Homotopy

In this section we further develop the homotopy theory of the given structure onssSet.

Definition 3.3.5. Let𝑓 , 𝑔 ∶ 𝑋 → 𝑌be semisimplicial maps. Thenℎ ∶ 𝑋

⊗

𝐼 → 𝑌is ahomotopy from𝑓 to𝑔 if the following diagram commutes.

𝑋 𝑋

⊗

𝐼 𝑋 𝑌 𝑓 𝑋⊗𝜖1 ℎ 𝑋⊗𝜖0 𝑔

Given a homotopyℎ ∶ 𝑋

⊗

𝐼 → 𝑌, we writeℎ_𝑘for the restrictionℎ(𝑋

⊗

𝜖𝑘) ∶ 𝑋 → 𝑌 for 𝑘 ∈ {0, 1}.

Proposition 3.3.6([JT08, p. 45]). On the class of morphisms𝑓 ∶ 𝑋 → 𝐾 with fibrant codomain, homotopy is an equivalence relation.

Proof. By adjunction, a homotopyℎ ∶ 𝑋

⊗

𝐼 → 𝐾is equivalent to a pathℎ ∶ 𝐼 → [𝑋 , 𝐾]. The result follows from the fact that, by Corollary 2.7.7, the object[𝑋 , 𝐾] is fibrant and thus, by Proposition 3.3.4, path-connectedness on[𝑋 , 𝐾]is an equivalence relation.

Definition 3.3.7. A morphism 𝑓 ∶ 𝑋 → 𝑌 is called ahomotopy equivalence if there is a map 𝑔 ∶ 𝑌 → 𝑋 together with homotopiesℎ ∶ 𝑋

⊗

𝐼 → 𝑋 and ℎ′ ∶ 𝑌

⊗

𝐼 → 𝑌 such that

ℎ ∶ 𝑔𝑓 ∼ 𝑖𝑑𝑋 andℎ′∶ 𝑓 𝑔 ∼ 𝑖𝑑𝑌. ■

The following lemma records some properties of homotopy (equivalence) on morphisms between fibrant objects that will be useful later on.

Lemma 3.3.8. For morphisms between fibrant objects: (i) Homotopy is stable under composition.

(ii) Homotopy equivalence is stable under homotopy. (iii) Homotopy equivalences are closed under composition.

Proof. (i) Let𝑓 , 𝑔 ∶ 𝑋 → 𝑌 and𝑝, 𝑞 ∶ 𝑌 → 𝑍. Givenℎ ∶ 𝑓 ∼ 𝑔 and𝑘 ∶ 𝑝 ∼ 𝑞, we have 𝑝ℎ ∶ 𝑝𝑓 ∼ 𝑝𝑔and (by naturality) also𝑘(𝐼

⊗

𝑔) ∶ 𝑝𝑔 ∼ 𝑞𝑔. Transitivity gives𝑝𝑓 ∼ 𝑞𝑔.

(ii) Let𝑓 be a homotopy equivalence with homotopy inverse𝑓−1, and let 𝑔 ∼ 𝑓. By part (i), we have𝑔𝑓−1∼ 𝑓 𝑓−1∼ 𝑖𝑑and dually for𝑓−1𝑔.

(iii). Let𝑓 and𝑔 be homotopy equivalences with homotopy inverses𝑓−1and𝑔−1. Then, using part (i) and the fact that homotopy is an equivalence relation, we find

𝑓−1𝑔−1𝑔𝑓 ∼ 𝑓−1𝑓 ∼ 𝑖𝑑. It can be dually shown that𝑔𝑓 𝑓−1𝑔−1∼ 𝑖𝑑.

InsSetthe above homotopic notions are defined in the same way, but withΔ[0]andΔ[1] instead ofΔ_𝑖[0]andΔ_𝑖[1]as{∗}and𝐼, and with the categorical product instead of

⊗

. It can be easily seen that, since𝑖! preserves all three, homotopies and homotopy equivalences are

preserved by𝑖!.

Definition 3.3.9. A homotopy equivalence𝑓 is called astrongif there are homotopiesℎandℎ′ witnessing that𝑓 is a homotopy equivalence such that the diagram

𝑋

⊗

𝐼 𝑌

⊗

𝐼 𝑋 𝑌 𝑓⊗𝐼 ℎ ℎ′ 𝑓 commutes. ■

The proof of the next proposition is based on the proofs of propositions 3.2.5 and 3.2.6 of [JT08], but with some required adaptations because, unlike in the simplicial case, we do not have projections𝐾

⊗

𝐼 → 𝐾and𝐾

⊗

𝐼 → 𝐼.

Proposition 3.3.10. For𝑝a fibration with fibrant codomain, the following are equivalent: (i) 𝑝is a homotopy equivalence;

(ii) 𝑝is a strong homotopy equivalence; (iii) 𝑝is a trivial fibration.

Proof. Let 𝑝 ∶ 𝐸 → 𝐾 be a fibration with a fibrant codomain. (i) ⇒(ii). Let 𝑠 ∶ 𝐾 → 𝐸, ℎ ∶ 𝑠𝑝 ∼ 𝑖𝑑𝐸andℎ′ ∶ 𝑝𝑠 ∼ 𝑖𝑑𝐾 be the data that witness that𝑝is a homotopy equivalence. Then

the filler of the diagram

𝐾

⊗

{∗} 𝐸 𝐾

⊗

𝐼 𝐾 ∗𝐾⊗̂ 𝜖1 𝑠 𝑝 𝑡 ℎ′

is such that𝑝𝑡₀ = ℎ′₀= 𝑖𝑑_𝐾 and𝑡₁= 𝑠. By Lemma 3.3.8, we have𝑡₀𝑝 ∼ 𝑡₁𝑝 ∼ 𝑠𝑝 ∼ 𝑖𝑑_𝐸, witnessed by, say𝑘 ∶ 𝑡₀𝑝 ∼ 𝑖𝑑_𝐸. Furthermore, by reflexivity, there is a homotopy𝑘′ ∶ 𝑝𝑡₀ ∼ 𝑖𝑑_𝐾 and so the fact that𝑝is a homotopy equivalence is additionally witnessed by the section 𝑡0 and the

homotopies𝑘, 𝑘′. We will show that there is a𝑘∗∶ 𝑡0𝑝 ∼ 𝑖𝑑𝐸giving the required commutativity.

Let𝛼 ∶ 𝐼

⊗

𝐼 → [𝐸, 𝐾]be the diagonal filler

({∗}

⊗

𝐼 ) +_{∗}

_⊗

_{({∗}+{∗})}(𝐼

⊗

({∗} + {∗})) [𝐸, 𝐾] 𝐼

⊗

𝐼

[𝑘′_(𝑝⊗_{𝐼 ), [𝑝𝑘, 𝑝𝑘]]}

𝜖0⊗̂ _[𝜖1_{, 𝜖}0_]

𝛼

Then the diagram

({∗}

⊗

𝐼 ) +_{∗}

_⊗

_{({∗}+{∗})}(𝐼

⊗

({∗} + {∗})) [𝐸, 𝐸] 𝐼

⊗

𝐼 [𝐸, 𝐾] [[𝐸, 𝑡0]𝛼(𝜖1⊗𝐼 ), [𝑡0𝑝𝑘, 𝑘]] 𝜖1⊗̂ _[𝜖1_{, 𝜖}0_] [𝐸, 𝑝] 𝛽 𝛼

commutes. Let𝛽 ∶ 𝐼

⊗

𝐼 → 𝐸𝐸be the induced filler and consider the homotopy𝑘∗ = 𝛽(𝜖0

⊗

_{𝐼 ).}

We have𝑘₁∗ = (𝑡0𝑝𝑘)0= 𝑡0𝑝𝑡0𝑝 = 𝑡0𝑝and𝑘0∗ = 𝑘0= 𝑖𝑑𝐸, as well as𝑝𝑘∗= 𝑘′(𝐼

⊗

𝑝), as required.

(ii)⇒(iii). Letℎ ∶ 𝑠𝑝 ∼ 𝑖𝑑𝐸 andℎ′ ∶ 𝑝𝑠 ∼ 𝑖𝑑𝐾 be the homotopies that witness that𝑝is a

strong homotopy equivalence. Suppose then that we are given a lifting problem

𝐴 𝐸 𝐵 𝐾 𝑖 𝑎 𝑝 𝑏

with𝑖 ∶ 𝐴 → 𝐵a cofibration. It follows that both the squares 𝐴

⊗

𝐼 𝐸 𝐵

⊗

𝐼 𝐾 𝑖⊗𝐼 ℎ(𝑎⊗𝐼 ) 𝑝 ℎ′_{(𝑏⊗𝐼 )} and 𝐵

⊗

{∗} 𝐸 𝐵

⊗

𝐼 𝐾 𝐵⊗𝜖1 𝑠𝑏 𝑝 ℎ′_{(𝑏⊗𝐼 )} commute. Combining the two squares, we get a commuting square

(𝐴

⊗

𝐼 ) +_𝐴

_⊗

_{∗}(𝐵

⊗

{∗}) 𝐸

𝐵

⊗

𝐼 𝐾,

𝑖 ̂⊗𝜖1 𝑝

ℎ ℎ′_{(𝑏⊗𝐼 )}

which has a liftingℎ ∶ 𝐵

⊗

𝐼 → 𝐸such that𝑝ℎ = ℎ′(𝑏

⊗

𝐼 )andℎ(𝑖

⊗

𝐼 ) = ℎ(𝑎

⊗

𝐼 ). But then 𝑝ℎ0 = 𝑏andℎ0𝑖 = 𝑎, which means thatℎ1provides the desired lifting.

(iii)⇒(i). Suppose that𝑝is trivial. Then it has a section𝑠 ∶ 𝐾 → 𝐸obtained as the diagonal filler

0 𝐸

𝐾 𝐾

𝑝 𝑠

Letℎ ∶ 𝑝𝑠 ∼ 𝑖𝑑_𝐾, a suitable homotopyℎ′∶ 𝑠𝑝 ∼ 𝑖𝑑_𝐸arises as the diagonal filler

(𝐸

⊗

{∗}) + (𝐸

⊗

{∗}) 𝐸 𝐸

⊗

𝐼 𝐾 [𝑠𝑝, 𝑖𝑑𝐸] [𝜖1_{, 𝜖}0_{] 𝑝} ℎ(𝑝⊗𝐼 ) ℎ′

Proposition 3.3.11([JT08, Prop. 3.2.3]). A trivial cofibration between fibrant objects is a strong homotopy equivalence.

Proof. Let𝑖 ∶ 𝐴 → 𝐵be a trivial cofibration between fibrant objects. We obtain a retract𝑟 of𝑖 as diagonal filler in the following diagram

𝐴 𝐴

𝐵

𝑖 _𝑟

Letℎ ∶ 𝑟𝑖 ∼ 𝑟𝑖, then a homotopy𝑖𝑟 ∼ 𝑖𝑑𝐵 is obtained as the indicated filler

(𝐴

⊗

𝐼 ) +_𝐴

_⊗

_{({∗}+{∗})}(𝐵

⊗

({∗} + {∗})) 𝐵 𝐵

⊗

𝐼

𝑖 ̂⊗[𝜖1_{, 𝜖}0_]

In document MoL 2018 34: A right semimodel structure on semisimplicial sets (Page 56-60)