Fibrant regular and cubical transition systems

The purpose of this last section is to describe completely the fibrant regular and cubical transition systems. We already know by Proposition 6.8 that the fibrant regular transition systems are exactly the fibrant cubical ones which are regular. Thus, we just have to give a combinatorial characterization of the fibrant objects of L_CubCTS. Corollary 7.16 encompasses the results of [8] and [9].

Definition 7.1. A cubical transition system X is combinatorially fibrant if for anyn ≥1, any stateαandβand any actionsu₁, v₁, . . . , u_n, v_nsuch that

µ(u_i) =µ(v_i)for1≤i≤ n, if the tuple(α, u₁, . . . , u_n, β)is a transition of X, then the tuple(α, v₁, . . . , v_n, β)is a transition ofXas well.

Proposition 7.2. LetX = (S, µ : L → Σ, T)be a combinatorially fibrant cubical transition system. Write Path : CTS → CTS for the right adjoint of the cartesian cylinder Cyl : CTS → CTS. Then the cubical transition systemPath(X)hasS as its set of states andL×_ΣL as its set of actions, the labelling map is the composite map µ : L ×_Σ L → L → Σ and a tuple(α,(u⁰₁, u¹₁), . . . ,(u⁰_n, u¹_n), β)ofS×(L×ΣL)ⁿ×S is a transition of Path(X) if and only if there exist ₁, . . . , _n ∈ {0,1} such that the tuple (α, u₁¹, . . . , u_nⁿ, β)is a transition ofX.

Proof. Let us recall that the cartesian cylinder Cyl : CTS → CTS is the restriction of an endofunctor ofWTS defined in the same way. The functor Cyl : WTS → WTS has a right adjointPath^WTS : WTS → WTS defined on objects as follows [8, Proposition 5.8]: for a weak transition systemX = (S, µ : L → Σ, T), the weak transition systemPath^WTS(X) has the same set of statesS, the set of actions isL×_ΣLand a tuple

(α,(u⁻₁, u⁺₁), . . . ,(u⁻_n, u⁺_n), β)

with n ≥ 1 is a transition of Path^WTS(X) if and only if the 2ⁿ tuples (α, u^±₁, . . . , u^±_n, β) are transitions of X. The right adjoint of the functor Cyl :CTS → CTSis equal to the composite functor

Path :CTS ⊂ WTS ^{PathWTS //}WTS ^//CTS, where the right-hand functor fromWTS toCTS is the coreflection.

Let (u, v) ∈ L×_Σ L. Since u is used in X, there exists a transition (α, u, β)of X. Sinceµ(u) = µ(v)and sinceX is combinatorially fibrant, the triple(α, v, β)is a transition ofX. This means that the couple(u, v) ∈ L×_ΣLis used by the transition(α,(u, v), β)ofPath^WTS(X). We deduce that all actions ofPath^WTS(X)are used. Consider a transition

(α,(u⁻₁, u⁺₁), . . . ,(u⁻_n, u⁺_n), β)

ofPath^WTS(X)withn ≥2. Let1≤p≤n−1. SinceXis cubical, there ex-ists a stateγ such that the tuples(α, u⁻₁, . . . , u⁻_p, γ)and(γ, u⁻_p+1, . . . , u⁻_n, β)

are two transitions ofX. ButXis combinatorially fibrant. This implies that all tuples (α, u^±₁, . . . , u^±_p, γ) and (γ, u^±_p+1, . . . , u^±_n, β) are transitions of X.

Therefore the two tuples

(α,(u⁻₁, u⁺₁), . . . ,(u⁻_p, u⁺_p), γ),(γ,(u⁻_p+1, u⁺_p+1), . . . ,(u⁻_n, u⁺_n), β)

are transitions of Path^WTS(X). This means that the weak transition system Path^WTS(X)satisfies the Intermediate state axiom. We have just proved that ifXis combinatorially fibrant, then the weak transition systemPath^WTS(X) is cubical: in other terms, one hasPath(X) = Path^WTS(X)in this case. Fi-nally and becauseX is combinatorially fibrant, all tuples(α, u^±₁, . . . , u^±_n, β) are transitions ofXif and only if there exist₁, . . . , _n ∈ {0,1}such that the tuple(α, u₁¹, . . . , u_nⁿ, β)is a transition ofX. This completes the proof.

Proposition 7.3. If the cubical transition system X is combinatorially fi-brant, then so is the cubical transition systemPath(X).

Proof. Let X = (S, µ : L → Σ, T) be a combinatorially fibrant cubical transition system. Let

(α,(u⁻₁, u⁺₁), . . . ,(u⁻_n, u⁺_n), β),(α,(v₁⁻, v₁⁺), . . . ,(v⁻_n, v_n⁺), β)

be two tuples ofS×(L×_ΣL)ⁿ×Swithn≥1andµ(u⁻_i , u⁺_i ) = µ(v⁻_i , v_i⁺)for 1≤i≤n. Let us suppose that(α,(u⁻₁, u⁺₁), . . . ,(u⁻_n, u⁺_n), β)is a transition ofPath(X). Then the tuple(α, u⁻₁, . . . , u⁻_n, β)is a transition ofX. But for all1≤i≤n, one has

µ(u⁻_i ) =µ(u⁺_i ) = µ(u⁻_i , u⁺_i ) = µ(v_i⁻, v_i⁺) = µ(v⁻_i ) =µ(v_i⁺).

So, all tuples (α, v₁^±, . . . , v^±_n, β)are transitions of X because X is combi-natorially fibrant. This implies that the tuple (α,(v₁⁻, v⁺₁), . . . ,(v_n⁻, v_n⁺), β) is a transition of Path(X). This is the definition of combinatorial fibrancy applied toPath(X).

Proposition 7.4. LetXbe a cubical transition system. IfX is combinatori-ally fibrant, then it is injective with respect to any map of the formf ? γ for = 0,1for any cofibration of cubical transition systemsf.

Proof. Letf :A →B be a map of cubical transition systems. LetLbe the set of actions ofX. By adjunction, the cubical transition systemX is injec-tive with respect tof ? γif and only if the mapπ: Path(X)→Xsatisfies the RLP with respect to f. Let us recall that the mapπ : Path(X) → X is the identity on states and the projection on the ( + 1)-th component L×_Σ L → Lon actions by Proposition 7.2. Consider a diagram of solid arrows of cubical transition systems:

A

f

φ //Path(X)

π

B ^{ψ //}

`

<<

X.

Since the right vertical map is onto on actions and the left vertical map is one-to-one on actions, there exists a set map `e: L_B → L×_Σ L from the set of actions ofB to the set of actions ofPath(X)such that the following diagram of sets is commutative,L_A being the set of actions ofA (note that πeis the projection on the(+ 1)-th component):

L_A

f

φ //L×_ΣL

π

L_B ^{ψ //}

`e

<<

L.

Let` :B → Path(X)defined on states by`(α) = ψ(α)and on actions by

`(u) =`(u). The diagrame

A

f

φ //Path(X)

π

B ^{ψ //}

`

<<

X

is commutative since its right vertical map is the identity on states. It just remains to prove that ` : B → Path(X)is a well-defined map of cubical transition systems. Let(α, u₁, . . . , u_n, β)be a transition ofB. It suffices to prove that the tuple

(α,`(ue <sub>1</sub>), . . . ,e`(u_n), β)

is a transition ofPath(X)to complete the proof. Without lack of generality, we can suppose that = 0, which means that `(u) = (ψ(u), χ(u)). Onee obtains

(α,`(ue <sub>1</sub>), . . . ,`(ue _n), β) = (α,(ψ(u₁), χ(u₁)), . . . ,(ψ(u_n), χ(u_n)), β).

Sinceψmaps the transitions ofBto transitions ofX, the tuple (α, ψ(u₁), . . . , ψ(u_n), β)

is a transition of X. Since µ(ψ(u)) = µ(u) = µ(χ(u))for all actionsu of B, and sinceX is combinatorially fibrant, the tuple

(α,(ψ(u₁), χ(u₁)), . . . ,(ψ(u_n), χ(u_n)), β) is then a transition ofPath(X)by Proposition 7.2.

Proposition 7.5. LetXbe a cubical transition system. IfX is combinatori-ally fibrant, then it is injective with respect to the maps ofS^cof.

Proof. Letx ∈ Σ. Consider a diagram of solid arrows of cubical transition systems

C₁[x]tC₁[x]

p^cofx

φ //X

Z_x^x1,x2,

`

;;

wherex₁ andx₂are the two actions ofC₁[x]tC₁[x]and whereZ_x^x1,x2 is the cubical transition system depicted in Figure 2. Define`on states by`(α) = φ(α), and on actions by`(x<sub>i</sub>) = φ(x<sub>i</sub>)for i = 1,2 and`(x) = φ(x₁). Let (α_i, φ(x_i), β_i)fori= 1,2be the images byφof the two transitions ofC₁[x]t C₁[x]. SinceXis combinatorially fibrant, the two triples(α_i, φ(x3−i), β_i)for i= 1,2are two transitions ofX. The map`is therefore a well-defined map of cubical transition systems.

Proposition 7.6. Let X = (S, µ : L → Σ, T) and X⁰ = (S⁰, µ⁰ : L⁰ → Σ, T⁰)be two cubical transition systems. The binary product X ×X⁰ has S×S⁰ as its set of states, L×_ΣL⁰ ={(x, x⁰)∈ L×L⁰, µ(x) =µ⁰(x⁰)}as its set of actions and the labelling map µ×_Σ µ⁰ : L×_Σ L⁰ → Σ. A tuple ((α, α⁰),(u₁, u⁰₁), . . . ,(u_n, u⁰_n),(β, β⁰))is a transition ofX×X⁰ if and only ifµ(u_i) = µ⁰(u⁰_i)for1 ≤i≤ nwithn ≥ 1, the tuple(α, u₁, . . . , u_n, β)is a transition ofXand(α⁰, u⁰₁, . . . , u⁰_n, β⁰)a transition ofX⁰.

Proof. The binary product is the same in CTS and inWTS becauseCTS is a small-injectivity class ofWTS. The theorem is then a consequence of [8, Proposition 5.5].

Proposition 7.7. LetXbe a cubical transition system. IfX is combinatori-ally fibrant, then it is injective with respect to any map of the formf ? γ for any map of cubical transition systemsf which is onto on states.

Proof. Let f : A → B be a map of cubical transition systems. By adjunc-tion, the cubical transition systemX is injective with respect tof ? γif and only if the mapπ : Path(X)→X×X satisfies the RLP with respect tof.

Consider a diagram of solid arrows of cubical transition systems:

A

f

φ //Path(X)

π

B ^{ψ=(ψ0,ψ1) //}

`

<<

X×X.

Since the set map f : A⁰ → B⁰ is onto by hypothesis, for any state α of B, there existss(α) ∈ A⁰ such that f(s(α)) = α. Let` : B → Path(X) defined on states by `(α) = φ(s(α))and on actions by `(u) = ψ(u)(since X is combinatorially fibrant, the mapπ : Path(X)→X×Xis the identity on actions by Proposition 7.2). We are going to prove that`is a well-defined map of cubical transition systems and that it is a lift of the diagram above.

`is a lift for the sets of actions. One has the following diagram of solid

arrows between the sets of actions:

L_A

f

φ //L_X ×_ΣL_X

L_B ^{ψ //}

ψ

;;

L_X ×_ΣL_X.

It is evident that the two triangles commute since the square of solid arrows commutes.

`is a lift for the sets of states. One has the diagram of solid arrows be-tween the sets of states:

A⁰

f

φ //X⁰

∆

B^{0 ψ=(ψ0,ψ1) //}

φ◦s

;;

s

@@

X⁰×X⁰,

where∆ :s7→(s, s)is the codiagonal map. For any stateβ ofB⁰, one has ψ(β)

=ψ(f(s(β))) sincesis a section off

=π(φ(s(β))) sinceψ◦f =π◦φ

= (φ(s(β)), φ(s(β))) by Proposition 7.6.

Hence we obtainψ₀ =ψ₁ =φ◦son states, and therefore∆◦φ◦s =ψ on states. We deduce that the bottom triangle commutes on states. For any state αofA⁰, one has

∆(φ(s(f(α))))

=ψ(f(s(f(α)))) since∆◦φ=ψ◦f

=ψ(f(α)) sincesis a section off

= ∆(φ(α)) because the square above is commutative.

Hence we obtain φ◦s◦f = φ on states. We obtain that the top triangle commutes.

`maps a transition ofB to a transition ofPath(X). Let (α, u₁, . . . , u_n, β)

be a transition ofB. Then one has

(`(α), `(u₁), . . . , `(u<sub>n</sub>), `(β)) = (φ(s(α)), ψ(u₁), . . . , ψ(u_n), φ(s(β)))

= (ψ0(α), ψ(u1), . . . , ψ(un), ψ0(β)).

The tuple(ψ₀(α), ψ₀(u₁), . . . , ψ₀(u_n), ψ₀(β))is a transition ofX since it is the image by the composite map of cubical transition systems ψ₀ : B → X×X →X of the transition(α, u₁, . . . , u_n, β)ofB. Therefore by Propo-sition 7.2 applied with₁ =· · ·=_n= 0, the tuple

(`(α), `(u₁), . . . , `(u<sub>n</sub>), `(β))

is a transition of Path(X) sinceX is combinatorially fibrant. This means that

`:B −→Path(X) is a well-defined map of cubical transition systems.

Proposition 7.8. LetXbe a cubical transition system. IfX is combinatori-ally fibrant, then it is injective with respect to any map of the form(f ? γ)? γ for any map of cubical transition systemsf.

Proof. Letf :A→Bbe a map of cubical transition systems. The mapf ? γ goes from(BtB)tAtACyl(A)toCyl(B). Since the forgetful functor from CTS toSet taking a cubical transition system to its underlying set of states is colimit-preserving, the set of states of the source off ? γ is B⁰ t_A⁰ B⁰. Hence the mapf ?γis onto on states. Then by Proposition 7.7,Xis injective with respect to(f ? γ)? γ.

Notation 7.9. LetIandSbe two sets of maps of a locally presentable cate-goryK. LetCyl :K → Kbe a cylinder. Denote byΛK(Cyl, S, I)the set of maps defined as follows:

• Λ⁰_K(Cyl, S, I) = S∪(I ? γ⁰)∪(I ? γ¹)

• Λⁿ⁺¹_K (Cyl, S, I) = Λⁿ_K(Cyl, S, I)? γ

• ΛK(Cyl, S, I) = S

n≥0Λⁿ_K(Cyl, S, I).

Theorem 7.10. LetXbe a cubical transition system. IfXis combinatorially fibrant, then it is fibrant.

Proof. Let X be a combinatorially fibrant cubical transition system. By Proposition 7.4 and Proposition 7.5, it is Λ⁰(Cyl,S^cof,I)-injective. Let f : A → B be a map of cubical transition systems. Let ∈ {0,1}. The map f ? γ goes from B tACyl(A) to Cyl(B). Since the forgetful func-tor fromCTS toSet taking a cubical transition system to its underlying set of states is colimit-preserving, the set of states of the source off ? γisB⁰. Hencef ?γis bijective on states. Therefore all maps ofΛ⁰(Cyl,S^cof,I)are bijective on states. Then, by Proposition 7.7,XisΛ¹(Cyl,S^cof,I)-injective.

The cubical transition system X isΛⁿ(Cyl,S^cof,I)-injective for all n ≥ 2 by Proposition 7.8. Hence X is fibrant in the Bousfield localization ofCTS by the cofibrations ofS^cof by [8, Corollary 6.8] and [10, Theorem 4.6]. But Bousfield localizing byS^cof is the same as Bousfield localizing byS, which is the same as Bousfield localizing by the cubification functor. Hence the proof is complete.

Notation 7.11. Letx∈Σ. The two maps fromC₁[x]to↑x↑are denoted by c_xfor= 0,1. One hasp_x =c⁰_xtc¹_xfor allx∈Σ.

Proposition 7.12. Letx∈Σ. Consider the pushout diagram ofCTS

C₁[x] ^c

0x //

γ_C⁰

1[x]

↑x↑

Cyl(C₁[x]) ^//Cyl(C₁[x])t_0,0 ↑x↑. The composite

θ_x : C₁[x]tC₁[x]

γ¹_C

1[x]tc¹_x

//Cyl(C₁[x])t ↑x↑ ^//Cyl(C₁[x])t_0,0 ↑x↑

is a trivial cofibration ofL_CubCTS.















C₁[x]tC₁[x]

α ^{x1 //}β

γ ^{x2 //}δ

θx

−→















Cyl(C1[x])t0,0 ↑x↑

α

x1

++

x2

33β

γ ^{x2 //}δ

Figure 3: Cofibrationθ_xwithµ(x₁) =µ(x₂) =x

Proof. The mapθxis depicted in Figure 3. It is bijective on actions, therefore it is a cofibration. One hasL^CTS_S (C₁[x]tC₁[x])∼=L^CTS_S (Cyl(C₁[x])t_0,0 ↑x↑

) ∼=↑x↑. Hence it is a weak equivalence ofL_CubCTS by [8, Theorem 8.10].

Proposition 7.13. In the following, the notationt 0n= 0n 1_n= 1_n

means the identi-fication of the initial states (the final states resp.) of the two summands. Let n ≥2andx₁, . . . , x_n∈Σ. Then the map

η_x1,...,xn :∂C_n[x₁, . . . , x_n]t 0_n= 0_n 1n= 1n

C_n[x₁, . . . , x_n]

−→C_n[x₁, . . . , x_n]t 0_n= 0_n 1n= 1n

C_n[x₁, . . . , x_n]

induced by the inclusion∂Cn[x1, . . . , xn] ⊂ Cn[x1, . . . , xn]is a trivial cofi-bration ofL_CubCTS.

Proof. The mapη_x1,...,xn is bijective on actions: the set of actions is {(x1,1), . . . ,(xn, n)} × {0,1},

with for example 0 for the left-hand term and 1 for the right-hand term.

Hence it is a cofibration. The mapη_x1,...,xn is also bijective on states: the set of states is a set denoted by{0,1}ⁿt 0n= 0n

1_n= 1_n

{0,1}ⁿ, which means the quo-tient of the coproduct{0,1}ⁿt {0,1}ⁿby the identifications of0_n(1_nresp.) of the left-hand term with 0_n (1_n resp.) of the right-hand term. Since the map X → L^CTS_S (X)is bijective on states for all cubical transition systems

X, the map of cubical transition systems

L^CTS_S (η_x1,...,xn) :L^CTS_S

∂C_n[x₁, . . . , x_n]t 0n= 0n 1_n= 1_n

C_n[x₁, . . . , x_n]

−→

L^CTS_S

C_n[x₁, . . . , x_n]t 0n= 0n 1_n= 1_n

C_n[x₁, . . . , x_n]

is bijective on states as well. The set of actions of the source and tar-get of L^CTS_S (ηx1,...,xn) is {x1, . . . , xn}. Since L^CTS_S (ηx1,...,xn) is one-to-one on action by [8, Remark 8.8], it is bijective on actions. By [10, Proposi-tion 4.4], the map L^CTS_S (η_x1,...,xn) is one-to-one on transitions by. To see that the map L^CTS_S (ηx1,...,xn)is also onto on transitions, it suffice to see that the n! n-transitions of the left-hand n-cube of the target are the n! tuples (0_n, x_σ(1), . . . , x_σ(n),1_n)which are actually transitions of the source because of the identifications of the two initial states and the two final states. So L^CTS_S (η_x1,...,xn)is an isomorphism. Therefore by [8, Theorem 8.10], the map η_x1,...,xn is a weak equivalence ofL_CubCTS.

Proposition 7.14. A cubical transition system is combinatorially fibrant if and only if it is injective with respect toθ_xandη_x1,...,xnfor allx, x₁, . . . , x_n ∈ Σ.

Proof. LetXa combinatorially fibrant cubical transition system. ThenX is fibrant by Theorem 7.10. Since the mapsθ_xandη_x1,...,xnfor all

x, x₁, . . . , x_n∈Σ

are trivial cofibrations by Proposition 7.12 and Proposition 7.13,Xis injec-tive with respect to these maps. Conversely, letXbe a cubical transition sys-tem which is injective with respect toθ_x andη_x1,...,xn for allx, x₁, . . . , x_n ∈ Σ. Let (α, x₁, β) be a transition of X and let x₂ an action ofX such that µ(x₁) = µ(x₂). The injectivity ofX with respect to θ_µ(x1) proves that the triple(α, x₂, β)is a transition ofX. Let(α, x₁, . . . , x_n, β)be a transition of X with n ≥ 2. Let y₁, . . . , y_n be n actions of X with µ(x_i) = µ(y_i) for 1 ≤ i ≤ n. The injectivity of X with respect toη_{µ(x1),...,µ(xn)} proves that the triple (α, y₁, . . . , y_n, β) is a transition of X. So, X is combinatorially fibrant.

Corollary 7.15. LetXbe a cubical transition system. IfXis fibrant, then it is combinatorially fibrant.

Proof. LetXbe a fibrant cubical transition system. Then it is injective with respect to any trivial cofibration ofL_CubCTS. By Proposition 7.12, Proposi-tion 7.13 and ProposiProposi-tion 7.14, it is then combinatorially fibrant.

Corollary 7.16. A cubical transition systemX is fibrant inL_CubCTSif and only it is combinatorially fibrant.

Corollary 7.17. Every S-injective cubical transition system is fibrant in L_CubCTS.

Proof. Let (α, u₁, . . . , u_n, β) and (α, v₁, . . . , v_n, β) as in the statement of Theorem 7.16. Since XisS-injective, the labelling mapµis one-to-one by Proposition 6.2. Thereforeu_i =v_i for1≤i≤n.

In particular, all cubical transition systems of the formL^CTS_S (X)and all regular transition systems of the formL^RTS_S (X)are fibrant because they are S-injective.

Dans le document VOLUME LVI-4 (Page 39-50)