Aja viide. Aja lugu.

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\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 [1]. 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.^[2] 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.^[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 $X$ 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 $gf\sim \operatorname {id} _{K}$ .^[3]

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 $n\geq 0$ .^[4] 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 $X$ .

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^[5]

\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 $n\geq 1$ . By the Eckmann-Hilton argument, $\pi _{n}X$ is abelian for $n\geq 2$ .

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 $n\geq 0$ , $\pi _{n}(f)$ is bijective.^[6]

References

^ Goerss, Paul G.; Jardin, John F. (2009). Simplicial Homotopy Theory. Birkhäuser Basel. ISBN 978-3-0346-0188-7. OCLC 837507571.
^ Joyal & Tierney 2008, § 2.4.
^ ^a ^b Joyal & Tierney 2008, § 3.2.
^ Joyal & Tierney 2008, § 4.2.
^ Cisinski 2023, (3.8.8.6)
^ Joyal & Tierney 2008, Theorem 4.4.2.

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

[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.

[homotopy-3] Joyal & Tierney 2008, § 3.2.

[4] Joyal & Tierney 2008, § 4.2.

[5] Cisinski 2023, (3.8.8.6)

[6] Joyal & Tierney 2008, Theorem 4.4.2.

[1]

[2]

[3]

[4]

[5]

[6]

E	T	K	N	R	L	P
	1	2	3	4	5	6
7	8	9	10	11	12	13
14	15	16	17	18	19	20
21	22	23	24	25	26	27
28	29	30

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Simplicial homotopy equivalence

Simplicial homotopy group

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References

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