Simplicial homotopy

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

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

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

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

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

Using the adjunction

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

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

A simplicial homotopy is in general not an equivalence relation.^[2] However, if ${\displaystyle {\underline {\operatorname {Hom} }}(X,Y)}$ is a Kan complex (e.g., if ${\displaystyle Y}$ is a Kan complex), then a homotopy from ${\displaystyle f:X\to Y}$ to ${\displaystyle g:X\to Y}$ 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 ${\displaystyle X}$  is a simplicial set and ${\displaystyle K}$  a Kan complex, then we form the quotient

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

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

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

An obvious pointed version of the above consideration also holds.

Simplicial homotopy group

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

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

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

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

${\displaystyle \Omega X=\operatorname {Map} _{X}(x_{0},x_{0})}$ 

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

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

as pointed Kan complexes. Thus,

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

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

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

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

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

• 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