Homotopy Theory

Abstract

Homotopy theory is the branch of algebraic topology that studies topological spaces and continuous function up to continuous deformation. It replaces strict equality with the existence of a continuous path or deformation between objects, forming the geometric intuition that is generalized by categorical homotopy and homotopy type theory.

Definition

Two continuous maps $f,g:X\to Y$ are homotopic if there exists a continuous map $H:X\times I\to Y$, where $I=[0,1]$, such that $H(x,0)=f(x)$ and $H(x,1)=g(x)$ for all $x\in X$.

We denote this relation by $f\simeq g$. A continuous map $f:X\to Y$ is a homotopy equivalence if there exists a map $g:Y\to X$ such that $g\circ f\simeq {\mathrm{id}}_X$ and $f\circ g\simeq {\mathrm{id}}_Y$. Spaces $X$ and $Y$ are homotopy equivalent if such maps exist, which we interpret as them having the same underlying homotopy type.

Algebraic Invariants

To classify spaces up to homotopy equivalence, we construct algebraic invariants that are preserved under homotopy.

Homotopy Groups

The fundamental group ${\pi }_1(X,x_0)$ captures one-dimensional loops up to homotopy. Higher homotopy groups ${\pi }_n(X,x_0)$ consist of homotopy classes of basepoint-preserving maps from the $n$-sphere $S^n$ to $X$.

$\begin{array}{rl}{\pi }_1(X,x_0)& =[(S^1,\ast ),(X,x_0)]\\ {\pi }_n(X,x_0)& =[(S^n,\ast ),(X,x_0)]\end{array}$

Whitehead’s theorem establishes that a map between CW complexes is a homotopy equivalence if and only if it induces isomorphisms on all homotopy groups.

References

• Hatcher, A. Algebraic Topology.

• May, J. P. A Concise Course in Algebraic Topology.

Would you like me to construct the formal correspondence between the fundamental $\infty$-groupoid of a space and the identity types of a type in HoTT?

See also

Homotopy Type Theory Homotopy Group Path Equivalence Homotopy Equivalence

Source

https://www.youtube.com/watch?v=gVx5KS0kuOg riehl2014-categorical-homotopy-theory