Doctrine:Homotopy terminology

Terminology

Before we can define terms, here are the definitions we work with:

• Let $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$ be continuous spaces
• Let $A\in\mathcal{P}(X)$ be an arbitrary subset of $X$
• Let $C^0(X,Y)$ denote the set of continuous maps between $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$
• Let $f,g,h\in C^0(X,Y)$ be continuous maps of the form $f,g,h:X\rightarrow Y$

Terms

1. Homotopy $\mathbf{(\text{rel }A)}$ - Any continuous map of the form $H:X\times I\rightarrow Y$ such that:
   • $\forall a\in A\forall s,t\in I[H(a,t)=H(a,s)]$ - the homotopy is fixed on $A$.
     • Note: if $A=$$\emptyset$ then this represents no constraint, it is vacuously true
2. Stages of a homotopy - family of maps, $\{h_t:X\rightarrow Y\}_{t\in I}$ given by $h_t:x\mapsto H(x,t)$
   • Initial stage - $h_0:X\rightarrow Y$ with $h_0:x\mapsto H(x,0)$
   • Final stage - $h_1:X\rightarrow Y$ with $h_1:x\mapsto H(x,1)$
3. Homotopy of maps - A homotopy, $H:X\times I\rightarrow Y$ is a homotopy of $f:X\rightarrow Y$ and $g:X\rightarrow Y$ if its initial stage is $f$ and its final stage is $g$. That is to say there exists a homotopy of maps between $f$ and $g$ (relative to $A$) if:
   • There exists a homotopy, $H:X\times I\rightarrow Y$ such that:
     1. $\forall x\in X[f(x)=H(x,0)]$,
     2. $\forall x\in X[g(x)=H(x,1)]$ and
     3. $\forall a\in A\forall s,t\in I[H(a,s)=H(a,t)]$ - obviously, in the case of $s=0$ and $t=1$ we see $f(a)=g(a)$ too, so:
        • $\forall a\in A\forall s,t\in I[H(a,s)=H(a,t)=f(a)=g(a)]$ - often said as the "homotopy is fixed on $A$"
4. Homotopic maps - $f$ and $g$ are homotopic maps (written $f\simeq g\ (\text{rel }A)$ and said "$f$ is homotopic to $g$ relative to $A$") if there exists a homotopy of maps between $f$ and $g$
5. Homotopy relation - refers to $\big({\small(\cdot)}\simeq{\small(\cdot)}\ (\text{rel }A)\big)\subseteq C^0(X,Y)\times C^0(X,Y)$