# Homotopy (object)

From Maths

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## Contents

## Definition

Given topological spaces [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath], and any set [ilmath]A\in\mathcal{P}(X)[/ilmath]^{[Note 1]} a *homotopy (relative to [ilmath]A[/ilmath])* is any continuous function:

- [ilmath]H:X\times I\rightarrow Y[/ilmath] (where [ilmath]I:=[0,1]\subset\mathbb{R}[/ilmath]) such that:
- [ilmath]\forall s,t\in I\ \forall a\in A[H(a,s)=H(a,t)][/ilmath]
^{[Note 2]}

- [ilmath]\forall s,t\in I\ \forall a\in A[H(a,s)=H(a,t)][/ilmath]

If [ilmath]A=\emptyset[/ilmath]^{[Note 2]} then we say [ilmath]H[/ilmath] is a *free homotopy* (or just a *homotopy*).

If [ilmath]A\neq \emptyset[/ilmath] then we speak of a *homotopy rel [ilmath]A[/ilmath]* or *homotopy relative to [ilmath]A[/ilmath]*.

### Stages of a homotopy

For a homotopy, [ilmath]H:X\times I\rightarrow Y\ (\text{rel }A)[/ilmath], a *stage of the homotopy [ilmath]H[/ilmath]* is a map:

- [ilmath]h_t:X\rightarrow Y[/ilmath] for some [ilmath]t\in I[/ilmath] given by [ilmath]h_t:x\mapsto H(x,t)[/ilmath]

The family of maps, [ilmath]\{h_t:X\rightarrow Y\}_{t\in I} [/ilmath], are collectively called the *stages of a homotopy*

## Homotopy of maps

## Notes

- ↑ Recall [ilmath]\mathcal{P}(X)[/ilmath] denotes the power set of [ilmath]X[/ilmath] - the set containing all subsets of [ilmath]X[/ilmath]; [ilmath]A\subseteq X\iff A\in\mathcal{P}(X)[/ilmath].
- ↑
^{2.0}^{2.1}Note that if [ilmath]A=\emptyset[/ilmath] then [ilmath]\forall s,t\in I\ \forall a\in\emptyset[H(a,s)=H(a,t)][/ilmath] is trivially satisfied; it represents no condition. As there is no [ilmath]a\in\emptyset[/ilmath] we never require [ilmath]H(a,s)=H(a,t)[/ilmath].