Difference between revisions of "Doctrine:Homotopy terminology"
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# '''Homotopy of maps''' - A homotopy, {{M|H:X\times I\rightarrow Y}} is a homotopy of {{M|f:X\rightarrow Y}} and {{M|g:X\rightarrow Y}} if its initial stage is {{M|f}} and its final stage is {{M|g}}. That is to say there exists a homotopy of maps between {{M|f}} and {{M|g}} (relative to {{M|A}}) if: | # '''Homotopy of maps''' - A homotopy, {{M|H:X\times I\rightarrow Y}} is a homotopy of {{M|f:X\rightarrow Y}} and {{M|g:X\rightarrow Y}} if its initial stage is {{M|f}} and its final stage is {{M|g}}. That is to say there exists a homotopy of maps between {{M|f}} and {{M|g}} (relative to {{M|A}}) if: | ||
#* There exists a homotopy, {{M|H:X\times I\rightarrow Y}} such that: | #* There exists a homotopy, {{M|H:X\times I\rightarrow Y}} such that: | ||
| − | #* | + | #** {{M|1=\forall x\in X[f(x)=H(x,0)]}}, {{M|1=\forall x\in X[g(x)=H(x,1)]}} and {{M|1=\forall a\in A\forall s,t\in I[H(a,s)=H(a,t)]}} |
| − | + | #*** obviously, in the case of {{M|1=s=0}} and {{M|1=t=1}} we see {{M|1=f(a)=g(a)}} too, so: | |
| − | + | #**** {{M|1=\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 {{M|A}}" | |
| − | #* | + | |
# '''Homotopic maps''' - {{M|f}} and {{M|g}} are homotopic maps (written {{M|1=f\simeq g\ (\text{rel }A)}} and said "''{{M|f}} is homotopic to {{M|g}} relative to {{M|A}}''") if there exists a homotopy of maps between {{M|f}} and {{M|g}} | # '''Homotopic maps''' - {{M|f}} and {{M|g}} are homotopic maps (written {{M|1=f\simeq g\ (\text{rel }A)}} and said "''{{M|f}} is homotopic to {{M|g}} relative to {{M|A}}''") if there exists a homotopy of maps between {{M|f}} and {{M|g}} | ||
# '''Homotopy relation''' - refers to {{M|\big({\small(\cdot)}\simeq{\small(\cdot)}\ (\text{rel }A)\big)\subseteq C^0(X,Y)\times C^0(X,Y)}} | # '''Homotopy relation''' - refers to {{M|\big({\small(\cdot)}\simeq{\small(\cdot)}\ (\text{rel }A)\big)\subseteq C^0(X,Y)\times C^0(X,Y)}} | ||
| + | # '''Homotopy class''' - [[equivalence classes]] of maps under the homotopy relation. | ||
==Notes== | ==Notes== | ||
<references group="Note"/> | <references group="Note"/> | ||
Latest revision as of 09:01, 31 October 2016
Contents
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
- 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
- [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].
- 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]
- 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:
- [ilmath]\forall x\in X[f(x)=H(x,0)][/ilmath], [ilmath]\forall x\in X[g(x)=H(x,1)][/ilmath] and [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]"
- 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 x\in X[f(x)=H(x,0)][/ilmath], [ilmath]\forall x\in X[g(x)=H(x,1)][/ilmath] and [ilmath]\forall a\in A\forall s,t\in I[H(a,s)=H(a,t)][/ilmath]
- There exists a homotopy, [ilmath]H:X\times I\rightarrow Y[/ilmath] such that:
- 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]
- 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]
- Homotopy class - equivalence classes of maps under the homotopy relation.
Notes
- ↑ 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]
