Difference between revisions of "Doctrine:Homotopy terminology"
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==Terminology== | ==Terminology== | ||
Before we can define terms, here are the definitions we work with: | Before we can define terms, here are the definitions we work with: | ||
| − | * Let {{Top.|X|J}} and {{Top.|Y|K}} be [[ | + | * Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]] |
* Let {{M|A\in\mathcal{P}(X)}} be an [[arbitrary subset]] of {{M|X}} | * Let {{M|A\in\mathcal{P}(X)}} be an [[arbitrary subset]] of {{M|X}} | ||
* Let {{M|C^0(X,Y)}} denote [[the set of continuous maps between spaces|the set of continuous maps between {{Top.|X|J}} and {{Top.|Y|K}}]]<ref group="Note">The 0 comes from this being notation being used for [[classes of continuously differentiable functions]], {{M|C^1}} means all continuous functions whose first-order [[partial derivatives]] are continuous, {{M|C^2}} means continuous with continuous first and second derivatives, so forth, {{M|C^\infty}} means [[smooth]]. | * Let {{M|C^0(X,Y)}} denote [[the set of continuous maps between spaces|the set of continuous maps between {{Top.|X|J}} and {{Top.|Y|K}}]]<ref group="Note">The 0 comes from this being notation being used for [[classes of continuously differentiable functions]], {{M|C^1}} means all continuous functions whose first-order [[partial derivatives]] are continuous, {{M|C^2}} means continuous with continuous first and second derivatives, so forth, {{M|C^\infty}} means [[smooth]]. | ||
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#* {{M|1=\forall a\in A\forall s,t\in I[H(a,t)=H(a,s)]}} - the homotopy is fixed on {{M|A}}. | #* {{M|1=\forall a\in A\forall s,t\in I[H(a,t)=H(a,s)]}} - the homotopy is fixed on {{M|A}}. | ||
#** '''Note: ''' if {{M|1=A=}}{{emptyset}} then this represents no constraint, it is [[vacuously true]] | #** '''Note: ''' if {{M|1=A=}}{{emptyset}} then this represents no constraint, it is [[vacuously true]] | ||
| − | # '''Stages of a homotopy''' | + | # '''Stages of a homotopy''' - family of maps, {{M|\{h_t:X\rightarrow Y\}_{t\in I} }} given by {{M|h_t:x\mapsto H(x,t)}} |
| + | #* '''Initial stage''' - {{M|h_0:X\rightarrow Y}} with {{M|h_0:x\mapsto H(x,0)}} | ||
| + | #* '''Final stage''' - {{M|h_1:X\rightarrow Y}} with {{M|h_1:x\mapsto H(x,1)}} | ||
| + | # '''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: | ||
| + | #** {{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}} | ||
| + | # '''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
[hide]Terminology
Before we can define terms, here are the definitions we work with:
- Let (X,J) and (Y,K) be topological spaces
- Let A∈P(X) be an arbitrary subset of X
- Let C0(X,Y) denote the set of continuous maps between (X,J) and (Y,K)[Note 1]
- Let f,g,h∈C0(X,Y) be continuous maps of the form f,g,h:X→Y
Terms
- Homotopy (rel A) - Any continuous map of the form H:X×I→Y such that:
- ∀a∈A∀s,t∈I[H(a,t)=H(a,s)] - the homotopy is fixed on A.
- Note: if A=∅ then this represents no constraint, it is vacuously true
- ∀a∈A∀s,t∈I[H(a,t)=H(a,s)] - the homotopy is fixed on A.
- Stages of a homotopy - family of maps, {ht:X→Y}t∈I given by ht:x↦H(x,t)
- Initial stage - h0:X→Y with h0:x↦H(x,0)
- Final stage - h1:X→Y with h1:x↦H(x,1)
- Homotopy of maps - A homotopy, H:X×I→Y is a homotopy of f:X→Y and g:X→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×I→Y such that:
- ∀x∈X[f(x)=H(x,0)], ∀x∈X[g(x)=H(x,1)] and ∀a∈A∀s,t∈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:
- ∀a∈A∀s,t∈I[H(a,s)=H(a,t)=f(a)=g(a)] - often said as the "homotopy is fixed on A"
- obviously, in the case of s=0 and t=1 we see f(a)=g(a) too, so:
- ∀x∈X[f(x)=H(x,0)], ∀x∈X[g(x)=H(x,1)] and ∀a∈A∀s,t∈I[H(a,s)=H(a,t)]
- There exists a homotopy, H:X×I→Y such that:
- Homotopic maps - f and g are homotopic maps (written f≃g (rel A) and said "f is homotopic to g relative to A") if there exists a homotopy of maps between f and g
- Homotopy relation - refers to ((⋅)≃(⋅) (rel A))⊆C0(X,Y)×C0(X,Y)
- Homotopy class - equivalence classes of maps under the homotopy relation.
Notes
- Jump up ↑ The 0 comes from this being notation being used for classes of continuously differentiable functions, C1 means all continuous functions whose first-order partial derivatives are continuous, C2 means continuous with continuous first and second derivatives, so forth, C∞ means smooth.
Of course C0 means all continuous functions; and we have C0⊃C1⊃C2⊃⋯⊃C∞
