Difference between revisions of "Doctrine:Homotopy terminology"

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Terminology

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

• Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be topological spaces
• Let [ilmath]A\in\mathcal{P}(X)[/ilmath] be an arbitrary subset of [ilmath]X[/ilmath]
• Let [ilmath]C^0(X,Y)[/ilmath] denote the set of continuous maps between [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath]^{[Note 1]}
• Let [ilmath]f,g,h\in C^0(X,Y)[/ilmath] be continuous maps of the form [ilmath]f,g,h:X\rightarrow Y[/ilmath]

Terms

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

Notes

1. ↑ The 0 comes from this being notation being used for classes of continuously differentiable functions, [ilmath]C^1[/ilmath] means all continuous functions whose first-order partial derivatives are continuous, [ilmath]C^2[/ilmath] means continuous with continuous first and second derivatives, so forth, [ilmath]C^\infty[/ilmath] means smooth.
   Of course [ilmath]C^0[/ilmath] means all continuous functions; and we have [ilmath]C^0\supset C^1\supset C^2\supset\cdots\supset C^\infty[/ilmath]

References

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