Homotopy

Idea

A homotopy is a continuous deformation between continuous maps.

Definition

Let $\mathcal{T}op$ be the category of topological spaces
Given $f,g:X\to Y$ in $\mathcal{T}op$, a homotopy between $f$ and $g$ is a continuous map,

$$H:X\times \mathbb{I}\to Y$$

such that,

$$H(x,0)=f(x)H(x,1)=g(x)\forall x\in X.$$

where $\mathbb{I}$ is the Unit Interval.

See also

Homotopy Type Theory
Homotopy Group
Path Equivalence
Homotopy Equivalence

Source

https://www.youtube.com/watch?v=gVx5KS0kuOg
hott2013-book