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单词 44ContractibleFibers
释义

4.4 Contractible fibers


Note that our proofs about 𝗂𝗌𝗁𝖺𝖾(f) and 𝖻𝗂𝗂𝗇𝗏(f) made essential use of the fact that the fibers of an equivalence are contractibleMathworldPlanetmath.In fact, it turns out that this property is itself a sufficient definition of equivalence.

Definition 4.4.1 (Contractible maps).

A map f:AB is contractibleif for all y:B, the fiber fibf(y) is contractible.

Thus, the type 𝗂𝗌𝖢𝗈𝗇𝗍𝗋(f) is defined to be

𝗂𝗌𝖢𝗈𝗇𝗍𝗋(f):y:B𝗂𝗌𝖢𝗈𝗇𝗍𝗋(𝖿𝗂𝖻f(y))(1)

Note that in §3.11 (http://planetmath.org/311contractibility) we defined what it means for a type to be contractible.Here we are defining what it means for a map to be contractible.Our terminology follows the general homotopy-theoretic practice of saying that a map has a certain property if all of its (homotopyMathworldPlanetmath) fibers have that property.Thus, a type A is contractible just when the map A𝟏 is contractible.From http://planetmath.org/node/87580Chapter 7 onwards we will also call contractible maps and types (-2)-truncated.

We have already shown in Theorem 4.2.6 (http://planetmath.org/42halfadjointequivalences#Thmprethm2) that 𝗂𝗌𝗁𝖺𝖾(f)𝗂𝗌𝖢𝗈𝗇𝗍𝗋(f).Conversely:

Theorem 4.4.2.

For any f:AB we have isContr(f)ishae(f).

Proof.

Let P:𝗂𝗌𝖢𝗈𝗇𝗍𝗋(f). We define an inverse mapping g:BA by sending each y:B to the center of contraction of the fiber at y:

g(y):𝗉𝗋1(𝗉𝗋1(Py))

We can thus define the homotopy ϵ by mapping y to the witness that g(y) indeed belongs to the fiber at y:

ϵ(y):𝗉𝗋2(𝗉𝗋1(Py))

It remains to define η and τ. This of course amounts to giving an element of 𝗋𝖼𝗈𝗁f(g,ϵ). By Lemma 4.2.11 (http://planetmath.org/42halfadjointequivalences#Thmprelem5), this is the same as giving for each x:A a path from (gfx,ϵ(fx)) to (x,𝗋𝖾𝖿𝗅fx) in the fiber of f over fx. But this is easy: for any x:A, the type 𝖿𝗂𝖻f(fx)is contractible by assumptionPlanetmathPlanetmath, hence such a path must exist. We can construct it explicitly as

(𝗉𝗋2(P(fx))(gfx,ϵ(fx)))-1\\centerdot(𝗉𝗋2(P(fx))(x,𝗋𝖾𝖿𝗅fx)).

It is also easy to see:

Lemma 4.4.3.

For any f, the type isContr(f) is a mere proposition.

Proof.

By Lemma 3.11.4 (http://planetmath.org/311contractibility#Thmprelem2), each type 𝗂𝗌𝖢𝗈𝗇𝗍𝗋(𝖿𝗂𝖻f(y)) is a mere proposition.Thus, by Example 3.6.2 (http://planetmath.org/36thelogicofmerepropositions#Thmpreeg2), so is (1).∎

Theorem 4.4.4.

For any f:AB we have isContr(f)ishae(f).

Proof.

We have already established a logical equivalence 𝗂𝗌𝖢𝗈𝗇𝗍𝗋(f)𝗂𝗌𝗁𝖺𝖾(f), and both are mere propositions (Lemma 4.4.3 (http://planetmath.org/44contractiblefibers#Thmprelem1),Theorem 4.2.13 (http://planetmath.org/42halfadjointequivalences#Thmprethm3)).Thus, Lemma 3.3.3 (http://planetmath.org/33merepropositions#Thmprelem2) applies.∎

Usually, we prove that a function is an equivalence by exhibiting a quasi-inverse, but sometimes this definition is more convenient.For instance, it implies that when proving a function to be an equivalence, we are free to assume that its codomain is inhabited.

Corollary 4.4.5.

If f:AB is such that Bisequiv(f), then f is an equivalence.

Proof.

To show f is an equivalence, it suffices to show that 𝖿𝗂𝖻f(y) is contractible for any y:B.But if e:B𝗂𝗌𝖾𝗊𝗎𝗂𝗏(f), then given any such y we have e(y):𝗂𝗌𝖾𝗊𝗎𝗂𝗏(f), so that f is an equivalence and hence 𝖿𝗂𝖻f(y) is contractible, as desired.∎

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