Doctrine:Homotopy terminology

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 continuous 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]

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
Stages of a homotopy

↑ 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]