicial+homotopy Simplicial homotopy]

Category:Homotopy theory

Category:Simplicial sets