Doctrine:Homotopy terminology

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Terminology

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

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].
  2. Stages of a homotopy

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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