Cédric Ho

Thanh Chaitanya

Leena Subramaniam

OPETOPIC ALGEBRAS III: PRESHEAF MODELS OF HOMOTOPY-COHERENT OPETOPIC ALGEBRAS

Keywords: Mathematics Subject Classification. Primary 18G55; Secondary 18C20 Opetope, Higher categoy, Higher operad, Complete Segal space, Complete Dendroidal Segal space, Model category

For Λ the category of free opetopic algebras, we construct a model structure à la Cisinski on the category of presheaves over Λ, and show that is is equivalent to opetopic complete Segal spaces. This generalizes the results of Joyal and Tierney in the simplicial case, and of Cisinski and Moerdijk in the planar dendroidal case.

Introduction

This work is a continuation of [START_REF] Ho | Opetopic algebras ii: Homotopy-coherent opetopic algebras[END_REF] and [START_REF] Ho | Opetopic algebras iii: Presheaf models of homotopy-coherent opetopic algebras[END_REF]. In the former, we introduced the notion of opetopic algebra, which can be thought of as algebras whose operations have higher dimensional arities. In the latter, we lifted the notion of Segal and complete Segal space from the simplicial and dendroidal setting [START_REF] Rezk | A model for the homotopy theory of homotopy theory[END_REF][START_REF] Cisinski | Dendroidal sets as models for homotopy operads[END_REF] to our opetopic framework. In this paper, we provide a presheaf model for complete Segal spaces, namely opetopic ∞-algebras. Much like in the simplicial case, where they are called quasi-categories [START_REF] Boardman | Homotopy invariant algebraic structures on topological spaces[END_REF][START_REF] Joyal | Notes on quasi-categories[END_REF], ∞-algebras are opetopic algebras where laws only hold up to homotopy, instead of strictly. Our strategy to prove this equivalence is largely adapted from that of Joyal and Tierney [START_REF] Joyal | Quasi-categories vs Segal spaces[END_REF].

1.1. Related work. Complete Segal spaces, introduced in [START_REF] Rezk | A model for the homotopy theory of homotopy theory[END_REF], are simplicial spaces that have compositions and identities up to homotopy. In [START_REF] Joyal | Quasi-categories vs Segal spaces[END_REF], Joyal and Tierney showed that they can be modeled by quasi-categories, that is, fibrant objects in a suitable model structure on the category Psh(∆) of simplicial sets. Specifically, they show that the discrete space functor is a left Quillen equivalence (-) disc ∶ Psh(∆) Joyal → ← Sp(∆) Rezk ∶ (-) -,0 where the right adjoint is the "first row" functor.

The notion of complete Segal space has been lifted to the dendroidal setting by Cisinski and Moerdijk in [START_REF] Cisinski | Dendroidal Segal spaces and ∞-operads[END_REF]. In this setting, they are dendroidal spaces having operadic compositions and identities up to homotopy. In [START_REF] Cisinski | Dendroidal sets as models for homotopy operads[END_REF], they show that ∞-operads, model complete dendroidal Segal spaces in the same sense as before. Furthermode, since the category of simplicial sets (resp. spaces) can be recovered as a slice of the category of dendroidal sets (resp. spaces), the equivalence in the simplicial case can be recovered as a special case.

1.2. Plan. In section section 2, we recall elements of Cisinski's homotopical machinery from [START_REF] Cisinski | Les préfaisceaux comme modèles des types d'homotopie[END_REF], and specify the so-called Joyal-Tierney calculus [START_REF] Joyal | Quasi-categories vs Segal spaces[END_REF] to an arbitrary category of simplicial presheaves. In section 3, we survey some results of [START_REF] Ho | Opetopic algebras ii: Homotopy-coherent opetopic algebras[END_REF], and construct the folk model structure for opetopic algebras. In section 4, we consider presheaves over Λ, the category of free opetopic algebras, we construct the model structure for ∞-algebras following the methods of Cisinski presented in section 2.1. Lastly, in section 5, we establish a Quillen equivalence between ∞-algebras and complete Segal spaces of [START_REF] Ho | Opetopic algebras iii: Presheaf models of homotopy-coherent opetopic algebras[END_REF].

1.3. Acknowledgments. We are grateful to Pierre-Louis Curien for his patient guidance and reviews. The first named author has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 665850.

Preliminaries

2.1. Cisinski homotopy theory. We recall some results and constructions of [START_REF] Cisinski | Les préfaisceaux comme modèles des types d'homotopie[END_REF].

Definition 2.1.1 (Lifting property). Let C be a category, and l, r ∈ C → . We say that l has the left lifting property against r (equivalently, r has the right lifting property against l), written l ⋔ r, if for any solid commutative square as follows, there exists a (non necessarily unique) dashed arrow making the two triangles commute:

⋅ ⋅ ⋅ ⋅ l r (2.1.2)
If C has a terminal object 1, then for all X ∈ C, we write l ⋔ X for l ⋔ (X ↠ 1). Let L and R be two classes of morphisms of C. We write L ⋔ R if for all l ∈ L and r ∈ R we have l ⋔ r. The class of all morphisms r (resp. l) such that L ⋔ r (resp. l ⋔ R) is denoted L ⋔ (resp ⋔ R).

Definition 2.1.3 (Orthogonality). We say that l is left orthogonal to r (equivalently, r is right orthogonal to l), written l ⊥ r, if for any solid commutative square as in equation (2.1.2), there exists a unique dashed arrow making the two triangles commute. The relation ⊥ is also known as the unique lifting property. Note that notations of definition 2.1.1 also make sense when ⋔ is replaced by ⊥.

Notation 2.1.4. For C a small category, let Psh(C) ∶= Cat(C op , Set) be the category of Set-presheaves over C, and Sp(C) ∶= Cat(C op , Psh(∆)) be the category of simplicial presheaves over C.

Definition 2.1.5 (Cylinder object, [Cis06, definition 1.3.1]). Let C be a small category, and X ∈ Psh(C) be a presheaf over C. A cylinder of X is a factorization of the fold map

X + X X IX, ∇ (i0,i1) ∇
such that (i 0 , i 1 ) ∶ X + X → IX is a monomorphism. We write X (e) for the image of i e ∶ X → IX.

Definition 2.1.6 (I-homotopy, [Cis06, definition 1.3.3, remark 1.3.4]). Let C be a small category, f, g ∶ X → Y be two parallel maps in Psh(C), and IX be a cylinder of X (definition 2.1.5). An elementary I-homotopy from f to g is a morphism H ∶ IA → B such that the following triangle commutes:

A + A IA B.
(i0,i1) (f,g)

H

Let ≃ I (or just ≃ is the context is clear), the I-homotopy relation, be the equivalence relation spanned by this relation on Psh(C)(A, B).

On readily checks that ≃ is a congruence on the category Psh(C), and let Ho Psh(C) be the quotient category. A morphism f ∶ X → Y is a I-homotopy equivalence (or just homotopy equivalence if the context is clear) if it is invertible in Ho Psh(C).

Definition 2.1.7 (Elementary homotopical data, [Cis06, definition 1.3.6]). Let C be a small category. An elementary homotopical data on Psh(C) is a functorial cylinder I ∶ Psh(C) → Psh(C) (definition 2.1.5) such that

(1) (DH1) I preserves small colimits and monomorphisms;

(2) (DH2) for all monomorphism f ∶ X → Y in Psh(C), and for e = 0, 1, the following square is a pullback:

X Y IX IY. f ie ie If
Definition 2.1.8 (Anodyne extension). Let C be a small category, and I ∶ Psh(C) → Psh(C) be an elementary homotopical data on Psh(C) (definition 2.1.7). A class of anodyne extensions relative to I is a class An ⊆ Psh(C) → such that

(1) (An0) there exists a set A ⊆ Psh(C) → of monomorphisms such that An = ⋔ (A ⋔ ) (definition 2.1.1); (2) (An1) for all monomorphism m ∶ X ↪ → Y in Psh(C), and e = 0, 1, the cocartesian gap map g is in An:

X Y IX IX ∪ Y (e)
IY ;

m ie ⌜ ie g
(3) (An2) for all m ∶ X ↪ → Y in An, the cocartesian gap map g is in An:

X + X Y + Y IX IX ∪ (Y + Y ) IY. m+m ⌜ g
Definition 2.1.9 (Homotopical structure, [Cis06, definition 1.3.14]). Let C be a small category. A homotopical structure on Psh(C) is a pair (I, An), where I is a functorial cylinder on Psh(C) (definition 2.1.5), and An is a class of anodyne extension relative to I definition 2.1.8.

Definition 2.1.10 (Cisinski model category). A Cisinski model category is a model structure on a presheaf category over a small category, that is cofibrantly generated, and where cofibrations are the monomorphisms.

A notable source of such structures is the following theorem:

Theorem 2.1.11 ([Cis06, definition 1.3.21, theorem 1.3.22]). Let C be a small category, and (I, An) be a homotopical structure on Psh(C) (definition 2.1.9). There is a model structure on Psh(C) such that:

(1) a morphism f is a naive fibration if An ⋔ f ; a presheaf X ∈ Psh(Λ) is fibrant if the terminal morphism X → 1 is a naive fibration; (2) a morphism f ∶ X → Y is a weak equivalence if for all fibrant object P ∈ Psh(Λ), the induced map

f * ∶ Ho Psh(Λ)(Y, P ) → Ho Psh(Λ)(Y, P ) is a bijection (definition 2.1.6); (3) a morphism f is a cofibrations if it is a monomorphisms, it is a acyclic cofibrations if it
• (Sq0) (Invariance) Isomorphisms are in C + and C -. Furthermore, if c, c ′ ∈ C are isomorphic, then deg c = deg c ′ . • (Sq1) (Dimension) If f ∶ c → c ′ is an arrow in C + (resp. C -) that is not an isomorphism, then deg c < deg c ′ (resp. deg c > deg c ′ ). • (Sq2) (Factorization) Every arrow f of C can be essentially uniquely factored as f = f + f -, with f + ∈ C + and f -∈ C -. • (Sq3) (Section)

Joyal-Tierney calculus. Consider the fully faithful functor

i C ∶ C ↪ → C × ∆ mapping c ∈ C to the tuple (c, [0]). It induces an adjunction (-) disc ∶ Psh(C) → ← Sp(C) ∶ (-) -,0
, where (-) disc is the left Kan extension of the composite C → C × ∆ → Sp(C) along the Yoneda embedding, and where (-) -,0 is the precomposition by i C , i.e. the "evaluation at 0". We call (-) disc the discrete space functor, as for X ∈ Psh(C) and c ∈ C, the space

X disc c is discrete at X c . Dually, the projection p ∆ ∶ C × ∆ → ∆ induces an adjunction (-) const ∶ Psh(∆) → ← Sp(C) ∶ r,
where r is the right Kan extension of C × ∆ → ∆ → Psh(∆) along the Yoneda embedding, and (-) const is the precomposition by p ∆ . We call (-) const the constant space functor, as for Y ∈ Psh(∆), the functor Y const ∶ C op → Psh(∆) is constant at Y . The functor r ∶ Sp(C) → Psh(∆) provides a simplicial enrichment on Sp(C) as follows:

Map(X, Y ) ∶= r(Y X ). Note that Map(X, Y ) n = Sp(C)(X × ∆[n] const , Y ).
Proposition 2.2.1 (Generalization of [JT07, proposition 2.4]). The simplicially enriched category Sp(C) is tensored and cotensored over Psh(∆): for K ∈ Psh(∆) and X ∈ Sp(C), we define

K ⊗ X ∶= K const × X, X K ∶= X K const . Proof. Let Y ∈ Sp(C).
(1) We have

Map(K ⊗ X, Y ) = Map(K const × X, Y ) = ∫ c∈C Map(K × X c , Y c ) ≅ ∫ c∈C Map (K, Map(X c , Y c )) ≅ Map(K, ∫ c∈C Map(X c , Y c )) = Map (K, Map(X, Y )) ,
naturally in all variables, and thus -⊗ X is an enriched left adjoint to Map(X, -).

(2) We have

Map(X, Y K ) = Map(X, Y K const ) ≅ Map(K const × X, Y ) = Map(K ⊗ X, Y )
naturally in all variables, and thus (-) K is an enriched right adjoint to K ⊗ -. □

We now specify the so-called Joyal-Tierney calculus [JT07, section 2] and [RV14, section 4] to the category Sp(C) of simplicial presheaves over C. This convenient formalism will be heavily used throughout this work. Definition 2.2.2 (Box product). The box product

1 functor ⊠ ∶ Psh(C) × Psh(∆) → Sp(C) is defined as (X ⊠ Y ) c,n ∶= X c × Y n , for X ∈ Psh(C), Y ∈ Psh(∆), c ∈ C, and [n] ∈ ∆.
This functor is divisible on both sides, meaning that is is left adjoint in each variable. The right adjoint to X ⊠-∶ Psh(∆) → Sp(C) will be denoted by X/-∶ Sp(C) → Psh(∆), and the right adjoint to -⊠ Y ∶ Psh(C) → Sp(C) will be denoted by -/Y ∶ Sp(()C) → Psh(C). Note that X/-and -/Y are contravariant in X and Y respectively. Consequently, for W ∈ Sp(()C), the functors -/W ∶ Psh(C) → Psh(∆) and W /-∶ Psh(∆) → Psh(C) are mutually right adjoint.

Lemma 2.2.3.

(1) For X ∈ Psh(C), we have

X disc = X ⊠∆[0]. Dually, for Y ∈ Psh(∆), we have Y const = 1⊠Y , where 1 is the terminal presheaf in Psh(C). (2) Let W ∈ Sp(C). For c ∈ C, we have c/W = W c ∈ Psh(∆). Dually, for [n] ∈ ∆, we have W /∆[n] = W -,n .
(3) Let h be a morphism in Sp(C), and K ∈ Psh(C).

Then K/h = ⟨(∅ ↪ K)/h⟩. Similarly, if L ∈ Psh(∆), then h/L = ⟨h/(∅ ↪ L)⟩. (4) Let X ∈ Sp(C), and f be a morphism in Psh(C). Then f /X = ⟨f /(X ↠ 1)⟩. Likewise, if g is a morphism in Psh(∆), then X/g = ⟨(X ↠ 1)/g⟩. Definition 2.2.4 (Leibniz construction, [RV14, definition 4.4]). Consider a functor -⊗ -∶ A × B → C, where C has pushouts. Its Leibniz construction is the functor -⊗ -∶ A → × B → → C → which maps an arrow f ∶ A 1 → A 2 in A and g ∶ B 1 → B 2 in B
to the cocartesian gap map below:

A 1 ⊗ B 1 A 1 ⊗ B 2 A 2 ⊗ B 1 P A 2 ⊗ B 2 , A1⊗g f ⊗B1 ⌜ f ′ f ⊗B2 g ′ A2⊗g f ⊗g P = A 1 ⊗ B 2 ∐ A1⊗B1 A 2 ⊗ B 1 .
1 The box product is denoted by ◻ in [JT07, section 2]. In [START_REF] Riehl | The theory and practice of Reedy categories[END_REF]notation 4.1], it is called the exterior product and written ×.

The Leibniz construction -⊗ -essentially has the same properties as -⊗ -, see [RV14, section 4].

Definition 2.2.5 (Leibniz box product). The Leibniz box product

2 -⊠ -∶ Psh(C) → × Psh(∆) → → Sp(C)
→ is simply the Leibniz construction of definition 2.2.4 applied to the box product of definition 2.2.2. If K and L are classes of morphisms of Psh(C) and Psh(∆) respectively, let

K ⊠ L ∶= {k ⊠ l | k ∈ K, l ∈ L}.
Akin to the box product, the Leibniz box product is divisible on both sides. Specifically, if h ∶ W 1 → W 2 is a morphism in Sp(C), let ⟨f /h⟩, be the morphism induced by the universal property of the pullback:

X 2 /W 1 P X 1 /W 1 X 2 /W 2 X 1 /W 2 , f /W1 ⟨f /h⟩ X2/h ⌟ X1/h f /W2 P = (X 1 /W 1 ) ∏ X1/W2 (X 2 /W 2 ) .
The morphism ⟨f /h⟩ is also called the cartesian gap map of the square above. Then the functor ⟨f /-⟩ is right adjoint to f ⊠ -. Dually, let ⟨h/g⟩, on the left, be the cartesian gap map of the square on the right

W 1 /Y 2 P W 2 /Y 2 W 1 /Y 1 W 2 /Y 1 h/Y2 ⟨h/g⟩ W1/g ⌟ W2/g h/Y1 P = (W 1 /Y 1 ) ∏ W2/Y1 (W 2 /Y 2 ) .
Then the functor ⟨-/g⟩ is right adjoint to -⊠ g.

2.3.

(c ⊠ ∂∆[n]) ∐ ∂c⊠∂∆[n] (∂c ⊠ ∆[n]) bc ⊠bn → c ⊠ ∆[n].
We apply proposition 2.1.15 to conclude. □ Definition 2.3.2 (Trivial fibration). We say that a morphism f (in some category) is a trivial fibration if it has the right lifting property with respect to all monomorphisms.

In model category theory, a trivial fibration is usually a fibration that is also a weak equivalence. In fact, the motivation for the terminology of definition 2.3.2 is that in the familiar Quillen model structure on Psh(∆), both notions coincide. More generally, they coincide in all Cisinski model category (definition 2.1.10). In general however, there is no reason for it to be the case, and both terminologies clash. To remedy this, we resort to a classical alternative: fibrations that are weak equivalences will be called acyclic fibrations throughout this paper. Likewise, we will favor the name acyclic cofibration instead of trivial cofibration.

Proposition 2.3.3 (Generalization of [JT07, proposition 2.3]). Let f ∶ X → Y be a morphism in Sp(C). The following are equivalent:

(1) f is a trivial fibration;

(2) the map ⟨b c /f ⟩ is a trivial fibration, for all c ∈ C;

(3) the map ⟨u/f ⟩ is a acyclic fibtation, for all monomorphism u in Psh(C);

(4) the map ⟨f /b n ⟩ is a trivial fibration, for all n ∈ N;

(5) the map ⟨f /v⟩ is a trivial fibration, for all monomorphism v in Psh(∆).

Proof

/f ⟩ is a fibration in Psh(E). (3) A map f ∶ X → Y in Psh(E) D op
is an acyclic Reedy fibration if and only for all d ∈ D, the relative matching map ⟨b d /f ⟩ is an acyclic fibration in Psh(E).

Proof. For the fist point, observe that

∂d/X = ⎛ ⎜ ⎝ colim d ′ →d in D+,not iso d ′ ⎞ ⎟ ⎠ /X ≅ lim d ′ →d in D+,not iso (d ′ /X) ≅ lim d ′ →d in D+,not iso X d ′ ,
which is the matching object of X at d. The second claim is by definition, and the third is [START_REF] Philip S Hirschhorn | Model categories and their localizations[END_REF]theorem 15.3.15]. □

For the the rest of this section, we assume that Psh(C) is endowed with a Cisinski model structure (definition 2.1.10).

Definition 2.3.5. Since C is normal skeletal, it is a Reedy category. Let Sp(C) v , the vertical model structure on

Sp(C) ≅ Sp(∆) C op
, be the Reedy structure induced by the Quillen model structure on Psh(∆). In this structure, a map f ∶ X → Y is a weak equivalence (also called column-wise weak equivalence) if for all c ∈ C, the map of simplicial sets c/f = f c ∶ X c → Y c is a weak equivalence. It is a fibration (also called vertical fibration) if for all c ∈ C, the relative matching map ⟨b c /f ⟩ is a Kan fibration, where b c ∶ ∂c → c is the boundary inclusion of c. (2) the map ⟨u/f ⟩ is a Kan fibration, for all monomorphism u ∈ Psh(C);

(3) the map ⟨f /h k n ⟩ is a trivial fibration, where

h k n ∶ Λ k [n] → ∆[n] is the k-th horn inclusion of [n]
, for all n ∈ N and 0 ≤ k ≤ n; (4) the map ⟨f /v⟩ is a trivial fibration, for all anodyne extension v ∈ Psh(∆).

Definition 2.3.8 (Homotopically constant space [RSS01, section 3]

). A space X ∈ Sp(C) is homotopically constant if for all map f ∶ [k] → [l] in ∆, the structure map X/f ∶ X -,l → X -,k is a weak equivalence 4 .
Lemma 2.3.9. Let X ∈ Sp(C). The following are equivalent:

(1) X is homotopically constant;

(2) for all k ∈ N, writing s ∶ [k] → [0] the terminal map in ∆, the structure map X/s ∶ X -,0 → X -,k a weak equivalence;

(3) for all codegeneracy

s i ∶ [k] → [k-1] in ∆, the structure map X/s i ∶ X -,k-1 → X -,k is a weak equivalence; (4) for all k ∈ N and all map d ∶ [0] → [k] in ∆, the structure map X/d ∶ X -,k → X -,0 is a weak equivalence; (5) for all coface map d i ∶ [k] → [k + 1] in ∆, the structure map X/d i ∶ X -,k+1 → X -,k is a weak equivalence.
Proof.

• (1) ⇒ (2) ⇒ (3) and (1) ⇒ (4) ⇒ (5) are trivial.

• (2) ⇒ (1) Take a map f ∶ [k] → [l] in ∆. Clearly, s = sf , so X/s = (X/f )(X/s). By 3-for-2, X/f is a weak equivalence. • (3) ⇒ (2) Note that the terminal map s ∶ [k] → [0] is a composite of codegeneracies [k] → [k -1] → ⋯ → [1] → [0]. • (4) ⇒ (2) The terminal map s ∶ [k] → [0] is a retraction of any map d ∶ [0] → [k]
, thus X/s is a section of X/d. By 3-for-2, X/s is a weak equivalence.

• (5) ⇒ (4) Note that all map d ∶ [0] → [k] is a composite of coface
-,0 ∶ X -,0 → Y -,0 is a weak equivalence.
Proof. Let n ∈ N and s ∶ [n] → [0] be the terminal map in ∆. We have a commutative square

X -,0 Y -,0 X -,n Y -,n f-,0 X/s Y /s f-,n
where by proposition 2.3.10, the vertical morphisms are weak equivalences. The result follows by 3-for-2. □

Opetopic algebras

3.1. Remiders. In this section, we fix once and for all a parameter n ≥ 1 and k ≥ 0, and write Z for Z n , Alg for Alg k (Z n ), A for A k,n , etc. Recall the main construction of [START_REF] Ho | Opetopic algebras i: Algebraic structures on opetopic sets[END_REF], namely the reflective adjunction h ∶ Psh(O ≥n-k ) → ← Alg ∶ N between the category of truncated opetopic sets (i.e. trivial below dimension n-k), and k-colored n-opetopic algebras. It exhibits Alg as the localization A -1 k,n Psh(O), or equivalently, as the orthoginality class A ⊥ k,n (definition 2.1.3). Let Λ be the full subcategory of Alg spanned by the image of h, i.e. the full subcategory of free algebras. Taking h ! ∶ Psh(O ≥n-k ) → Psh(Λ) to be the left Kan extension of O ≥n-k h → Λ → Psh(Λ) along the Yoneda embedding, and v ∶ Psh(Λ) → Alg to be the left Kan extension of the inclusion Λ ↪ → Alg along the Yoneda embedding, the reflection h factors as h ≅ vh ! :

Alg Psh(O ≥n-k ) Psh(Λ), N ⊥ M ⊥ h h !

Proposition 3.1.2. The adjunction

v ∶ Psh(Λ) → ← Alg ∶ M is reflective, i.e.
M is an embedding of categories. Proof. Recall that N is an embedding, i.e. fully faithful and injective on objects. In particular, M must be injective on objects as well. By [Web07, theorem 4.10], M is fully faithful.

□ Definition 3.1.3 (Spine). Let ω ∈ O n+1 , let S ω ∶= O n-k,n /S[ω] be the category of elements of the spine S[ω] ∈ Psh(O n-k,n ) of ω.
Define the spine of hω to be the colimit

S[hω] ∶= colim (S ω → O n-k,n h → Λ → Psh(Λ)) .
We have an inclusion s hω ∶ S[hω] → hω called the spine inclusion of hω and let S ∶=

{s hω ∶ S[hω] → hω | ω ∈ O n+1 } be the set of spine inclusions of Psh(Λ). Lemma 3.1.4. The functor h ! ∶ Psh(O) → Psh(Λ) maps S n+1 ⊆ Psh(O) to S ⊆ Psh(Λ), and morphisms in S >n+1 ⊆ Psh(O) to S-local isomorphisms. Proof. Recall from [HTLS19, lemma 4.5.1] that ḣ! ∶ Psh(O n-k,n+2 ) → Psh(Λ) maps S n+1 to S and S n+2 to S-local isomorphisms. Thus, so does h ! ∶ Sp(O ≥n-k ) → Sp(Λ).
Take ω ∈ O m with m ≥ n + 3 and consider the following square in Psh(O): Proof. By [Web07, theorem 4.10], a presheaf X ∈ Psh(∆) is in the essential image of M if and only if h * X is in the essential image of N . In other words, X is an algebra if and only if S ≥n+1 ⊥ h * X, which under the adjunction h ! ⊣ h * , is equivalent to S ⊥ X. □

S[t ω] S[ω] O[t ω] O[ω].

Free opetopic algebras.

Recall that Λ = Λ 1,n is the full subcategory of Alg spanned by free algebras on opetopes in O n-1,n+1 . However, if ϕ ∈ O n-1 , then hϕ = hI ϕ , and if ψ ∈ O n , then hψ = hY ψ . So we may consider Λ to be the full subcategory of Alg spanned by free algebras over (n + 1)-opetopes.

Definition 3.2.1 (Diagram). For ω, ω ′ ∈ O n+1 and f ∶ hω → hω ′ in Λ, a diagram [HTLS19] of f is the datum of a cospan ω s [p] → ξ t ← ω ′ , with ξ ∈ O n+2 , such that f = (h t) -1 (h s [p]
). The situation is summarized as follows:

ξ ω ω ′ s [ p ] t hω hω ′ f (3.2.2)
The morphism f is said to be diagrammatic if it admits a diagram.

Lemma 3.2.3 (Diagrammatic lemma, [HTLS19]). Let f ∶ hω → hω ′ be a morphism in Λ. If ω is not degenerate, then f is diagrammatic. Lemma 3.2.4. Let ω ∈ O n+1 = tr Z n-1 and ψ ∈ O n . Then hω ψ = (N hω) ψ is the set of sub-Z n-1 -trees ν of ω such that t ν = ψ. Proof. By definition, hω ψ = ∑ ν∈On+1 t ν=ψ Psh(O)(S[ν], O[ω]),
and morphisms S[ν] → O[ω] are precisely the Z n-1 -tree embeddings of ν in ω. □ Lemma 3.2.5. Let f 1 , f 2 ∶ hω → hω ′ be two morphism in Λ, with ω, ω ′ ∈ O n+1 . Then f 1 = f 2 if and only if for all [p] ∈ ω • , f 1 c ω,[p] = f 2 c ω,[p]
. In other words, f 1 = f 2 if and only if f 1 and f 2 agree on all one-node subtrees of ω.

Proof. Under the adjunction Psh(O ≥n-1 ) → ← Alg, f i corresponds to a morphism fi ∶ O[ω] → hω ′ , and since S ≥n+1 ⊥ hω ′ , if is uniquely determined by its restriction fi ∶ S[hω] → hω ′ . □ Notation 3.2.6. As a consequence of lemma 3.2.5, if ω ∈ O n+1 is not degenerate, the algebra hω is freely generated by its one-node subtrees. We denote these generators by c ω, [p] , where [p] ranges over ω • .

3.3. Quotients. Let X ∈ Alg. Recall that X can be seen as an opetopic set X ∈ Psh(O) such that A k,n ⊥ X. We now present a notation that shows how solutions of lifting problems S[hω] → X can really be understood as compositions of "tree-shaped arities".

For ψ ∈ O n-1 and x ∈ X ψ , write id x ∈ X Y ψ for the target of the solution of the lifting problem

S[I ψ ] = O[ψ] X O[I ψ ]. x ∃!f Explicitely, if f is the solution, let id x be the cell of X Y ψ selected by O[Y ψ ] t → O[I ψ ] f → X. Let ω, ω ′ ∈ O n , and [p] ∈ ω • such that s [p] ω = t ω ′ , so that Y ω ○ [[p]] Y ω ′ is a well defined (n + 1)-opetope. For x ∈ X ω , x ′ ∈ X ω ′ such that s [p] x = t x ′ , let x ○ [p]
x ′ be the target of the solution of the following lifting problem:

S[Y ω ○ [[p]] Y ω ′ ] []↦x,[[p]]↦x ′ → X.
An iterated composition as on the right can be concisely written as on the left:

x ◯ [pi] y i ∶=(⋯(x ○ [p1] y 1 ) ○ [p2] y 2 ⋯) ○ [p k ] y k , where x ∈ X ω , ω • = {[p 1 ], . . . , [p k ]}, y i ∈ X ωi , and t ω i = s [pi] ω.
Let X ∈ Alg be an opetopic algebra, ω ∈ O n , and x, y ∈ X ω . We say that x and y are parallel if the following two composites are equal:

∂O[ω] O[ω] X. bω x y
In that case, let the quotient of X by the equality x = y be the following coequalizer in Alg:

hω X X/ {x = y} . x y
For each ω ∈ O n , take a set K ω ⊆ X n × X n of pairs of parallel cells, and let K ∶= ∑ ω K ω . Then the quotient of X by K is the following coequalizer in Alg:

∑ ω∈On ∑ (x,y)∈Kω hω X X/K, p1 p2
where p 1 (x, y) ∶= x and p 2 (x, y) ∶= y.

3.4. The folk model structure. In this section, fix a parameter n ≥ 1, and take k = 1. As usual, we omit n and k from most notations, e.g. Alg = Alg 1 (Z n ). Let A ∈ Alg be an algebra, ψ ∈ O n , and d ∶ ∂hψ → A, and consider the following pullback

A d A hψ = Psh(Λ)(hψ, A) * Psh(Λ)(∂hψ, A). ⌟ b * hψ d Explicitely, A d is the subset of A hψ of all cell a such that s [q] a = d(s [q] ψ) for all [q] ∈ ψ • , and t a = d(t ψ).
Example 3.4.1. Take n = 2, so that Alg is the category of planar colored operads. For P ∈ Alg and

k ∈ O 2 , a morphism d ∶ ∂hk → P is just a (k + 1)-tuple of colors (d 0 , . . . , d k-1 ; d k ) of P , where d i = d s [ * i ] k for 0 ≤ i < k, and d k = d t k. The set P d is simply P (d 0 , . . . , d k-1 ; d k ). Definition 3.4.2 (Internal isomorphism). Let f ∶ A → B be a morphism of algebras. Let ϕ ∈ O n-1 and a ∈ A hY ϕ . If x = s [] a and y = t a, we write a ∶ x → y. We say that a is an internal isomorphism (or just isomorphism) if it is invertible, i.e. if there exists a cell a -1 ∈ A hY ϕ such that a ○ [] a -1 = id y , a -1 ○ [] a = id x .
In this case, we also say that x and y are isomorphic, and write x ≅ y.

Definition 3.4.3 (Natural transformation). Let f, g ∶ A → B be two parallel morphisms of algebras. A natural transformation α ∶ f → g is a collection of cells α a ∶ f (a) → g(a)
(called components, see definition 3.4.2 for notation) such that for all ψ ∈ O n and x ∈ A hψ , the following relation holds:

g(x) ◯ [q] α s [q] a = α t a ○ [] f (x).
Natural transformations can be composed in the obvious way, and the adequate exchange law holds. A natural isomorphism is an invertible natural transformation, or equivalently, on whose compoments are all isomorphisms.

Definition 3.4.4 (Algebraic equivalence). Let f ∶ A → B be a morphism of algebras.

(1) We say that f is fully faithful if for all ψ ∈ O n and for all morphism d ∶ ∂hψ → A, the induced map

f d ∶ A d → B f d is a bijection. (2)
We say that f is essentially surjective of for all b ∈ B n-1 , there exist an a ∈ A n-1 such that f (a) ≅ b.

(3) We say that f is an algebraic equivalence (or equivalence of algebras, or simply equivalence) if it is fully faithful and essentially surjective. Clearly, if n = 1, we recover the notion of fully faithful functor between categories, whereas if n = 2, this matches the definition of operadic equivalence of [START_REF] Moerdijk | Dendroidal sets[END_REF].

Proposition 3.4.5. A morphism f ∶ A → B of
. For b ∈ B n-1 , choose an a = g(b) ∈ A n-1 and an isomorphism η b ∶ b → f (g(b)). Let ψ ∈ O n and y ∈ B hψ be such that t y is in the image of f , say t y = f (a) for some a ∈ A n-1 . Then z ∶= y ◯ [q] η -1 s [q] y
where [q] ranges over ψ • , has all its faces in the image of f , as

s [q] z = s [] η -1 s [q] y = f (g(s [q] y)).
Since f is fully faithful, there exist a unique g(y) in A hψ such that f (g(y)) = z. This defines a morphism g ∶ B → A, and a natural isomorphism η ∶ id B → f g.

Let a ∈ A n-1 , and consider the isomorphism e ∶= η -1 f (a) . Note that e ∶ f gf (a) → f (a), and since f is fully faithful, there exists a unique ε a ∶ gf (a) → a such that e = f (ε a ). It is straightforward to check that the ε a s are the component of a natural isomorphism ε ∶ gf → id A . □ Lemma 3.4.6. Let f ∶ A → B be an equivalence that is injective on (n -1)-cells. Then f admits a retract up to isomorphism, i.e. a weak inverse g ∶ B → A together with natural isomorphisms ε ∶ gf → id A and η ∶ id B → f g as in proposition 3.4.5, but where gf = id A and ε is an identity (i.e. all its compoments are identities), and where f = f gf , and for a

∈ A n-1 , η f (a) = id f (a)
Proof. It suffices to amend the proof of proposition 3.4.5 so that g, ε and η have the desired properties. For a ∈ A n-1 , since f is injective on objects, we may choose g(f (a)) to be a, and η f (a) to be id f (a) . It follows that, after extending g to a morphism B → A, we have gf

= id A . Further, for a ∈ A n-1 , ε a ∶ gf (a) → a is the only n-cell of A such that f (ε a ) = η -1 f (a) = id f (a) , whence ε is the identity. □ Definition 3.4.7 (Isofibration). A morphism f ∶ A → B of algebras is an isofibration if for all a ∈ A n-1 , all isomorphism g ∶ f (a) → b in B, there exists an isomorphism g ′ ∶ a → a ′ in A such that f (g ′ ) = g.
Theorem 3.4.8 (Generalization of [JT07, theorem 1.4]). The category Alg = Alg 1 (Z n ) of 1-colored algebras admits a model structure where the weak equivalences are the algebraic equivalences (definition 3.4.4), the cofibrations are those morphism that are injective on (n -1)-cells, and where the fibrations are the isofibrations (definition 3.4.7).

We call this stucture the folk model structure, and denote it by Alg folk . Furthermore, acyclic fibrations are the algebraic equivalences that are surjective on (n -1)-cells, and every object is both fibrant and cofibrant.

The second claim can easily be checked once the model structure is established. The rest of this section is dedicated to proving this. To that end, we verify each of Quillen's axioms, cf [Hir03, definition 7.1.3].

Proof of theorem 3.4.8, (M1): limit axiom. Since Alg is the category of models of a small projective sketch [START_REF] Ho | Opetopic algebras i: Algebraic structures on opetopic sets[END_REF], it is locally presentable [AK80, corollary 1.52], and therefore has all small limits and colimits.

□ where i is a cofibration and p is a fibration. We show that a lift exists whenever i or p is a weak equivalence.

Proof
(1) Assume that i is a weak equivalence, and let r ∶ C → A be a weak retract of i as in lemma 3.4.6, together with the natural isomorphism η ∶ id C → ir. Let c ∈ C n-1 , and consider gir(c) = pf r(c) ∈ D n-1 . Since p is an isobibration, there exist an isomorphism

β c ∶ f r(c) → b in B such that p(β c ) = g(η -1 c ) ∶ gir(c) = pf r(c) → g(c). In particular, p(b) = g(c), and define l(c) ∶= b. This defined a lift l ∶ C n-1 → B n-1 .
The construction above also provides a natural isomorphism β ∶ f r → l. Without loss of generality, we choose l and the components of β such that for all a ∈ A n-1 , li(a) = f (a), and

β i(a) = id f (a) . For ψ ∈ O n and x ∈ C hψ , let l(x) ∶= β t x ○ [] f r(x) ◯ [q] β -1 s [q] x .
It can easily be checked that l is then a morphism of algebras C → B, and finally the desired lift. (2) Assume that p is a weak equivalence. In particular, on (n -1)-cells, i is an injection, and p a surjection, and so a lift l ∶ C n-1 → B n-1 can be found. We now extend l to a morphism of algebras (1) We decompose f as f = pi, where i is an acyclic cofibation, and where p is fibration. Define C ∈ Psh(Λ) as follows. For ϕ ∈ O n-1 , let

l ∶ C → B. Let ψ ∈ O n , x ∈ C ψ ,
C hϕ ∶= {(a, v, b) | a ∈ A hϕ , b ∈ B hϕ , v ∈ B Y ϕ isomorphism f (a) → b} . There is ab obvious projection proj ∶ C n-1 → A n-1 , and if ψ ∈ O n and d ∶ ∂hψ → C n-1 , let C d ∶= A proj d .
At this stage, C clearly extends as an algebra, essentially inheriting the same law as A.

Let i ∶ A → C map an (n -1)-cell a to (a, id f (a) , f (a)). This completely determines i, as indeed, for an n-cell x ∈ A ≥n we necessarily have i(x) = x. Clearly, i is a fully faithful cofibration. It remains to show that it is essentially surjective.

If (a, v, b) ∈ C n-1 , note that id a exhibits an isomorphism i(a) = (a, id f (a) , f (a)) → (a, v, b) in C. Therefore, i is an acyclic cofibration. Let p ∶ C → B be the obvious projection (a, v, b) → b on (n -1)-cells. Let ψ ∈ O n and x ∈ C hψ . Write t x = (a t , v t , b t ), for a source address [q] ∈ ψ • , write s [q] x = (a [q] , v [q] , b [q]
), and define

p(x) ∶= v t ○ [] f (x) ◯ [q] v -1 [q] .
This defines a morphism p ∶ C → B which we claim to be a fibration. Indeed, if

(a, v, b) ∈ C n-1 and w ∶ b → b ′ is an isomorphism in C, we have an isomorphism id a ∶ (a, v, b) → (a, w ○ [] v, b ′ ) in C, and p(id f (a) ) = (w ○ [] v) ○ [] f (id a ) ○ [] v = w.
Lastly, it is clear that f = pi, and thus f decomposes as a acyclic cofibration followed by a fibration. (2) We decompose f as f = pi, where i is an cofibation, and where p is an acyclic fibration. Define D ∈ Psh(Λ) as follows. On (n -1)-cells, it is given as on the left, and let f be defined as on the right: Let i ∶ A → D map an (n -1)-cell a to a, and an n-cell x to f (x). Obviously, this is a cofibration. Let p ∶ D → A map an (n -1)-cell d to f (d), and an n-cell x to x. This can easily be seen to be an acyclic fibration. Lastly, f = pi, so that f can be decomposed into a cofibration followed by an acyclic fibration. □ Definition 3.4.9 (Rezk interval). Let ϕ ∈ O n-1 . The Rezk interval of shape ϕ is the algebra J ϕ ∈ Alg generated by one invertible operation j ϕ of shape Y ϕ . Explicitely, J ϕ has two (n -1)-cells 0 ϕ , 1 ϕ ∈ (J ϕ ) ϕ , and four n-cells j ϕ , j -1 ϕ , id 0 , id 1 ∈ (J ϕ ) Y ϕ , satisfying the following equalities

D n-1 ∶= A n-1 + B n-1 , f ∶=(f, id Bn-1 ) ∶ D n-1 → B n-1 . Explicitely, f maps a ∈ A n-1 to f (a)
s [] j ϕ = t j -1 ϕ = 0 ϕ , t j ϕ = s [] j -1 ϕ = 1 ϕ , j ϕ ○ [] j -1 ϕ = id 1 , j -1 ϕ ○ [] j ϕ = id 0 .
Note that up to isomorphism, there is a unique endpoint inclusion j ϕ ∶ hϕ → J ϕ , and let

E J ∶= {j ϕ | ϕ ∈ O n-1 }. If X = (X ψ | ψ ∈ O n-1 ) is a set over O n-1 , let J X ∶= ∑ ψ ∑ x∈X ψ J ψ .
In the x ∈ X ψ component, we write j x instead of j ψ , and similarly, j -1

x , 0 x , and 1 x . The Rezk interval (without any mention of shape) is the sum

J = J On-1 ∶= ∑ ψ J ψ in Alg.
Theorem 3.4.10. The model structure Alg folk is cofibrantly generated, and E J can be taken as a set of generating acyclic cofibrations.

Proof. Clearly, a morphism f is an isofibration if and only if E J ⋔ f . It is straightforward to check that f is an acyclic fibration of and only if J ⋔ f , where J is

{∅ ↪ → hϕ | ϕ ∈ O n-1 } ∪ vB ∪ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ hψ ∐ ∂hψ hψ → hψ | ψ ∈ O n ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ ,
(see definition 4.2.1 for the definition of B). □ 3.5. Homotopy equivalences.

Definition 3.5.1 (Rezk cylinder of an algebra). Let A ∈ Alg, and consider the following pushout in Alg:

hA n-1 + hA n-1 A + A J An-1 A ′ . ⌜ (i0,i1)
For a a cell of A, write a (e) ∶= i e (a), for e = 0, 1. Explicitely, A ′ is generated by two copies of A, and for each a ∈ A ψ , ψ ∈ O n-1 , an additional cell j a of shape Y ψ with s [] j a = a (0) and t j a = a (1) , which can informally be written j a ∶ a (0) → a (1) .

The Rezk cylinder of A is the quotient JA ∶= A ′ /K, where

K ∶= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ a (1) ◯ [pi] j s [p i ] a = j t a ○ [] a (0) | a ∈ A ω , ω ∈ O n , ω • = {[p 1 ], . . .} ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ . (3.5.2)
See section 3.3 for the ○ notation. It is a cylinder object in the sense of definition 2.1.5, i.e. we have a canonical factorization of the codiagonal map

A + A JA A. ∇ (i0,i1) ∇ Explicitly, ∇ ∶ JA → A maps a (e)
to a, for a cell a ∈ A, and j a to id a , if a is (n -1)-dimensional. Note that (i 0 , i 1 ) ∶ A + A → JA is a monomorphism, since the relation K of equation (3.5.2) does not identify cells of eA, for e = 0, 1. We write A (e) the image of i e , and by abuse of notation, i e ∶ A (e) ↪ → JA the obvious inclusion.

Example 3.5.3. Let ψ ∈ O n . Then Jhψ is generated by (1) ψ (0) , ψ (1) ∈ (Jhψ) ψ , i.e. two copies of hψ, (2) for each [p] ∈ ψ • , and writing ϕ ∶= s [p] ψ, two cells j ϕ , j -1 ϕ ∈ (Ihψ) Y ϕ , i.e. one copy of J ϕ , (3) j t ψ , j t ψ , i.e. one copy of J t ψ , subject to the relation equation (3.5.2). Likewise, for ω ∈ O n+1 , the cylinder Ihω is generated by two copies of hω, and one copy J e [q] ω for all edge address [q] of ω. Remark 3.5.4. The cylinder JA of an algebra A can be thought of the Boardman-Vogt tensor product [Wei11, BV73, May72] J ⊗ BV A. Unfortunately, in the planar case, a general such construction is not possible, as there is no way to "shuffle the inputs". Since this is the only obstruction, one could still define a A ⊗ BV B as long as either A of B only have unary (i.e. endotopic) operations. Lemma 3.5.5. Let f, g ∶ A → B be two parallel morphisms of algebras. The following are equivalent:

(1) f ≃ g (definition 2.1.6);

(2) there is an elementary J-homotopy from f to g;

(3) there exist a natural isomorphism f → g.

Proof.

• (1) ⇒ (3). A homotopy H from f to g as in definition 2.1.6 induces a natural isomorphism Ĥ with components Ĥa = H(j a ) (see definition 3.5.1 for notations), for a ∈ A n-1 . Proof. Follows from proposition 3.4.5 and lemma 3.5.5. □

• (3) ⇒ (2). A natural isomorphism α ∶ f → g induces a homotopy α ∶ JA → B from f to g,

The homotopy theory of ∞-opetopic algebra

We fix a parameter n ≥ 1, and assume that k = 1. In [START_REF] Ho | Opetopic algebras i: Algebraic structures on opetopic sets[END_REF], we built a reflective adjunction h ∶ Psh(O ≥n-1 ) → ← Alg ∶ N. We consider N to be the inclusion of a full subcategory Alg ↪ → Psh(O ≥n-1 ), and omit it if unambiguous. 4.1. A skeletal structure on Λ. In this section, we endow Λ = Λ 1,n with the structure of a skeletal category (cf. section 2.1). We first need to asign a notion of degree to any objects λ ∈ Λ, denoted by deg λ ∈ N. Next, we need to specify two casses of morphisms Λ + and Λ -satisfying axioms (Sq0) to (Sq3). Definition 4.1.1. Recall that the objects of Λ are the free algebras on (n + 1)-opetopes.

For ω ∈ O n+1 , let deg hω ∶= # ω • . Definition 4.1.2. Let f ∶ hω → hω ′ be a morphism in Λ, with ω, ω ′ ∈ O n+1 .
In particular, it induces a set map between (n -1)-cells:

f n-1 ∶ hω n-1 → hω ′ n-1 . Let Λ + (resp. Λ -)
be the wide subcategory spanned by those morphisms f such that f n-1 is an monomorphism (resp. epimorphism).

The rest of this section is dedicated to prove the following result. Proof of theorem 4.1.3, axiom (Sq0). Since Λ is rigid (i.e. has no isomorphisms beside the identities), this axiom holds trivially.

□ Lemma 4.1.4. Let f ∶ hω → hω ′ be a morphism in Λ, with ω, ω ′ ∈ O n+1 . The following are equivalent: (1) f ∈ Λ + (resp. Λ -); (2) the set map between n-cells f n ∶ hω n → hω ′ n is an monomorphism (resp. epimorphism); (3) f is a monomorphism in Psh(O ≥n-1 ) (resp. epimorphism).
Proof. Recall that h ∶ Psh(O ≥n-1 ) → Alg is the localization with respect to the set s ≥n+1 . Therefore, f ≥n is an monomorphism (resp. epimorphism) if and only if f n is. It remains to show that f n-1 is a monomorphism (resp. epimorphism) if and only if f n is. But this is a direct consequence of lemma 3.2.4, stating that hω n is the set of sub-Z n-1 -trees of ω. □ Corollary 4.1.5. Let f ∶ hω → hω ′ be a morphism in Λ, with ω, ω ′ ∈ O n+1 . The following are equivalent:

(1) f ∈ Λ + ∩ Λ -, i.e. f n-1 is an isomorphism;

(2) f n is an isomorphism;

(3) f is an isomorphism;

Proof. Recall that in any presheaf category, the epi-monos are exactly the isomorphisms. □

Proof of theorem 4.1.3, axiom (Sq1). Let f ∶ hω → hω ′ be a morphism in Λ that is not an isomorphism, with ω, ω ′ ∈ O n+1 . If f ∈ Λ + , then by lemma 4.1.4 and corollary 4.1.5, f n is a monomorphism that is not an isomirphism, thus # hω n < # hω ′ n , i.e. the number of subtrees of ω is strictly less than that of ω ′ . In particular, t he number of nodes of ω is strictly less than that of ω ′ , i.e. deg hω < deg hω ′ .

The case where f ∈ Λ -is treated similarly. □

Proof of theorem 4.1.3, axiom (Sq2). Let f ∶ hω → hω ′ be a morphism in Λ, with ω, ω ′ ∈ O n+1 . It maps the maximal subtree ω ⊆ ω to a subtree of ω ′ , say ν ⊆ ω ′ . Then the following is the desired factorization:

hω hω ′ hν. f f
Uniqueness comes from the fact that Λ + only contains inclusions (of Psh(O ≥n-1 )). □

Proof of theorem 4.1.3, axiom (Sq3). Let f ∶ hω → hω ′ be a morphism in Λ -. We define a section g of f . By adjointness, this is equivalent to specifying a map

ḡ ∶ O[ω ′ ] → hω in Psh(O ≥n-1 ). Since S ≥n+1 ⊥ hω, it is enough to define ḡ on the spine S[ω ′ ] of ω ′ . For [p] ∈ (ω ′ ) • , let ḡ(s [p] ω ′ ) ∶= s [q] ω, [q] ∶= min {[r] ∈ ω • | f (s [r] ω) = s [p] ω ′ } .
In other words, g maps the node s [p] ω ′ to the lexicographically minimal node in the fiber f -1 (s [p] ω ′ ). The composite

S[ω ′ ] ḡ → hω f → hω ′ maps a node s [p] ω ′ to s [p] ω ′ , thus g is a section of f .
Let now f 1 , f 2 ∶ hω → hω ′ be a morphism in Λ -having the same sections. In particular, for g i the section of f i constructed as above, where i = 1, 2, we have (2) Otherwise, for f an inner face of λ, define Λ f λ, the inner horn of λ at f , as the colimit on the left, or equivalently, as the union on the right

f 1 g 2 = id hω ′ . Thus for [p] ∈ (ω ′ ) • , we have g 2 (s [p] ω ′ ) ⪯ g 1 (s [p] ω ′ ), meaning that the node g 2 (s [p] ω ′ ) is lexicographically inferior to (or "below") g 1 (s [p] ω ′ ) in ω. Conversely, since f 2 g 1 = id hω ′ , we have g 1 (s [p] ω ′ ) ⪯ g 2 (s [p] ω ′ ), and finally, g 1 (s [p] ω ′ ) = g 2 (s [p] ω ′ ). By lemma 3.2.5, g 1 = g 2 , so clearly, f 1 = f 2 . □ 4.
λ ′ = ⋃ f ∶λ ′ →λ in Λ+, deg λ ′ =deg λ-1 im f. Explicitely, if λ = hω, for ω ∈ O n+1 ,
Λ f λ ∶= colim g∶λ ′ →λ elem. face g≠f λ ′ = ⋃ g∶λ ′ →λ elem. face g≠f im g. Let h f λ ∶ Λ f λ → λ
Λ J hω u ↪ → Λ I hω h J hω
↪ → hω, and u is a cell complex of inner horn inclusions of opetopes with at most d -1 nodes.

Proof. We proceed by induction on m ∶= #(J -I). If m = 0, then u is an identity, and the result holds trivially. Assume the lemma holds up to m -1. Take f ∈ J -I, and let J ′ ∶= J -{f }. The inclusion u decomposes as

Λ J hω v ↪ → Λ J ′ hω w ↪ → Λ I hω
and by induction, w is a cell complex of inner horn inclusions of opetopes with at most d -1 nodes. It remains to show that v is too.

The inner face f of hω exhibits a subtree ν ⊆ ω with two nodes, or equivalently, an inner edge of ω, say at address [e]. Let ω /[e] be ω where this inner edge has been contracted. Explicitely, decomposing ω on the left so as to exhibits the subtree ν, the opetope ω /[e] is defined on the right:

ω = α ○ [p] ν ◯ [li] β i , ω /[e] ∶= α ○ [p] Y t ν ◯ [℘ν [li]] β i ,
where [l i ] ranges over ν | . Then the inclusion w ∶ Λ J hω → Λ J ′ hω above is obtained as the following pushout

Λ J ′ hω /[e] Λ J hω hω /[e] Λ J ′ hω. ⌜ h J ′ ω/[e]
Λ I hω u ↪ → Λ J hω h J hω ↪ → hω.
By induction on m, h J hω is an inner anodyne extension. By lemma 4.2.8, u is a cell complex of inner horn inclusions of opetopes of at most d -1 nodes, and so by induction on d, u is an inner anodyne extension as well. □ Lemma 4.2.10. Let ω ∈ O n+1 have d ≥ 2 nodes, and I be the set of all inner faces of ω. Note that Λ I hω contains all the generators c ω, [p] , and thus the spine inclusion s hω decomposes as

S[hω] u ↪ → Λ I hω h I hω ↪ → hω.
Then the inclusion u is a cell complex of spine inclusions of opetopes of at most d -1 nodes.

Proof. Note that an outer face (all elementary faces that are not inner) of hω are inclusions ω ′ ⊆ ω of subtrees of ω of d -1 nodes. Hence Λ I hω is the union on the left, while S[hω] can be expressed as on the right

Λ I hω = ⋃ ω ′ ⊆ω deg hω ′ =d-1 hω ′ , S[hω] = ⋃ ω ′ ⊆ω deg hω ′ =d-1 S[hω ′ ]. Thus u ∶ ⋃ ω ′ ⊆ω deg hω ′ =d-1 S[hω ′ ] ↪ → ⋃ ω ′ ⊆ω deg hω ′ =d-1 hω ′
can be obtained as a composition of pushouts of the s ω ′ 's. Explicitely, write ω 1 , . . . , ω m ⊆ ω be the subtrees of ω with d -1 nodes, and g 1 , . . . , g m the associated outer face inclusions of hω. Define X 0 ∶= S[ω], ι 0 be the identity on X 0 , and for 1 ≤ i ≤ m, let X i be the pushout

S[ω i ] X i-1 hω i X i , ⌜ ιi-1gisω i v and ι i ∶= vι i-1 . Then Λ I hω = X m . □ Proposition 4.2.

(Generalization of [CM13, proposition 2.4]).

We have an inclusion ⋔ (S ⋔ ) ⊆ An inner .

Proof. It is enough to show that spine inclusions s hω ∶ S[hω] ↪ → hω are inner anodyne extensions.

(1) If deg hω = # ω • = 0, 1, then s hω is an identity, thus an inner anodyne extension.

(2) If deg hω = 2, then hω admits a unique inner face f ∶ hY t ω → hω, and note that Λ f hω = S [hω]. Thus in this case, the spine inclusion is an inner horn inclusion. (3) Assume d = deg hω ≥ 3, and let I be the set of all inner faces of hω. Note that since Λ I hω contains all generators of hω (i.e. subtrees of one node), the spine inclusion s hω decomposes as

S[hω] u ↪ → Λ I hω h f hω ↪ → hω,
and in order to show that s hω is an inner anodyne extension, it suffices to show that u is. By lemma 4.2.10, it is a cell complex of spine inclusions of opetopes of at most d -1 nodes, thus by induction, it is an inner anodyne extension. □ Proposition 4.2.12. Inner anodyne extensions are S-local isomorphisms.

S[hω] u ↪ → Λ I hω v ↪ → Λ f hω h f hω ↪ → hω.
By lemma 4.2.10, u is a spine complex, thus an S local-isomorphism. By lemma 4.2.8, v is a cell complex of inner horn inclusions of opetopes with at most d -1 nodes. By induction, v is an S local-isomorphism. Thus vu and s hω = h f hω ⋅ (vu) are S local-isomorphisms, and by 3-for-2, so is h f hω . □ In dimension (n -1) and n, IX is the following pushout

X n-1 + X n-1 X n-1,n + X n-1,n M J Xn-1 (IX) n-1,n , ⌜ b
where b maps a x ∈ X ψ in the first (resp. second) component to 0 x (resp. 1 x ) in M J x ⊆ M J Xn-1 . The (n + 1)-cells of IX are so that in vIX, the following relation (analogous to equation (3.5.2)) holds: Lemma 4.4.3. Let X ∈ Psh(∆), be such that the unit map η X ∶ X → M vX is a monomorphism. Then η X is an inner horn complex, and in particular an inner anodyne extension.

x (1) ◯ [pi] j s [p i ] x = j t x ○ [] x (0) for a cell x ∈ X ω , ω ∈ O n ,
Proof. Let X (0) ∶= X and ι (0) ∶= η X . If x ∈ X (0) , we write deg x = 0. Assume by induction that we have an inclusion ι (α) ∶ X (α) ↪ → M vX for all ordinal α < β, and a degree function deg ∶ X (α) → α + 1.

If β is a limit ordinal, simply set X (β) ∶= ⋃ α<β X (α) , and ι (β) to be the induced inclusion. Assume that β is a successor ordinal, say β = α + 1, and choose a horn lifting problem l ∶ Λ f hω → X (α) such that

(1) (L1) l is unsolved in X (α) , (2) (L2) l has degree ≤ 2 only if there is no unsolved horn lifting problem of degree ≥ 3. If such an l does not exist, simply set X (α+1) ∶= X (α) and ι (α+1) ∶= ι (α) . If it does, set X (α+1) to be the following pushout

Λ f hω X (α) hω X (α+1) , ⌜ l u
and ι (α+1) to be the induced map X (α+1) → M vX. For x ∈ X (α+1) -X (α) , set deg x ∶= α + 1. We claim that ι (α+1) is a monomorphism. Towards a contradiction, assume that it is not. For ν ⊆ ω the two-nodes subtree corresponding to the inner face f , write x ∈ X (α+1) hν the cell selected by the arrow

hν ↪ → hω u → X (α+1)
Then there exist y ∈ X (α) hν such that ι (α+1) (x) = ι (α+1) (y).

(1) If l has degree ≤ 2, then the lifting problem l was not unsolved in X (α) , a contradiction with condition (L1).

(2) If the degree of l is ≥ 3, then the cell y fives a factorization of l as

Λ f hω X (α) ∂hω, l l ′
with l ′ unsolved in X (α) . But this contradicts condition (L2). Indeed, the step of the induction that created y but not u(hω) only considered a lifting problem of degree 2, whereas l could have been considered instead. Therefore, ι (α+1) is a monomorphism.

If κ is the cardinal of the set of horn lifting problems of M vX. Then the sequence X α stabilizes after κ + 1, as all lifting problems have been exhausted. Clearly,

M vX ≅ colim α<κ+1 X (α) ,
and by construction, η X is an inner horn complex. □ Corollary 4.4.4 (Generation lemma). Take X ∈ Psh(Λ), A ∈ Alg, and m ∶ X ↪ → M A be a monomorphism such that its transpose m ∶ vX → A is an isomorphism. Then m is an inner horn complex, and in particular, an inner anodyne extension.

Proof. The condition states that up to isomorphism, m is the unit map η X ∶ X → M vX. We can apply lemma 4.

a (1) = j t a ○ [] a (0) ◯ [qi] j -1 s [q i ] a
where [q i ] ranges over (s [p] ω)

∶ X = M hϕ ↪ → Y = M J ϕ , for a ϕ ∈ O n-1 . We thus have IX ∪ (Y + Y ) = IM hϕ ∪ (M J ϕ + M J ϕ ) = M J ϕ ∪ (M J ϕ + M J ϕ ).
We have

IY = I(M J ϕ ) = M (Jv(M J ϕ )) ≅ M (JJ ϕ ).
The algebra JJ ϕ contains four (n -1)-cells 00 ϕ , 01 ϕ , 10 ϕ , and 11 ϕ of shape ϕ, and is generated by the n-cells j

(0 * ) ϕ ∶ 00 ϕ → 01 ϕ , j (1 * ) ϕ ∶ 10 ϕ → 11 ϕ , j ( * 0) ϕ ∶ 00 ϕ → 10 ϕ , j (1 * ) ϕ
∶ 10 ϕ → 11 ϕ and their inverses. Further, the equality on the right holds, which can be depicted as a commutative square of invertible arrows on the right:

j ( * 1) ϕ ○ [] j (0 * ) ϕ = j (1 * ) ϕ ○ [] j ( * 0) ϕ , 00 ϕ 01 ϕ 10 ϕ 11 ϕ . j (0 * ) ϕ j ( * 0) ϕ j ( * 1) ϕ j (1 * ) ϕ
On the other hand, the pushout

M J ϕ ∪ (M J ϕ + M J ϕ ) contains j (0 * ) ϕ , j ( * 0) ϕ , j ( * 1) ϕ
, and their inverses. Thus it generates JJ ϕ , i.e. the cocartesian gap map g satisfies the conditions of corollary 4.4.4. Consequently, it is an inner anodyne extension. □ Theorem 4.4.7. The category Λ endowed with the functorial cylinder I ∶ Psh(Λ) → Psh(Λ) and the class An I of I-anodyne extension forms a homotopical structure (definition 2.1.9) Using theorem 2.1.11, we obtain the following model structure for ∞-algebras Psh(Λ) ∞ on Psh(Λ):

(1) a morphism f is a naive fibration if An I ⋔ f (definition 4.4.1); a presheaf X ∈ Psh(Λ) is fibrant if the terminal morphism X → 1 is a naive fibration; (2) a morphism f ∶ X → Y is a weak equivalence if for all fibrant object P ∈ Psh(Λ), the induced map f * ∶ Ho Psh(Λ)(Y, P ) → Ho Psh(Λ)(Y, P ) is a bijection, where ≃ is the I-homotopy relation of definitions 2.1.6 and 4.3.

2; (3) a morphism f is a cofibrations if it is a monomorphisms, it is a acyclic cofibrations if it is a cofibration and a weak equivalence; (4) a morphism f is a fibration if it has the right lifting property with respect to acyclic cofibrations, it is an

acyclic fibration if it has the right lifting property with respect to all cofibrations. In particular, Psh(Λ) ∞ is of Cisinski type (definition 2.1.10), cellular, and proper. Fibrant objects in Psh(Λ) ∞ are called ∞-algebras (or inner Kan complexes).

Proof. By definition, An I is the class of cell complexes over a set of monomorphisms, thus it satisfies axiom (An0). Axioms (An1) and (An2) are checked by propositions 4.4.5 and 4.4.6 respectively. □

This construction is a direct generalization of the model structure Psh(∆) Joyal for quasi-categories [JT07, theorem 1.9], and of Psh(Ω) CM for planar ∞-operads [CM11, theorem 2.4]. Lemma 4.4.8. For all X ∈ Psh(Λ) we have E I ⋔ X. In particular, if H inner ⋔ X if and only X is an ∞-algebra.

Proof. Let ϕ ∈ O n-1 , and consider a lifting problem f ∶ hϕ → X. In particular, f exhibits a cell x ∈ X hϕ , but also a cell id x ∈ X hY ϕ . Then a lift f of f as in hϕ X

I ϕ f i ϕ f
can be obtained by mapping 0 ϕ and 1 ϕ to x, and j ϕ and j -1 ϕ to id x (see definition 3.4.9 for notations). □

The following results comes as a "sanity-check" for the model structure of theorem 4.4.7:

Proposition 4.4.9. For A ∈ Alg, its nerve M A is an ∞-algebra.

Proof. By theorem 3.1.5, S ⊥ M A, and thus by proposition 4.2.12, H inner ⊥ M A. We apply lemma 4.4.8 to conclude. □ Proposition 4.4.10. We have a Quillen

adjunction v ∶ Psh(Λ) ∞ → ← Alg folk ∶ M . Proof. Trivially, if f ∶ X → Y is a monomorphism in Psh(Λ)
, then vf is injective on (n -1)-cells. Therefore, v preserves cofibrations. To conclude, it suffices to show that M preserves fibrations.

First, note that v maps H inner to isomorphisms, and since the adjunction is reflective, it maps E I = M E J (see definition 4.3.1) to E J up to isomorphism. Therefore, v maps An I to acyclic cofibrations. Let now f be a fibration in Alg folk . Then f has the right lifting property against all acyclic cofibrations, and in particular, vAn I ⋔ f . By adjointness, An I ⋔ M f , and thus M f is a naive fibration. By proposition 4.4.9, the codomain of M f is an ∞-algebra, and we apply lemma 2.1.12 to conclude that M f is a fibration. □ Note that Psh(Λ) Quillen is still of Cisinski type, cellular, and proper. In this structure, fibrant objects are called Kan complexes, and fibrations are Kan fibrations. For instance, in the case (k, n) = (1, 1), we recover the classical Quillen structure on simplicial sets. 4.5. Underlying categories. This section is devoted to prove the technical lemma 4.5.7. If n = 1, i.e. Λ 1,1 = ∆, then the result is stated in [JT07, proposition 1.13] and [START_REF] Joyal | Notes on quasi-categories[END_REF]2.13]. The bulk of the work is to study the underlying category adjunction

C ϕ! ∶ Psh(∆) → ← Psh(Λ) ∶ C * ϕ , ( 4
.5.1) where ϕ ∈ O n-1 , to reduce a similar statements for an arbitrary n ≥ 1 to n = 1.

In this section, we assume n ≥ 1, and for once, do not omit if from various notations, e.g. Λ 1,n , Alg 1,n , etc. Recall that Λ 1,1 = ∆ and Alg 1,1 = Cat. In this case, the adjunction

v 1,n ∶ Psh(Λ 1,n ) → ← Alg 1,n ∶ M 1,n is more commonly denoted by τ ∶ Psh(∆) → ← Cat ∶ N . Pick ϕ ∈ O n-1 ,

and define an functor

C ϕ ∶ ∆ → Λ 1,n as [0] → hϕ, [1] → hY ϕ , [i] → h (Y ϕ ○ [] Y ϕ ○ [] ⋯ ○ [] Y ϕ ) ,
where on the right, there are i instances of Y ϕ . In other words, C ϕ maps [i] to hω, where ω ∈ O n+1 is the linear tree with i nodes, all decorated by Y ϕ ∈ O n . Note that C ϕ is an embedding that exhibits ∆ as an sieve of Λ 1,n in that of for all morphism λ → λ ′ in Λ 1,n , if λ ′ ∈ im C ϕ , then so is λ. This functor induces an adjunction

C ϕ! ∶ Psh(∆) → ← Psh(Λ 1,n ) ∶ C * ϕ , (4.5.2)
where

C ϕ! is the left Kan extension of ∆ C ϕ → Λ → Psh(Λ 1,n
) along the Yoneda embedding, and C * ϕ is the precomposition by C ϕ . Since C ϕ is an embedding, so is C ϕ! , and furthermore, it exhibits Psh(∆) as an sieve of Psh(Λ 1,n ). The following result follows directly from this observation:

Lemma 4.5.3. Let k ∈ N. (1) We have C ϕ! S[k] ≅ S[C ϕ [k]]. (2) If Λ l [k] is an inner horn of [k], then C ϕ! Λ l [k] is an inner horn of C ϕ [k]
, and all inner horns of C ϕ [k] are obtained in this way.

Proof. The spine and inner horns of C ϕ [k] are obtained as colimits of presheaves over C ϕ [k]. Since C ϕ! exhibits Psh(∆) as an sieve of Psh(Λ 1,n ), those colimits can be computed in Psh(∆). □ Lemma 4.5.4. The adjunction equation (4.5.2) restrics and corestricts as an adjunction C ϕ! ∶ Cat → ← Alg 1,n ∶ C * ϕ . We thus have three commutative squares, where the last one is obtained by adjunction from the second one:

Psh(∆) Psh(Λ 1,n ) Cat Alg 1,n , C ϕ! N M1,n C ϕ! Psh(∆) Psh(Λ 1,n ) Cat Alg 1,n , C * ϕ N M1,n C * ϕ Psh(∆) Psh(Λ 1,n ) Cat Alg 1,n . C ϕ! τ v1,n C ϕ! Proof. (1) Take K ∈ Psh(∆). Note that if ϕ ′ ∈ O n-1 , ϕ ′ ≠ ϕ, then (C ϕ! K) ϕ ′ = ∅, i.e. C ϕ! K does not have cells of shape ϕ ′ ≠ ϕ. If ω ∈ O ≥n-1 , ω ∉ im C ϕ , then ω has a (n -1)-dimensional face different from ϕ, whence Psh(Λ 1,n )(S[hω], C ϕ! K) = ∅. By lemma 4.5.3 and the fact that C ϕ! is fully faithful, if S 1,1 ⊥ K, then S 1,n ⊥ d ϕ,! K, and C ϕ! restrics and corestricts as a functor Cat → Alg 1,n . (2) Take A ∈ Alg 1,n , i.e. a presheaf A ∈ Psh(Λ 1,n ) such that S 1,n ⊥ A. By lemma 4.5.3, C ϕ! S 1,1 ⊆ S 1,n , so in particular, C ϕ! S 1,1 ⊥ A, whence S 1,1 ⊥ C * ϕ A, i.e. C * ϕ A ∈ Cat. □ Lemma 4.5.5. (1) For A ∈ Cat, we have C ϕ! JA ≅ JC ϕ! A. (2) For K ∈ Psh(∆), we have C ϕ! IK ≅ IC ϕ! K. (3) Let f, g ∶ K → L be two parallel maps in Psh(∆). If f ≃ g (definition 2.1.6), then C ϕ! f ≃ C ϕ! g. Proof.
Point (1) is by definition, and (3) follows from (2). To prove (2), consider

C ϕ! IK = C ϕ! N Jτ K definition 4.3.2 ≅ M 1,n C ϕ! Jτ K lemma 4.5.4 ≅ M 1,n JC ϕ! τ K lemma 4.5.4 ≅ M 1,n Jv 1,n C ϕ! K lemma 4.5.4 = IC ϕ! K definition 4.3.2.
□ Proposition 4.5.6. We have a Quillen adjunction

C ϕ! ∶ Psh(∆) ∞ → ← Psh(Λ 1,n ) ∞ ∶ C * ϕ . Proof. Clearly, C ϕ!
∆[k] ⊗ λ ∶= M (Jλ ∐ λ Jλ ∐ λ Jλ ∐ λ ⋯ ∐ λ Jλ) ,
where there is k instances of Jλ. In other words, it is the nerve of k instances of the cylinder Jλ "glued end-to-end". Extending in both variables by colimits yields a tensor product -⊗ -∶ Psh(∆) × Psh(Λ) → Psh(Λ).

Let us unfold the definition a little bit. Take X ∈ Psh(Λ) and K ∈ Psh(∆). Then for each x ∈ X hϕ with ϕ ∈ O n-1 , and k ∈ K 0 , there is a cell k ⊗ x ∈ (K ⊗ X) hϕ . For all edge e ∈ K 1 , there is an isomorphism e ⊗ x ∈ (K ⊗ X) hY ϕ with source d 1 e ⊗ x and target d 0 e ⊗ x. More generally, for every m-cell k ∈ K m , writing k 0 , . . . , k m ∈ K 0 its vertices, and k i,j its edge from k i to k j , where 0 ≤ i < j ≤ m, we have a cell

k ⊗ x ∈ (K ⊗ X) hω , ω ∶= C ϕ [m] = Y ϕ ○ [] Y ϕ ○ [] ⋯ ○ [] Y ϕ , such that s [ * i ] (k ⊗ x) = k i,i+1 ⊗ x.
With this description, it is clear that for I ◾ ∈ Psh(∆) the nerve of the groupoid generated by one isomorphism (definition 4.3.1), we have IX ≅ I ◾ ⊗ X for all X ∈ Psh(Λ). Definition 4.6.2. A mapping space and cotensor can be constructed from the tensor product ⊗ of definition 4.6.1 so as to make Psh(Λ) tensored and cotensored over Psh(∆):

Map(X, Y ) k ∶= Psh(Λ)(∆[k] ⊗ X, Y ), (Y K ) λ ∶= Psh(Λ)(K ⊗ λ, Y )
, where X, Y ∈ Psh(Λ), K ∈ Psh(∆), and k ∈ N.

Lemma 4.6.3. For K ∈ Psh(∆) and X, Y ∈ Psh(Λ), consider the natural hom-set isomorphism

Φ ∶ Psh(Λ)(K ⊗ X, Y ) → Psh(Λ)(X, Y K ) of the adjunction K ⊗ -∶ Psh(Λ) → ← Psh(Λ) ∶ (-) K .
The map Φ preserves and reflects the I-homotopy relation (definitions 2.1.6 and 4.3.2), i.e. it induces an isomorphism

Φ ∶ Psh(Λ)(K ⊗ X, Y )/≃ → Psh(Λ)(X, Y K )/≃.
Proof. It is enough to show that Φ preserves and reflects the elementary I-homotopy relation. Let f, g ∶ K ⊗X → Y be elementary homotopic maps, i.e. such that there exist a homotopy H ∶ I(K ⊗ X) → Y making the following triangle commute:

(K ⊗ X) + (K ⊗ X) I(K ⊗ X) Y. (i0,i1) f +g H Note that (K ⊗ X) + (K ⊗ X) ≅ K ⊗ (X + X), and I(K ⊗ X) ≅ I ◾ ⊗ K ⊗ X ≅ (I ◾ × K) ⊗ X ≅ (K × I ◾ ) ⊗ X ≅ K ⊗ IX.
Under the adjunction, the triangle above transposes as

X + X IX Y K , (i0,i1)
Φf +Φg ΦH exhibiting a homotopy from Φf to Φg. Reflection of homotopies is done similarly. □ Lemma 4.6.4. Let K ∈ Psh(∆).

(1) For ω ∈ O ≥n-1 , and e an inner face of hω, the map

K ⊗ h e hω ∶ K ⊗ Λ e hω → K ⊗ hω is an anodyne extension. (2) For ϕ ∈ O n-1 , the map K ⊗ i ϕ ∶ K ⊗ hϕ → K ⊗ I ϕ is an anodyne extension.
Proof. In both cases, it is enough to check the claim where K is representable, say K = ∆[m] for some m ∈ N.

( Let T ∈ Psh(Λ) be the sum of m + 1 copies of hϕ, and write T hϕ = { * 0 , . . . , * m }. Clearly, IT is the sum of m + 1 copies of I ϕ , and the inclusion i 0 ∶ T → IT is the sum of m + 1 copies of the endpoint inclusion i ϕ ∶ hϕ → I ϕ . There is an obvious inclusion t ∶ T → ∆[m] ⊗ hϕ mapping the * i to i m ⊗ * . Let X ∈ Psh(Λ) be defined by the following pushout:

T ∆[m] ⊗ hϕ IT X. ⌜ t i0 u
In other words, X is ∆[m] ⊗ hϕ where an instance of I ϕ has been glued to each cell i m ⊗ * . The map ∆[m] ⊗ i ϕ factors as

∆[m] ⊗ hϕ u ↪ → X v ↪ → ∆[m] ⊗ I ϕ ,
and by construction, u is an anodyne extension. On the other hand, v satisfies the conditions of corollary 4.4.4, as the cells not in its image can be obtained as composites of cells in X. Therefore, v is an anodyne extension, and so is ∆[m] ⊗ i ϕ . □ Corollary 4.6.5. For K ∈ Psh(∆), the functor (-) K preserves naive fibration (definition 2.1.10), and in particular, ∞-algebras.

Proof. Let L ∶= H inner ∪ E I , so that An I = ⋔ (L ⋔ ). Let p be a naive fibration, i.e. a map such that An I ⋔ p, or equivalently, such that L ⋔ p. By lemma 4.6.4, we have K ⊗ L ⋔ p, and by adjunction, L ⋔ p K . Consequently, p K is a naive fibration. □ Lemma 4.6.6.

(1) For K ∈ Psh(∆), the tensor K ⊗ -∶ Psh(Λ) ∞ → Psh(Λ) ∞ preserves cofibrations and weak equivalences.

(2) Let X ∈ Psh(Λ) be an ∞-algebra. Then -⊗ X ∶ Psh(∆) Quillen → Psh(Λ) ∞ preserves cofibrations, and weak equivalences between Kan complexes.

Proof.

(1) Clearly, K ⊗ -preserves monomorphisms. Let u ∶ X → Y be a weak equivalence, and P ∈ Psh(Λ) be an ∞-algebra. Recall from lemma 4.6.3 that we have a natural isomorphism

Φ ∶ Psh(Λ)(K ⊗ X, Y )/≃ → Psh(Λ)(X, Y K )/≃,
where Φ is the natural hom-set isomorphism of the adjunction K ⊗ -⊣ (-) K . Therefore, we have the following naturality square:

Psh(Λ)(K ⊗ X, P )/≃ Psh(Λ)(K ⊗ Y, P )/≃ Psh(Λ)(X, P K )/≃ Psh(Λ)(Y, P K )/≃. (K⊗u) * Φ Φ u *
The vertical maps are bijections. By corollary 4.6.5, P K is an ∞-algebra, thus u * is a bijection as well. Therefore, (K ⊗ u) * is a bijection for all ∞-algebra P . By definition, K ⊗ u is a weak equivalence. (2) Clearly, -⊗ X preserves monomorphisms. Let w ∶ K → L be a weak equivalence between Kan complexes.

By [Hov99, theorem 1.2.10], it is a homotopy equivalence, meaning that it admits an inverse w -1 ∶ L → K up to homotopy, and w -1 ⊗ X is an inverse of w ⊗ X up to homotopy. □ Corollary 4.6.7. Let u ∶ K → L be a cofibration between Kan complexes, v ∶ X → Y be a cofibration in Psh(Λ), and consider the Leibniz tensor u ⊗ v (definition 2.2.4). Then u ⊗ v is a cofibration. If either u or v is an acyclic cofibration, then so is u ⊗ v.

Proof. Surely, since u and v are monomorphisms, u ⊗ v is too. Assume that u is an acyclic cofibration. By lemma 4.6.6, u ⊗ X and u ⊗ Y are too, and so is the pushout u ′ of u ⊗ X along K ⊗ v. By 3-for-2, u ⊗ v is a weak equivalence. The case where v is an acyclic cofibration instead of u is done similarly. □ Proposition 4.6.8. Let X ∈ Psh(Λ) be an ∞-algebra, and v ∶ K → K be a cofibration (resp. acyclic cofibration) between Kan complexes. Then X v ∶ X L → X K is a fibration (resp. acyclic fibration).

Proof. Assume that v is an cofibration (resp. acyclic cofibration). In order to show that X v is an fibration (resp. acyclic fibration), we must show that u ⋔ X v for all acyclic cofibration (resp. cofibration) u in Psh(Λ) ∞ . This is equivalent to u ⊗ v ⋔ X. By corollary 4.6.7, u ⊗ v is an acyclic cofibration, and since X is an ∞-algebra, the result holds. □ Remark 4.6.9. Unfortunately, the model structure Psh(Λ) ∞ , together with the tensor and cotensor of definitions 4.6.1 and 4.6.2 cannot be promoted into a simplicial model category. In fact, this already fails if n = 1 [JT07, section 6].

∞-algebras vs. complete Segal spaces

In [START_REF] Ho | Opetopic algebras ii: Homotopy-coherent opetopic algebras[END_REF], we introduced the Segal and Rezk model structure on Sp(Λ) as Bousfield localization of the projective structure. Here, we take a slightly different approach by starting with the vertical (or Reedy) structure. This corresponds to the theory we generalize [JT07, CM13], while still being equivalent to the notions of [START_REF] Ho | Opetopic algebras ii: Homotopy-coherent opetopic algebras[END_REF] as there is a Quillen equivalence id ∶ Sp(Λ) proj → ← Sp(Λ) v ∶ id. Fibrant objects (resp. weak equivalences) in Sp(Λ) Segal are called Segal spaces (resp. Segal weak equivalences). Explicitely, a Segal space X ∈ Psh(Λ) is a vertically fibrant space such that for all ω ∈ O n-1 , the map s hω /X ∶ hω/X = X hω → S[hω]/X is a weak equivalence.

Lemma 5.1.2. Let K be a saturated class of monomorphism of Psh(Λ) having the right cancellation property, i.e. such that for all composable pair or morphisms f, g ∈ Psh(Λ), if f g, g ∈ K, then f ∈ K.

(1) (Generalization of [JT07, lemma 3.5] and [CM13, proposition 2.5]) If S ⊆ K, then An inner ⊆ K.

(2) (Generalization of [JT07, lemma 3.7]) If K contains all elementary face embeddings, then An ⊆ K.

Proof.

(1) Since K is saturated, we have Cell (S) ⊆ K, so by lemma 4.2.10, H inner ⊆ K. By saturation again, An inner ⊆ K.

(2) For ω ∈ O n+1 , let F (ω) be the set of elementary faces of hω. It suffices to show that for all ω ∈ O n+1 and non empty set I ⊊ F (ω), the inclusion h I hω ∶ Λ I hω → (cf. section 4.2 for notations) is in K. Take a set I ⊊ F (ω) and f ∈ F (ω) -I, say f ∶ hω ′ → hω. If I = F (ω) -{f }, then clearly, Λ I hω = im f ≅ hω ′ , and h I hω ≅ f ∈ K. Otherwise, let I ⊊ J ⊊ F (ω) be a set of elementary faces of hω not containing f . Then f factors as

hω ′ u ↪ → Λ J hω v ↪ → Λ I hω h I hω ↪ → hω.
Since f ∈ K be assumption, and since K has the right cancellation property, in order to prove that h I hω , it suffices to show that u, v ∈ K. (a) Assuming v ∈ K for all J ⊋ I not containing f , it is enough to show that u ∈ K in the case where J = F (ω) -{f }. But in this case, as before, Λ J hω = im f ≅ hω ′ , and u is an isomorphism. (b) It is enough to consider the case J = I + {g}, for some elementary face g ∈ F (ω) different from f . Since Λ I hω = im f ∪ Λ J hω, we have the following pushout square:

im f ∩ Λ J hω Λ J hω im f Λ I hω. ⌜ w v
Recall that im f ≅ hω ′ , and it is easy to see that the inclusion w ∶ im f ∩ Λ J hω ↪ → im f is isomorphic to an elementary face of hω ′ . Consequently, v is the pushout of w, which is in K by assumption, and thus v ∈ K. □ and likewise, (X/K) hϕ ≅ Psh(∆)(K, X hϕ ). By assumption, X hϕ is a Kan complex, and since v is an anodyne extension, the precomposition map on top is surjective

Psh(∆)(L, X hϕ ) Psh(∆)(K, X hϕ ) (X/L) hϕ (X/K) hϕ . v * ≅ ≅ (X/v) hϕ
Therefore, X/v is surjective on (n -1)-cells, and thus an acyclic fibration as claimed.

We now show that X is vertically fibrant. By proposition 2.3.7, this is equivalent to X/v being an acyclic fibration for all anodyne extension v ∈ Psh(∆). Let K be the class of anodyne extensions v such that X/v is an acyclic fibration. Note that by the first claim, this is equivalent to X/v being a weak equivalence. Using lemma 5.1.2, it suffices to show that K has the right cancellation property, is saturated, and that it contains all simplicial face maps.

(1) Right cancellation follows from 3-for-2.

(2) For staturation, note that v ∈ K if and only if b hω ⋔ X/v for all ω ∈ O ≥n-1 by definition. This is

equivalent to v ⋔ b hω /X, thus K = ⋔ {b hω /X) | ω ∈ O ≥n-1 } is saturated. (3) Lastly, let us show that the simplicial face maps d i ∶ ∆[n -1] → ∆[n] belong to K. Since X is
homotopically constant, the simplicial map X -,n → X -,0 is a weak equivalence for all n ∈ N, so by 3-for-2, X/d i is too, as displayed by the following triangle

X -,n X -,n-1 X -,0 . X/d i ∼ ∼
Therefore, by proposition 2.3.7, K contains all anodyne extensions, and by proposition 2.3.7, X is vertically fibrant. Lastly, we show that X satisfies the Segal condition definition 5.1.1. It suffices to show that for all ω ∈ O ≥n-1 , the map s hω /X is an acyclic fibration, i.e. that for all n ∈ N, we have b n ⋔ s hω /X. This is equivalent to s hω ⋔ X/b n , which holds since s hω is an inner anodyne extension by proposition 4.2.11. □ Proposition 5.1.7 (Generalization of [JT07, proposition 3.10]). Let f ∶ X → Y be a vertical fibration between two Segal spaces.

(1

) If u ∶ A → B is an inner anodyne extension in Psh(Λ) (definition 4.2.6), then ⟨u/f ⟩ ∶ B/X → B/Y ∏ A/Y A/X is an acyclic fibration. (2) If v ∶ K → L is a monomorphism in Psh(∆), then ⟨f /v⟩ ∶ X/L → Y /L ∏ Y /K X/K is an inner fibration between ∞-algebras.
Proof.

(1) By proposition 2.3.7, the map ⟨v/f ⟩ is a Kan fibration. It remains to show that it is a weak equivalence. Consider the following diagram:

B/X ⋅ A/X B/Y A/Y. u/X ⟨u/f ⟩ B/f p2 p1 ⌟ A/f u/Y
By proposition 5.1.3, u/X and u/Y are trivial fibrations, and so is the pullback map p 2 . By 3-for-2, ⟨u/f ⟩ is a weak equivalence.

(2) By proposition 5.1.3, X/S and X/L are ∞-algebras. In the pullback square

Y /L ∏ Y /K X/K X/K Y /L Y /K, ⌟ p2 Y /v
the bottom map Y /v is an inner fibration by proposition 5.1.3, and thus p 2 is too. Since X/K is an ∞-algebra, so is Y /L ∏ Y /K X/K. We now show that ⟨f /v⟩ is an inner fibration, i.e. that u ⋔ ⟨f /v⟩ for all u ∈ An inner . By (1), ⟨u/f ⟩ is an acyclic fibration, so v ⋔ ⟨u/f ⟩, and by adjunction, u ⋔ ⟨f /v⟩ as desired. □

∈ Psh(Λ) is complete if for all ϕ ∈ O n-1 , the map r ϕ /X ∶ X hϕ → I ϕ /X is a weak equivalence. Lemma 5.2.3 (Generalization of [JT07, lemma 4.2]). A Segal space X ∈ Sp(Λ) is complete if and only if for all ϕ ∈ O n-1 , the map i ϕ /X is a trivial fibration, where i ϕ ∶ hϕ → I ϕ is the endpoint inclusion of the Rezk interval I ϕ (definition 4.3.1).
Proof. By definition, X is complete if and only if for all ϕ ∈ O n-1 , the map r ϕ /X is a weak equivalence. Since r ϕ i ϕ = id hϕ , we have (i ϕ /X)(r ϕ /X) = id X hϕ , and by 3-for-2, i ϕ /X is a weak equivalence if and only if r ϕ /X is. On the other hand, i ϕ /X is always a Kan fibration by proposition 2. For X ∈ Psh(Λ), let ΓX ∈ Sp(Λ) be given by ΓX -,k ∶= X J(∆[k]) , where the cotensor is given in definition 4.6.2.

Lemma 5.2.8 (Generalization of [JT07, lemma 4.8]). Let C be a small category, and F ∶ Psh(∆) op → Psh(C) be a continuous functor, i.e. mapping colimits in Psh(∆) to limits in Psh(C). op under small limits, we are done. □ Proposition 5.2.9 (Generalization of [JT07, proposition 4.10]). Let X ∈ Sp(Λ) ∞ be an ∞-algebra. Then ΓX is a complete Segal space, and there is a canonical acyclic cofibration X disc → ΓX, thus exhibiting ΓX as a fibrant replacement of X disc in Sp(Λ) Rezk .

Proof.

(1) Note that X J(-) ∶ Psh(∆) op → Psh(Λ) is a continuous functor, thus by lemma 5.2.8, there exists G ∈ Sp(Λ) such that X J(-) ≅ G/-. In particular, for k ∈ N, we have G -,k ≅ G/∆[k] ≅ ΓX -,k , and therefore, G ≅ ΓX, and X J(-) ≅ ΓX/-. (2) As a consequence, and by proposition 4.6.8, ΓX/b k is a fibration, for all k ∈ N. Likewise, the map d ∶ ∆[0] → ∆[n] is an acyclic cofibration in Psh(∆) Quillen . Thereofre ΓX/d ∶ ΓX -,n → ΓX -,0 is a weak equivalence, and by lemma 2.3.9, ΓX is homotopically constant. By proposition 5.2.5, it is a complete Segal space. (3) Note that ΓX -,0 ≅ ΓX/∆[0] ≅ X J(∆[0]) = X, so we have an isomorphism f ∶ X → ΓX -,0 . We how extend f to a morphism γ ∶ X disc → ΓX. First, let γ 0 ∶= f ∶ X disc -,0 = X → ΓX -,0 . Next, note that in ∆, the terminal map s k ∶ [k] → [0] is a retraction of any iterated coface map d ∶ [0] → [k]. Thus, in addition to being a weak equivalence (as ΓX is homotopically constant), the structure map s k ∶ ΓX -,0 → ΓX -,k is a section of d ∶ ΓX -,k → ΓX -,0 , thus injective. Letting γ k ∶= s k f ∶ X disc -,k = X → ΓX -,k gives rise to the desired map γ ∶ X disc → ΓX which by construction is a horizontal acyclic cofibration. By theorem 5.2.6, it is a Rezk acyclic cofibration. □ Theorem 5.2.10 (Generalization of [JT07, proposition 4.7 and theorem 4.11]). We have a Quillen equivalence (-) disc ∶ Psh(Λ) ∞ → ← Sp(Λ) Rezk ∶ (-) -,0 (see section 2.2).

Proof.

(1) We first shows that the adjunction is Quillen. Clearly, (-) disc preserves monomorphisms and maps weak equivalences to horizontal weak equivalences, which by theorem 5.2.6 are Rezk weak equivalences. Therefore, (-) disac is left Quillen functor.

(2) We show that for a complete Segal space X ∈ Sp(Λ), the map (QX -,0 ) disc q disc → (X -,0 ) disc ε → X is a Rezk weak equivalence, where q ∶ QX -,0 → X -,0 a cofibrant replacement of X -,0 . First, since all objects are cofibrant in Psh(Λ) ∞ , whoose q to be an identity. Next, by theorem 5.2.6, it suffices to show that ε is a weak equivalence in Sp(Λ) h , i.e. that ε -,n ∶ (X -,0 ) disc -,n = X -,0 → X -,n is a weak equivalence. Clearly, ε -,n is induced by the terminal map [n] → [0] in ∆, and thus is a weak equivalence since X is homotopically constant by proposition 5.2.5.

(3) We show that for an ∞-algebra X ∈ Psh(Λ) ∞ , the map X ≅ → (X disc ) -,0 r-,0 → (RX disc ) -,0 is a Rezk weak equivalence, where r ∶ X disc → RX disc is a fibrant replacement of X disc . Choosing it to be γ ∶ X disc → ΓX (proposition 5.2.9) concludes the proof.

  d. The canonical map b c ∶ ∂c → c is a monomorphism and is called the boundary inclusion of c. Write B C (or just B if the context is clear) for the set of boundary inclusions of C. Proposition 2.1.15 ([Cis06, propositions 8.1.35 and 8.1.37]). Assume C is a normal skeletal category. Then the class of monomorphisms of Psh(C) is Cell (B C ).

4

  A vertically fibrant and homotopically constant space is called a simplicial resolution in [Dug01, definition 4.7].

  , lemma 3.4.9], i ∈ Cell (S n+1 ), thus a S-local isomorphism, and by induction, s t ω is a S-local isomorphism. By [HTLS19, corollary 3.4.10] the bottom target embedding is a S m-1,m -local isomorphism, thus a S-local isomorphism by induction. Consequently, s hω is a S-local isomorphism. □ Theorem 3.1.5. The left adjoint v ∶ Psh(Λ) → Alg is the Gabriel-Ulmer localization with respect to the set S of spine inclusions. Equivalently, M ∶ Alg → Psh(Λ) corestricts as an isomorphism Alg → S ⊥ .

  and d be the composite ∂hψ b hψ ↪ → hψ x → C. Since p is fully faithful, it induces a bijection p d ∶ C ld ≅ → D gd , and letting l(x) ∶= p -1 d g(x) extends l to an algebra morphism which is the desired lift. □ Proof of theorem 3.4.8, (M5): factorization axiom. Let f ∶ A → B be a morphism of algebra.

  , and b ∈ B n-1 to b. For ψ ∈ O n and d ∶ ∂hψ → D n-1 , let the fiber D d be simply B f d . At this stage, D clearly extends as an algebra, essentially inheriting the same law as B.

  where for a ∈ A, α(a (0) ) ∶= f (a) (see definition 3.5.1 for notations), α(a (1) ) ∶= g(a), α(j a ) = α a , and α(j-1 a ) ∶= α -1 a . • (2) ⇒ (1). By definition. □ Proposition 3.5.6. A morphism f ∶ A → B is a weak equivalence (for the folk structure of theorem 3.4.8) if and only if it is an isomorphism in Ho Alg (definition 2.1.6). Therefore, the category Ho Alg is the localization of Alg at the algebraic equivalences.

  Theorem 4.1.3. With the data above, Λ is a skeletal category (definition 2.1.13).

(

  Generalization of [MW09, lemma 5.1]). Let ω ∈ O n+1 , and I be a non empty set of inner faces of hω. Then the inclusion h I hω ∶ Λ I hω ↪ → hω is an inner anodyne extension. Proof. We proceed by induction d ∶= deg hω and on m ∶= # I. Since I is non empty, d ≥ 2. If d = 2, then hω has a unique inner face, and the corresponding inner horn inclusion is just the spine inclusion S[hω] ↪ → hω. Assume now that d ≥ 3. If m = 1, then Λ I hω is just an inner horn, and the claim holds. If m ≥ 2, take f ∈ I, and let J ∶= I -{f }. The inclusion h I hω decomposes as

Proof.

  It is enough to show that inner horn inclusions are S-local isomorphisms. Let ω ∈ O n+1 have d ≥ 2 nodes. If d = 2, the it has a unique inner face, and the corresponding inner horn inclusion is simply the spine inclusion S[hω] ↪ → hω. Assume d ≥ 3, let I be the set of all inner faces of hω, and f ∈ I. Then the spine inclusion s hω decomposes as

4. 3 .

 3 Cylinder objects. The Resk cylinder construction of definition 3.5.1 extends to Psh(Λ). Definition 4.3.1 (Rezk interval). For ϕ ∈ O n-1 , recall the definition of the Rezk interval J ϕ ∈ Alg. Define I ϕ ∶= M J ϕ . Write i ϕ ∶ hϕ → I ϕ for the endpoint inclusion, and E I ∶= {i ϕ | ϕ ∈ O n-1 }. Definition 4.3.2 (Rezk cylinder). For ω ∈ O ≥n-1 , let Ihω ∶= M (Jhω). Extend I by colimits to obtain a functor I ∶ Psh(Λ) → Psh(Λ). The Rezk cylinder IX of a presheaf X ∈ Psh(Λ) is a cylinder object in the sense of definition 2.1.5, i.e. we have a canonical factorization of the codiagonal map X ∈ Psh(Λ), IX ∶= colim hω→X M (Jhω).

  and ω • = {[p 1 ], . . .}. We readily deduce the following: Lemma 4.3.3. For X ∈ Psh(Λ) we have a canonical isomorphism JvX ≅ vIX. Proposition 4.3.4. The functorial cylinder I is an elementary homotopical data (definition 2.1.7). Proof. Straightforward unpacking of the definition of I. □ 4.4. Homotopical structure. Recall from definition 4.2.5 that the class An inner of inner anodyne extensions of Psh(Λ) is the class ⋔ (H ⋔ inner ) of retracts of cell complexes of inner horn inclusions. Definition 4.4.1 (I-anodyne extension). Recall from definition 4.3.1 the set E I of endpoint inclusions of the Rezk intervals in Psh(Λ). Let An I , the class of I-anodyne extensions, be the class ⋔ ((H inner ∪ E I ) ⋔ ).Definition 4.4.2 (Lifting problem). Let k ∶ K ↪ → hω be a subpresheaf of a representable presheaf, and f ∶ K → X a morphism. We say that f is a lifting problem of degree d (or d-lifting problem), where d = deg hω. We say that f is k-unsolved (or just unsolved if k is clear from the context) if f does not factor through k.

  4.3 to conclude. □ Proposition 4.4.5. The pair (I, An I ) satisfies condition (An1) of definition 2.1.8. Proof. By [Cis06, proposition 1.1.16], it is enough to check the claim when m is a boundary inclusion, say m = b hω ∶ X = ∂hω → Y = hω, where ω ∈ O n+1 . (1) Assume deg hω = 0, i.e. ω is degenerate, say ω = I ϕ . Then ∂hω = ∅, and since I preserves colimits, I∂hω = ∅ as well. Thus g is the inclusion hω (e) = hI ϕ → Ihω = M J ϕ , i.e. an endpoint inclusion of the Rezk interval J ϕ , which by definition is I-anodyne (2) Assume deg hω ≥ 1. We only treat the case e = 0, the other one being similar. Let [p] ∈ ω • , and consider the n-cell a = s [p] ω of hω. By equation (3.5.2), in Ihω, we have

Definition 4.4. 11 (

 11 Quillen model structure). Let H be the set of all horn inclusions of Λ, not only the inner ones. The Quillen model structure Psh(Λ) Quillen on Psh(Λ) is the Bousfield localization H -1 Psh(Λ) ∞ , which exists by [Hir03, theorem 4.1.1].

  preserves monomorphisms, and by lemma 4.5.5, it preserves the homotopy relation, thus the weak equivalences. □ Lemma 4.5.7 (Generalization of [JT07, proposition 1.13] and [Joy08, 2.13]). Let X, Y ∈ Psh(Λ) be ∞-algebras, and f ∶ X → Y be an inner fibration (definition 4.2.6). If vf is a fibration in Alg folk (i.e. E J ⋔ vf ), then E I ⋔ f . Proof. If n = 1, i.e. Λ = ∆, then the result holds by [JT07, proposition 1.13]. Assume n > 1, and take ϕ ∈ O n-1 . By proposition 4.5.6, C * ϕ is a right Quillen functor, thus C * ϕ X and C * ϕ Y are quasi-categories. By lemma 4.5.3, C * ϕ f is an inner fibration, since f is. By assumption, E J ⋔ v 1,n f , and in particular,C ϕ! j ◾ = j ϕ ⋔ v 1,n f . By adjunction and lemma 4.5.4, j ◾ ⋔ C * ϕ v 1,n f = τ C * ϕ f , i.e. C *ϕ f is a fibration in Cat folk . Finally, C * ϕ f satisfies the conditions of [JT07, proposition 1.13], hence i ◾ ⋔ C * ϕ f . By adjunction, i ϕ ⋔ f □ 4.6. Simplicial tensor and cotensor. Definition 4.6.1. For k ∈ N and λ ∈ Λ, let ∆[k] ⊗ λ be the nerve

  1) Clearly, the inclusion ∆[m] ⊗ h e hω ∶ ∆[m] ⊗ Λ e hω → ∆[m] ⊗ hω satisfies the condition of corollary 4.4.4, i.e. ∆[m] ⊗ Λ e hω generates v(∆[m] ⊗ hω). (2) Write * the unique element of hϕ hϕ (corresponding the the identity map hϕ → hϕ in Λ), and 0 m , . . . , m m the vertices of ∆[m]. In particular, (∆[m] ⊗ hϕ) hϕ = {0 m ⊗ * , . . . , m m ⊗ * }.

5. 1 .

 1 Segal spaces. Definition 5.1.1. Recall from definition 2.3.5 that Sp(Λ) v is the Reedy structure on Sp(Λ) = Psh(∆) Λ op induced by Psh(∆) Quillen . Let Sp(Λ) Segal , the Segal model structure on Sp(Λ), be the left Bousfield localization of Sp(Λ) v at the set S of spine inclusions (definition 3.1.3), which exists by [Hir03, theorem 4.1.1].

  Then F ≅ G/-, where G ∈ Sp(C) is the restriction of F to ∆, i.e. G -,k ∶= F ∆[k], for k ∈ N. Proof. For k ∈ N, we have G/∆[k] = G -,k = F ∆[k],thus G/-and F coincide on ∆ op . Since ∆ op freely generates Psh(∆)

  is a cofibration and a weak equivalence; (4) a morphism f is a fibration if it has the right lifting property with respect to acyclic cofibrations, it is an acyclic fibration if it has the right lifting property with respect to all cofibrations. This model structure is of Cisinski type, cellular, and proper. Let C be a small category, and Psh(C) be endowed with a model structure as in theorem 2.1.11. Let f ∶ X → Y be a morphism in Psh(C), and assume that Y is fibrant. Then f is a fibration if and only if it is a naive fibration. Skeletal category). A skeletal category [Cis06, definition 8.1.1] is a small category C endowed with a map deg ∶ ob C → N and two wide subcategories C + and C -such that the following axioms are satisfied.

	Lemma 2.1.12 ([Cis06, proposition 1.3.36]). Definition 2.1.13 (

  Every arrow in C -admits a section. Two arrows f, f ′ ∈ C -are equal if and only if they have the same sections. The skeletal category C is normal [Cis06, definition 8.1.36, proposition 8.1.37] if C is rigid, i.e. does not have non-trivial automorphisms. Note that in this case, C is a Reedy category [Hir03, defintion 15.1.12].

Definition 2.1.14 (Boundary). Let C be a skeletal category. The boundary [Cis06, paragraph 8.1.30] ∂c ∈ Psh(C) of an object c ∈ C is defined as ∂c ∶= colim f ∶d→c f ∈C+ not an iso

  + with codomain (c,[n]) are adequate pairs or morphisms f = (f C , f ∆ ) ∈ C + × ∆ + , and f is not an isomorphism if and only if f C or f ∆ is not 3 . Thus it is easy to see that the boundary and boundary inclusion of (c, [n]) are given by

Application to simplicial presheaves. In this section, we lift some technical results of [JT07, section 2] from the settings of simplicial spaces to simplicial presheaves over a normal skeletal category C (definition 2.1.13). Recall that by proposition 2.1.15, the set of boundary inclusions B C ∶= {b c ∶ ∂c ↪ → c | c ∈ C} generates the class of monomorphisms of Psh(C), in the sense that it is Cell (B C ). Proposition 2.3.1. The class of monomorphisms of Sp(C) is Cell (B C ⊠ B ∆ ). Proof. Observe that ∆ is normal skeletal, and thus so is the product C × ∆ in an evident way [Cis06, remark 8.1.7]. In particular, for (c, [n]) ∈ C × ∆, maps f ∈ (C × ∆)

  D, the relative matching map of f at d is ⟨b d /f ⟩. In particular, f is a Reedy fibration if and only if for all d ∈ D, the map ⟨b d

	. Simple consequence of proposition 2.3.1 and the adjunctions u ⊠ -⊣ ⟨u/-⟩ and -⊠ v ⊣ ⟨-/v⟩ of section 2.2.
	□
	Lemma 2.3.4. Let D be a normal skeletal category (particular, it is Reedy), and Psh(E) be endowed with a model structure. Consider the Reedy model structure on Psh(E) D op .
	(1) For X ∈ Psh(E) (2) For f ∶ X → Y in Psh(E) D op and d ∈ D, the matching object of X at d is ∂d/X. D op and d ∈

Fibrant spaces in Sp(C) v are also called vertically fibrant.

	Dually, let Sp(C) h , the horizontal model structure on Sp(C) ≅ Psh(C)

∆ op , be the Reedy structure induced by the model structure on Psh(C). The description of weak equivalence and fibrations transpose from the vertical model structure mutadis mutandis. Proposition 2.3.6. The

  model structures Sp(C) v and Sp(C) h are of Cisinsky type (definition 2.1.10).

	Proof. By [Hir03, theorem 15.6.27], both are cofibrantly generated. A map f ∈ Sp(C) is a vertical (resp. horizontal)
	acyclic fibration if and only if for all c ∈ C (resp. n ∈ N), the matching map ⟨b c /f ⟩ (resp. ⟨f /b n ⟩) is a acyclic fibration
	in Psh(∆) Quillen (resp. in the model structure on Psh(C), which is assumed to be of Cisinsly type), i.e. a trivial fibration. By proposition 2.3.3, f is a trivial fibration. Finally, f is a vertical or a horizontal acyclic fibration if
	and only if it is a trivial fibration. Therefore, vertical and horizontal cofibrations are the monomorphisms.	□
	Proposition 2.3.7 (Generalization of [JT07, proposition 2.5]). Let C be a normal skeletal category, and f ∶ X →
	Y be a morphism in Sp(C). The following are equivalent:
	(1) f is a vertical fibration, i.e. the map ⟨b c /f ⟩ is a Kan fibration, for c ∈ C;

  be the inner horn inclusion λ at f . Let H inner be the set of inner horn inclusions. (Inner anodyne extension). The class An inner of inner anodyne extensions if defined as An inner ∶= ⋔ (H ⋔ inner ). An inner fibration is a morphism f ∈ Psh(Λ) such that H inner ⋔ f , or equivalently, such that An inner ⋔ f . Generalizing definition 4.2.5, if ω ∈ O n+1 , and I is a set of (not necessarily inner) faces of hω, Let ω ∈ O n+1 have d ≥ 2 nodes, and ∅ ≠ I ⊆ J be two non empty sets of inner faces of hω. Then the inclusion h I hω factors as

	Λ I λ ∶= hω ∶ Λ I hω → hω the canonical inclusion. colim g∶λ ′ →λ elem. face g∉I and write h I Definition 4.2.6 Definition 4.2.7. we define Lemma 4.2.8.	λ ′ =	⋃ g∶λ ′ →λ elem. face g∉I	im g,

  Assume m is a inner horn inclusion, say m = h f hω ∶ X = Λ f hω → Y = hω, where ω ∈ O n+1 . Recall that by proposition 4.2.12, vΛ f hω = hω. Thus applying v to the diagram of definition 2.1.7 (An2) yeilds Jhω is an isomorphism. By corollary 4.4.4, g is an inner anodyne extension.(2) Assume m is an endpoint inclusion of a Rezk interval, say m

	(1) hω + hω	hω + hω
	Jhω	⌜	Jhω
			Jhω.
			corollary 4.4.4, g
	is an inner anodyne extension.		□
	Proposition 4.4.6. The pair (I, An I ) satisfies condition (An2) of definition 2.1.8.
	Proof. By [Cis06, proposition 1.1.16], it is enough to check the claim when m is an inner horn inclusion or an
	endpoint inclusion of a Rezk interval.

•

. Therefore, Ihω is freely generated by I∂hω ∪ hω

(0)

. By

vg Thus g ∶ IΛ f hω ∪ (hω + hω) → Ihω = M Jhω is such that its transpose Jhω vg → vM Jhω ≅

  .2. Complete Segal spaces. (Rezk map). For ϕ ∈ O n-1 , recall from definition 4.3.1 the definition of the Rezk interval I ϕ , and the endpoint inclusion i ϕ ∶ hϕ → I ϕ . There is a canonical morphism, called the Rezk map at ϕ, r ϕ ∶ I ϕ → hϕ, mapping j ϕ and j -1 ϕ to id ϕ (see definition 3.4.9 for notations), and let R = {r ϕ | ϕ ∈ O n-1 } be the set of Rezk maps. Definition 5.2.2. Let Sp(Λ) Rezk , the Rezk model structure on Sp(Λ), be the left Bousfield localization of Sp(Λ) Segal (definition 5.1.1) at the set of Rezk maps R, which exists by [Hir03, theorem 4.1.1].Fibrant objects (resp. weak equivalences) in Sp(Λ) Rezk are called complete Segal spaces (resp. Rezk weak equivalences). Explicitely, a Segal space X

	Definition 5.2.1

  3.7. Hence it is a trivial fibration if and only if it is a weak equivalence. By lemma 2.1.12, ⟨f /u⟩ us a fibration if and only if it is a naive fibration, so it remains to show that E I ⋔ ⟨f /u⟩.By adjunction, for ϕ ∈ O n-1 , we have i ϕ ⋔ ⟨f /u⟩ if and only if u ⋔ ⟨i ϕ /f ⟩. Thus, we must show that ⟨i ϕ /f ⟩ is an acyclic fibration. By proposition 2.3.7, it is a fibration. Consider the following commutative diagram: By lemma 5.2.3, i ϕ /X and i ϕ /Y are acyclic fibrations. Since f is a Rezk fibration, it is a vertical fibration, and its Recall from [JT07, proposition 1.16] the functor J ∶ qCat → Kan from quasi-categories to Kan complexes, that maps a quasi-category to its maximal sub Kan complex. It is right adjoint and preserves cofibrations, fibrations, and weak equivalences (from Psh(∆) ∞ ).

	I ϕ /X		i ϕ /X
		⟨i ϕ /f ⟩
	I ϕ /f	⋅	⌟	p2	hϕ/Y
			p1		hϕ/f
		I ϕ /Y	i ϕ /Y	hϕ/Y.

□

Lemma 5.2.4 (Generalization of [JT07, lemma 4.3]). Let f ∶ X → Y be a Rezk fibration (i.e. a fibration in Sp(Λ) Rezk ) between two complete Segal spaces, and u ∶ K ↪ → L be a monomorphism in Psh(∆). Then the map ⟨f /u⟩ ∶ X/L → (Y /L) ∏ Y /K (X/K) is a fibration. Proof. By proposition 5.1.7, ⟨f /u⟩ is an inner fibration between ∞-algebras. matching map ⟨(∂hϕ ↪ hϕ)/f ⟩ is a fibration by proposition 2.3.7. Since ϕ ∈ O n-1 , ∂hϕ = ∅, thus hϕ/f = ⟨(∂hϕ ↪ hϕ)/f ⟩ is a fibration. The pullback map p 2 is an acyclic fibration, and by 3-for-2, so is ⟨i ϕ /f ⟩. □ Definition 5.2.7.

□

  Index(-) const , see constant space (-) disc , see discrete space A(e) , seealgebraic Rezk cylinder M , 8 N , 8 S[ω], see spine S[hω], see spine Alg folk , see folk model structure Λ I λ, 17 Psh(Λ) ∞ , see model structure for ∞-algebras Psh(Λ) Quillen , see Quillen model structure Psh(C), 2 Sp(Λ) Rezk , see Rezk model structure Sp(Λ) Segal , see Segal model structure Sp(C), 2 Sω, see spine deg, see degree IX, see Rezk cylinder I ϕ , 18 JA, see algebraig Rezk cylinder J ϕ , see algebraic Rezk interval idx, 10 ∞-algebra, 21 ⟨-/-⟩, see Leibniz box product ⟨-/-⟩, see Leibniz box product ⊠, see Leibniz box product ⊗, see Leibniz construction An inner , see inner anodyne extension ∂λ, see boundary ⊥, see orthogonal ⋔, see lifting property B, see boundary inclusion b λ , see boundary inclusion h f λ , see inner horn inclusion h I hω , 17 i ϕ , 18 r ϕ , see Rezk map S, see spine inclusion s hω , see spine inclusion Rezk cylinder, 14, 19 Rezk interval, 13 Rezk map, 29 Rezk model structure, 29 Rezk weak equivalence, see Rezk model structure right cancellation property, 26 row-wise weak equivalence, see vertical model structure

		complete Segal space, 29
		component, see natural transformation
		constant space, 5
		cylinder, 2
		degree, 15
	Segal model structure, 26	diagram, 9
	Segal space, 26	diagrammatic morphism, see diagram
	Segal weak equivalence, see Segal model structure	discrete space, 4
	sieve, 22
	simplicial resolution, 8	elementary face, 16
	skeletal category, 4	elementary homotopical date, 3
	spine, 9	elementary homotopy, see homotopy
	spine inclusion, 9	equivalence of algebras, see algebraic equivalence
		essentially surjective, 11
	trivial cofibration, 6	exterior product, 5
	trivial fibration, 6
		fibrant object, 4
	underlying category, 22	fibration, 4
	unique lifting property, see orthogonal	acyclic, 4
	vertical fibration, see vertical model structure vertical model structure, 7 vertically fibrant, see vertical model structure	naive, 4 folk model structure, 12 fully faithful, 11
	weak equivalence, 4	homotopical structure, 3 homotopically constant, 8
		homotopy, 3
		homotopy equivalence, 3
		horizontal fibration, see vertical model structure
		horizontal model structure, 7
		horizontally fibrant, see vertical model structure
		I-homotopy, see homotopy
		inner anodyne extension, 16
		inner face, 16
		inner fibration, 16
		inner horn, 16
		inner horn inclusion, 16
		inner Kan complex, see ∞-algebra
		internal isomorphism, 11
		isofibration, 12
	h, 8	isomorphism, see internal isomorphism
	h * , 8
	v, 8	Kan complex, 22
	x ○ [p] x ′ , 10	Kan fibration, 22
	acyclic cofibration, 6	Leibniz box product, 6
	acyclic fibration, 6	Leibniz construction, 5
	algebraic equivalence, 11	Leibniz's formula, 6
	algebraic Rezk cylinder, 14	lifting problem, 19
	algebraic Rezk interval, 13	lifting property, 2
	anodyne extension, 3, 19	model structure for ∞-algebras, 21
	boundary, 16 boundary inclusion, 16 box product, 5	natural isomorphism, 11 natural transformation, 11 normal skeletal category, 4
	cartesian gap, 6 cellular model, 16	orthogonal, 2
	Cisinski model category, 4 cofibration, 4	parallel cells, 10
	acyclic, 4	Quillen model structure, 22
	column-wise weak equivalence, see vertical model structure	quotient, 10

We follow the terminology and notation of[START_REF] Riehl | The theory and practice of Reedy categories[END_REF][START_REF] Riehl | Homotopical categories: from model categories to (∞, 1)-categories[END_REF]. In [JT07, section 2], it is denoted by ◻ ′ .

This observation is called Leibniz's formula in [RV14, observation

4.2].

In particular, complete Segal spaces are exactly the simplicial resolutions [Dug01, definition 4.7].

Institut de Recherche en Informatique Fondamentale (IRIF), Université Paris Diderot, France Email address: cedric.hothanh@irif.fr URL: hothanh.fr/cedric Institut de Recherche en Informatique Fondamentale (IRIF), Université Paris Diderot, France Email address: chaitanya@irif.fr URL: www.irif.fr/~chaitanya

Proposition 5.1.3 (Generalization of [START_REF] Joyal | Quasi-categories vs Segal spaces[END_REF]proposition 3.4]). Let X ∈ Sp(Λ) be vertically fibrant. The following are equivalent:

(1) X is a Segal space;

(2) the map h f hω /X is an acyclic Kan fibration, for all ω ∈ O n+1 and all inner horn inclusion h f hω ∶ Λ f hω → hω;

(3) the map u/X is an acyclic Kan fibration, for all inner anodyne extension u ∈ Psh(Λ); (4) the map X/b n is a inner fibration (definition 4.2.6), for all n ∈ N;

(5) the map X/v is a inner fibration, for all monomorphism v ∈ Psh(∆).

Proof. All equivalences are straightforward, except for (1) ⇐⇒ (3).

• (1) ⇒ (3). Let K be the class of morphisms u ∈ Psh(Λ) such that u/X is an acyclic Kan fibration. Since X is a segal space, S ⊆ K, and clearly, K has the right cancellation property. We now show that it is saturated. By definition, u ∈ K if and only if u/X is an acyclic Kan fibration, i.e. b n ⋔ u/X for all n ∈ N. But this is equivalent to u ⋔ X/b n , and thus K = ⋔ {X/b n | n ∈ N}, and in particular, K is saturated. Finally, by lemma 5.1.2, K contains all inner anodyne extensions.

• (3) ⇒ (1). Recall that spine inclusions are inner anodyne extensions by proposition 4.2.11. □ Corollary 5.1.4 (Generalization of [JT07, corollary 3.6]). Take X ∈ Sp(Λ) be a Segal space, and K ∈ Psh(∆).

Then X/K is an inner Kan complex. In particular, X -,n = X/∆[n] is an inner Kan complex for all n ∈ N.

Proof. For k ∶ ∅ ↪ → K the initial map, the map X/k ∶ X/K → X/∅ = 1 is an inner fibration by proposition 5.1.3. □ Lemma 5.1.5 (Generalization of [JT07, lemma 3.8]). Let f ∶ X → Y be an inner fibration between ∞-algebras.

It is an acyclic fibration if and only if it is a weak equivalence surjective on (n -1)-cells.

Proof.

(1) ( ⇒ ) Surely, if f is an acyclic fibration, it is a weak equivalence. Moreover, f has the right lifting property against all monomorphisms (recall theorem 4.4.7), and in particular, against inclusions of the form ∅ ↪ → hϕ, for ϕ ∈ O n-1 . Therefore, f is surjective on (n -1)-cells.

(2) ( ⇐ ) By theorem 4.4.7 and lemma 2.1.12 f is a fibration if and only if it is an inner fibration, and E I ⋔ f . By lemma 4.5.7, it suffices to show that E J ⋔ vf . Note that f is a weak equivalence between cofibrant objects. By Ken Brown's lemma [Hov99, lemma 1.1.12], vf is a weak equivalence. Besides, it is also surjective on (n -1)-cells, so by theorem 3.4.8, it is an acyclic fibration. In particular, it is a fibration, and we apply lemma 4.5.7 to conclude. □ Proposition 5.1.6 (Generalization of [JT07, proposition 3.9]). A presheaf X ∈ Sp(Λ) is a Segal space if and only if the following conditions are satisfied:

(1) for all n ∈ N, the map X/b n is an inner fibration;

(2) X hϕ is a Kan complex for all ϕ ∈ O n-1 ;

(3) X is homotopically constant (see definition 2.3.8).

Proof.

In particular, it is vertically fibrant.

(1) This is proposition 5.1.3 (4).

(2) The terminal map ! ∶ X → 1 is a vertical fibration, so by definition 2.3.5, for ϕ ∈ O n-1 , the map

We claim that X/v is an acyclic fibration. By lemma 5.1.5, it suffices to show that

Proposition 5.2.5 (Generalization of [JT07, proposition 4.4]). A presheaf X ∈ Sp(Λ) is a complete Segal space if and only if the following conditions are satisfied 5 :

(1) X/b n is a fibration for all n ∈ N, i.e. X is horizontally fibrant (definition 2.3.5);

(2) X is homotopically constant (definition 2.3.8).

Proof.

• ( ⇒ ) By proposition 5.1.6, X is homotopically constant. By lemma 5.2.4, X/b n = ⟨(X → 1)/b n ⟩ is a fibration as X → 1 is a Rezk fibration.

• ( ⇐ ) We first show that X is vertically fibrant. By proposition 2.3.7, this is equivalent to X/u being an acyclic fibration for all anodyne extension u ∈ Psh(∆). Let K be the class of monomorphisms u such that X/u is an acyclic fibration. Using lemma 5.1.2, it suffices to show that K has the right cancellation property, is saturated, and that it contains all simplicial face maps.

(1) By condition (1), X/u is a fibration for any monomorphism u ∈ Psh(∆). Thus, X/u is an acyclic fibration if and if it is a weak equivalence. The right cancellation property of K then follows from 3-for-2. (2) For staturation, note that u ∈ K if and only if b hω ⋔ X/u for all ω ∈ O ≥n-1 by definition. This is

(3) Lastly, let us show that the simplicial face maps

homotopically constant, the structure map X -,n → X -,0 is a weak equivalence for all n ∈ N, so by 3-for-2, X/d i is too, as displayed by the following triangle

Since d i is a monomorphism, X/d i is a fibration by condition (1). Finally, X/d i is an acyclic fibration. Therefore, by proposition 2.3.7, K contains all anodyne extensions, and by proposition 2.3.7, X is vertically fibrant.

Since the terminal map ! ∶ X → 1 is a vertical fibration, by definition 2.3.5, for ϕ ∈ O n-1 , the map ⟨b hϕ /!⟩ ∶ X hϕ → 1 is a Kan fibration, and X hϕ is a Kan complex. By proposition 5.1.6, X is a Segal space.

Lastly, let us show that X is complete. By lemma 5.2.3, it suffices to show that i ϕ /X is an acyclic fibration, for all ϕ ∈ O n Clearly, the enpoint inclusion i ϕ ∶ hϕ → I ϕ is a weak equivalence, with the Rezklemma:saturation-rcp-rcpp to homotopy. Since i ϕ is a monomorphism, it is a an acyclic cofibration. By condition (1), i ϕ ⋔ X/b n , for all n ∈ N, so by adjunction, b n ⋔ i ϕ /X, and i ϕ /X is a trivial fibration. □ Theorem 5.2.6 (Generalization of [JT07, theorem 4.5]).

(1) The model structure Sp(Λ) Rezk is a Bousfield localization of the horizontal model structure Sp(Λ) h (definition 2.3.5). In particular, a weak equivalence in Sp(Λ) h is a Rezk weak equivalence.

(2) An horizontally fibrant space is a complete Segal space if and only if it is homotopically constant (definition 2.3.8).

Proof.

(1) By proposition 2.3.6, Sp(Λ) v and Sp(Λ) h are both of Cisinski type. Since Sp(Λ) Rezk is a left Bousfield localization of Sp(Λ) v , it is also of Cisinski type. In particular, Sp(Λ) Rezk and Sp(Λ) h have the same cofibrations, namely the monomorphisms. Thus, in order to prove the claim, it is enough to show that the identity functor induces a Quillen adjunction id ∶ Sp(Λ) h → ← Sp(Λ) Rezk ∶ id. By a result of Dugger [Dug01, corollary A.2] (also stated in [Hir03, proposition 8.5.4]), it suffices to show that id preserves cofibrations, and Rezk fibrations between complete Segal spaces. In both structures, cofibrations are the monomorphisms. By lemma 5.2.4, for f ∶ X → Y a Rezk fibration between complete Segal spaces, the matching map ⟨f /b n ⟩ is a fibration for all n ∈ N.

(2) Follows from proposition 5.2.5 □