Applying Theorem 3.3.1

3.4.1 Three saturated classes of morphisms

In this section we compare three classes of morphisms that may be used as trivial cofibrations in an application of Theorem 3.3.1. Let(TrivCof0,Fib0)be the weak factorisation system gen-

erated byI

_⊗

, which clearly is a set of morphisms with finitely presentable domains, and let (TrivCof1,Fib1)be generated by the semisimplicial horn inclusions.

The following well-known lemma for simplicial sets adapts to semisimplicial sets without problems.

Proof. We begin with the case𝑘 = 1. That is, we will show that every morphisms of the form 𝜖1

⊗

̂ 𝑖_𝑛 ∶ (Δ_𝑖[0]

⊗

Δ_𝑖[𝑛]) +_Δ𝑖[0]

_⊗

_{𝛿Δ𝑖[𝑛]}(Δ_𝑖[1]

⊗

𝛿Δ_𝑖[𝑛]) → Δ_𝑖[1]

⊗

Δ_𝑖[𝑛]

is in TrivCof₁. We will do so by showing that 𝜖1

⊗

̂ 𝑖_𝑛 can be written as a composition of inclusions obtained as certain pushouts of the horn inclusionΛ𝑘_𝑖[𝑛 + 1] → Δ_𝑖[𝑛 + 1]. We have seen thatΔ𝑖[1]

⊗

Δ𝑖[𝑛] ≅ 𝑁𝑖([1]

×

[𝑛]), which means that its𝑛 + 1-simplices can be described

by monotone injections𝜎𝑗 ∶ [𝑛 + 1] → [1]

×

[𝑛]of the form

(0, 0) → (0, 1) → ⋯ → (0, 𝑗) → (1, 𝑗) → ⋯ → (1, 𝑛),

for0 ≤ 𝑗 ≤ 𝑛. Note that𝜎0𝜖0 ∶ [𝑛] → {1}

×

[𝑛], so𝜎0𝜖0 ∈ Δ𝑖[0]

⊗

Δ𝑖[𝑛]. Moreover, it holds for

𝑟 ≥ 2that the simplex𝜎₀𝜖𝑟 ∶ [𝑛] → [1]

×

[𝑛](which skips the(𝑟 − 1)-th element of[𝑛]) is in Δ𝑖[1]

⊗

𝛿Δ𝑖[𝑛]. Indeed, we have

Δ𝑖[1]

⊗

Δ𝑖[𝑛] ≅ lim_−−→ 𝜖∶Δ𝑖[𝑚]→Δ𝑖[𝑛]

inΔ𝑖↓Δ𝑖[𝑛]

𝑁𝑖([1]

×

[𝑚])

andΔ𝑖[1]

⊗

𝛿Δ𝑖[𝑛]is the colimit of the restriction of this diagram toΔ𝑖↓ 𝛿Δ𝑖[𝑛]. It follows that

(𝑖𝑑_[𝑛], 𝜎₀𝜖𝑟) ∼ (𝜖𝑟−1, 𝜏) ∈ (Δ_𝑖[1]

⊗

𝛿Δ_𝑖[𝑛])_𝑛 where𝜏 ∶ (0, 0) → (1, 0) → ⋯ → (1, 𝑛 − 1).

Thus, we have the pushout diagram

Λ1_𝑖[𝑛 + 1] (Δ𝑖[0]

⊗

Δ𝑖[𝑛]) +Δ𝑖[0]

⊗

𝛿Δ𝑖[𝑛](Δ𝑖[1]

⊗

𝛿Δ𝑖[𝑛])

Δ𝑖[𝑛 + 1] (Δ𝑖[1]

⊗

Δ𝑖[𝑛])0

⟨𝜎0𝜖0, −, 𝜎0𝜖2, … , 𝜎0𝜖𝑛+1⟩

𝜎0

where(Δ𝑖[1]

⊗

𝛿Δ𝑖[𝑛])0 is the smallest subcomplex of Δ𝑖[1]

⊗

Δ𝑖[𝑛]that contains both the

image of𝜖1

⊗

̂ 𝑖_𝑛and the𝑛 + 1-simplex𝜎₀, together with its faces. Next we consider the simplex 𝜎1. We have that 𝜎1𝜖1 = 𝜎0𝜖0 ∈ (Δ𝑖[1]

⊗

𝛿Δ𝑖[𝑛])0 and, by the same reasoning as above, it

holds for every𝑟 ≠ 1, 2that𝜎1𝜖𝑟 ∈ Δ𝑖[1]

⊗

𝛿Δ𝑖[𝑛]. Thus, letting(Δ𝑖[1]

⊗

𝛿Δ𝑖[𝑛])1be the least

subcomplex of(Δ_𝑖[1]

⊗

𝛿Δ_𝑖[𝑛])0containing𝜎₁, the following is a pushout square:

Λ2_𝑖[𝑛 + 1] (Δ_𝑖[1]

⊗

Δ_𝑖[𝑛])0

Δ_𝑖[𝑛 + 1] (Δ_𝑖[1]

⊗

Δ_𝑖[𝑛])1

⟨𝜎1𝜖0, 𝜎1𝜖1, −, 𝜎1𝜖3, … , 𝜎1𝜖𝑛+1⟩

𝜎1

Repeating this process until we reach𝜎𝑛, we get a chain

(Δ_𝑖[0]

⊗

Δ_𝑖[𝑛]) +_Δ𝑖[0]

_⊗

_{𝛿Δ𝑖[𝑛]}(Δ_𝑖[1]

⊗

𝛿Δ_𝑖[𝑛]) → (Δ_𝑖[1]

⊗

Δ_𝑖[𝑛])0 → (Δ_𝑖[1]

⊗

Δ_𝑖[𝑛])1 → ⋯

as required. When𝑘 = 0, the same proof can be performed backwards, starting with𝜎_𝑛. It obviously follows thatTrivCof0

⊆

TrivCof1. Furthermore, because𝑖!sends the semisim-

plicial horn inclusions to the simplicial horn inclusions, it follows from Lemma 2.5.5 that

TrivCof₁

⊆

𝑖−1_! (TrivCof_Δ)and thus bothTrivCof₀andTrivCof₁lay within the bounds set in the hypothesis of Theorem 3.3.1. A third class of morphisms within these bounds is obtained by ap- plying Lemma 2.6.1 to the weak factorisation system(TrivCofΔ,FibΔ). we obtain a cofibrantly

generated weak factorisation systems onssSet, say(TrivCof_Δ𝑖,Fib_Δ𝑖). Note thatTrivCof₀is the minimal, andTrivCofΔ𝑖 the maximal class within the bounds.

Remark 3.4.2. In [JT08, Theorem 3.2.3], it is also proven that, insSet, it holds that the saturated class generated by𝑖_!(I

_⊗

)containsTrivCof_Δ. It follows that both𝑖_!(I

_⊗

)and the simplicial horn inclusions haveTrivCofΔas the least saturation. Thus, taking similar constraints on a saturated

classAas in the hypothesis of Theorem 3.3.1, but insSet,i.e. 𝑖!(I

⊗

)

⊆

A

⊆

TrivCofΔ,

uniquely determinesA. In contrast, with Proposition 3.5.7 we will show thatTrivCof1is strictly

contained in𝑖−1(TrivCof_Δ), showing that inssSetdistinct saturated classes lie between these constraints. It remains an open question whether, inssSet, the classI

_⊗

generates the same

saturated class as the semisimplicial horn inclusions.

▲

3.4.2 The left-induced right semimodel structure ssSet_𝑄

In this section we use Theorem 3.3.1 to obtain a right semimodel structure. Before we do that, let us first show that Theorem 3.3.1 can be used with the minimal classTrivCof0.

Proposition 3.4.3. TrivCof0satisfies the Leibniz axiom with respect toCof.

Proof. Let𝑘 be a trivial cofibration and letDbe the class of all cofibrations𝑚 ∶ 𝐴 → 𝐵such that𝑚 ̂

⊗

𝑘 is a trivial cofibration. It is easily verified thatDis saturated and thus it suffices to show thatI

_⊗

⊆

D. To that end, let𝜖𝑘

⊗

̂ 𝑖𝑛 ∈I

⊗

. Then, by the associativity of the Leibniz

product, we have

(𝜖𝑘

⊗

̂ 𝑖_𝑛) ̂

⊗

𝑘 = 𝜖𝑘

⊗

̂ (𝑖_𝑛

⊗

̂ 𝑘) ∈TrivCof₀

and thus, by Lemma 3.3.2,𝜖𝑘

⊗

̂ 𝑖𝑛∈D, as required. The full result follows from the symmetry

of the Leibniz product.

We invoke Theorem 3.3.1 to obtain a structure onssSet, sayssSet₀, that is almost a right semimodel structure.

The next application of Theorem 3.3.1 will provide a right semimodel structure, which we will callssSet_𝑄.

Proof. We use the fact that the statement holds for the standard model structure on the closed monoidal category (sSet,

×

, Δ[0]), where

×

is the categorical product. We write

×

̂ for the corresponding Leibniz product insSet. The strategy is to show that𝑖!(𝑗 ̂

⊗

𝑘) = 𝑖!(𝑗) ̂

×

𝑖!(𝑘),

from which the result can be easily seen to follow. Because𝑖!is a left adjoint, we have a pushout

square 𝑖_!𝐴

×

𝑖_!𝑋 𝑖_!𝐴

×

𝑖_!𝑌 𝑖!𝐵

×

𝑖!𝑋 𝑖!(𝐴

⊗

𝑌 +𝐴

⊗

𝑋 𝐵

⊗

𝑋 ) 𝑖!𝐴×𝑖!𝑘 𝑖!𝑗×𝑖!𝑋 𝑖!𝑟0 𝑖!𝑟1 ⌜

Now𝑖_!𝐵

×

𝑖_!𝑌 forms a cocone to the pushout diagram in the obvious way, and𝑖_!𝑗 ̂

×

𝑖_!𝑘 is the corresponding unique mediating morphism. However, it follows from the functoriality of 𝑖!

that𝑖!(𝑗 ̂

⊗

𝑘)also fits this role, hence we have found our equality.

Invoking Theorem 3.3.1 gives the structuressSet𝑄.

Theorem 3.4.5. ssSet_𝑄 is a right semimodel structure.

Proof. We only have to show that every acyclic cofibration is trivial, which follows directly from the fact that the weak equivalences are preserved by𝑖!. Indeed, for𝑓an acyclic cofibration

ofssSet_𝑄, we have that𝑖_!𝑓 is an acyclic cofibration of the model structure onsSet, and thus a trivial cofibration. It follows that𝑓 is a trivial cofibration.

Proposition 3.4.6. The functor𝑖!∶ssSet𝑄 →sSet𝑄 reflects weak equivalences.

Proof. Let𝑓 be such that its image under𝑖_!is a weak equivalence insSet. Take fibrant replace- ments and factor the obtained diagonal filler using(CofΔ𝑖,TrivFibΔ𝑖). Finally, apply𝑖!to get a

diagram

𝑖_!𝑋 𝑖_!𝑋 𝑖_!𝐸 𝑖_!𝑌 𝑖_!𝑌

𝑖!𝑓 𝑖!𝑓

Since𝑖!𝑓 is a weak equivalence in sSet, we have that 𝑖!𝑓 is as well. But𝐸 → 𝑌 is a weak

equivalence, as is𝑖!𝐸 → 𝑖!𝑌. It follows that𝑖!𝑋 → 𝑖!𝐸is an acyclic cofibration, hence trivial,

which means that𝑋 triv_{−→ 𝐸 −−↠ 𝑌}triv _{is, in fact, a factorisation into a trivial cofibration followed}

by a trivial fibration.

Since the left adjoint 𝑖! preserves and reflects cofibrations, trivial cofibrations and weak

equivalences,ssSet𝑄 is a left-induced right semimodel structure.

In document MoL 2018 34: A right semimodel structure on semisimplicial sets (Page 64-67)