Simplicial Homotopy Equivalence

In algebraic topology, a simplicial homotopy is an analog of a homotopy between topological spaces for simplicial sets. Precisely,^{pg 23} if
:$f, g: X \to Y$
are maps between simplicial sets, a simplicial homotopy from ''f'' to ''g'' is a map
:$h: X \times \Delta^ \to Y$
such that the restriction of $h$ along $X \simeq X \times \Delta^ \overset\hookrightarrow X \times \Delta^$ is $f$ and the restriction along $1$ is $g$; se

In particular, $f(x) = h(x, 0)$ and $g(x) = h(x, 1)$ for all ''x'' in ''X''.

Using the adjunction
:$\operatorname(X \times \Delta^1, Y) = \operatorname(\Delta^1 \times X, Y) = \operatorname(\Delta^1, \underline(X, Y))$,
the simplicial homotopy $h$ can also be thought of as a path in the simplicial set $\underline(X, Y).$

A simplicial homotopy is in general not an equivalence relation. However, if $\underline(X, Y)$ 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. 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
:$$, K= \operatorname(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 /math> of it is bijective; i.e., there is some $g$ such that $fg \sim \operatorname_L$ and $gf \sim \operatorname_K$.

An obvious pointed version of the above consideration also holds.

Simplicial homotopy group

Let $S^1$ be the pushout $\Delta^1 \sqcup_ 1$ 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 = ^n, X/math>
for each pointed Kan complex ''X'' and an integer $n \ge 0$. 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
:$\Omega X = \operatorname_X(x_0, x_0)$
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
:$\Omega^n X = \underline(S^n, X)$
as pointed Kan complexes. Thus,
:$\pi_n X = \pi_0 \Omega^n X.$
Now, we have the identification $\pi_0 \operatorname_C(x, y) = \operatorname_(x, y)$ 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.

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

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External links

Simplicial homotopy

Homotopy theory
Simplicial sets

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