Homotopy of continuous maps

Let $𝑓, 𝑔 : 𝑋 \to 𝑌$ be continuous maps. A homotopy $𝐻 : 𝑓 ≃ 𝑔$ of $𝑓$ into $𝑔$ is, loosely speaking, a continuous deformation of $𝑓$ into $𝑔$ . In point set topology, this is a continuous map

𝕀 \times 𝑋 \to 𝑌 : (𝑖, 𝑥) \mapsto 𝐻 𝑖 (𝑥)

where $𝕀$ is the real interval, satisfying the boundary constraints

\forall (𝑥 \in 𝑋) [𝐻 0 (𝑥) = 𝑓 (𝑥) \land 𝐻 1 (𝑥) = 𝑔 (𝑥)]

The maps $𝑓, 𝑔$ are thereby said to be homotopic, denoted with $𝐻 : 𝑓 ≃ 𝑔$ #m/def/homotopy The propositional truncation of $(≃)$ is a congruence relation on $𝖳 𝗈 𝗉 (𝑋, 𝑌)$ .

Proof

Clearly $𝑓 \in 𝖳 𝗈 𝗉 (𝑋, 𝑌)$ is homotopic to itself via $ℎ (𝑥, 𝑡) = 𝑓 (𝑥)$ , so $≃$ is reflexive. If $ℎ : 𝑋 \times [0, 1] \to 𝑌$ is a homotopy from $𝑓$ to $𝑔$ then $ℎ' : (𝑥, 𝑡) \mapsto ℎ (𝑥, 1 - 𝑡)$ is a homotopy from $𝑔$ to $𝑓$ , so $≃$ is symmetric. If $ℎ$ is a homotopy from $𝑓$ to $𝑔$ and $ℎ'$ is a homotopy from $𝑔$ to $𝑘$ , then

ℎ' \cdot ℎ = {ℎ (2 𝑡) 0 \leq 𝑡 \leq 12 ℎ' (2 𝑡 - 1) 12 \leq 𝑡 \leq 1

is a homotopy from $𝑓$ to $𝑘$ , so $≃$ is transitive. Therefore $≃$ is an equivalence relation. To show $≃$ is a congruence relation, let $𝑓 1, 𝑓 2 : 𝑋 \to 𝑌$ with $ℎ 1 : 𝑓 1 ≃ 𝑓 2$ and $𝑔 1, 𝑔 2 : 𝑌 \to 𝑍$ with $ℎ 2 : 𝑔 1 ≃ 𝑔 2$ . Then $𝑔 2 ℎ 1 : 𝑔 2 𝑓 1 ≃ 𝑔 2 𝑓 2$ , and similarly $ℎ 2 (𝑓 (-), -) : 𝑔 1 𝑓 1 ≃ 𝑔 2 𝑓 1$ . Thus $𝑔 1 𝑓 1 ≃ 𝑔 2 𝑓 2$ , as required.

Homotopy class

The congruence classes of homotopic maps are called homotopy classes of maps, and form the morphisms in $𝖳 𝗈 𝗉 / ≃$ which is a Quotient category $𝗁 𝖳 𝗈 𝗉 = 𝖳 𝗈 𝗉 / ≃$ .

Other kinds of topological homotopy

Further terminology

A map $𝑓$ is said to be null-homotopic iff it is homotopic to a Constant map.

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