Algebraic Quantum Field Theory

Home , Maxim Kontsevich, Steve Shnider

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1 Introduction

The problem of applying quantum mechanical principles to classical systems of fields was the foundational idea which motivates the emersion of Quantum Field Theories. However, a general formulation of what is precisely a Quantum Field Theory seems to be rather elusive. This fact gave rise to the appearance of a broad variety of competing definitions to deal with Quantum Field Theories formally. A very rough classification of such candidates could be: perturbative models, which use the knowledge about simpler free models ([26]); functorial approaches, which are centered on the study of spaces of fields ([23, 30]); and algebraic formulations, which deal with the possible algebraic structures that the collection of observables may carry ([20]). In this work, we focus on the algebraic picture, i.e. on the formal study of algebras of observables. A recent development in the field of Algebraic Quantum Field Theories (usu- ally denoted by AQFTs) has been undertaken in the program of Benini and collaborators [4, 5, 8, 9, 10]. Their fundamental contribution was to identify operadic constructions, and their importance, on previous axiomatics and foundations, e.g. interprete Haag-Kastler axioms and recognize generalizations [33] or justify the Fredenhagen’s local-to-global principle [14], as well as detecting its weakness. Reasonably, a good deal of new questions arises by the introduction of operad theory into the branch of AQFTs, as addresed in the survey [7]. Operad theory [24] has been proven to be a fruitful mathematical machinery to deal with algebraic structures. Its importance in mathematics is doubtless due to its multidisciplinarity, as it appears in homotopy theory, algebraic topology, geometry, category theory, mathematical physics or mathematical programming. For instance, this theory has also been a valuable tool in Kontsevich’s work on quantisation of Poisson manifolds [22] or Costello-Gwilliam’s factorization algebras in perturbative field theories [17]. One of the main advantages of working with operads is the possibility of ma- nipulate the homotopy theory of AQFTs quite explicitly, as observed in [9]. On the one hand, embracing homotopical mathematics permits a better understanding of non discrete phenomena such as gauge symmetries or the presence of anti-fields

2 and anti-ghosts in the BV-BRST formalism (e.g. [6, 8]). These facts motivate a deeper immersion into the homotopy theory of AQFTs. On the other hand, the presence of a Quillen model structure brings to our disposal an assorted toolkit to understand a particular homotopy theory. In [10, 13], the algebraic structure of an AQFT is encoded via a naturally de- fined operad, and so it is reasonable to define the homotopy theory of AQFTs as a canonical homotopy theory of operadic algebras [32]. However, such approx- imation is not well suited to introduce further axioms on AQFTs which are not structural. For example, it does not take into account dynamic axioms properly (time-slice axiom, Definitions 3.1 and 3.6), as noted in [7, Open Problem 4.14], or local-to-global principles (Definition 5.1). The present work exploits the operadic description of AQFTs one step further to find more advanced homotopy theories of Algebraic Quantum Field Theories which fix these problems. Our main results are the construction and comparisons of several model category structures summarized in Section 6. First, we propose two different model structures to deal with AQFTs satisfying a homotopy meaningful time-slice axiom (Definition 3.6) which turn out to be equivalent. This can be summarized in the following theorem (see Subsection 3.2). Theorem A (Propositions 3.11, 3.14 and Theorem 3.15). Let C⊥ be an orthogonal category (Definition 2.1) and S a set of maps in C. Then, the homotopy theory of algebraic quantum field theories over C⊥ satisfying the weak S-time-slice ax- ⊥ iom is presented by the Quillen equivalent model structures QFT w S(C ) and ⊥ LS QFT (C ). Both models provide different aspects of the homotopy theory of AQFTs satisfying such dynamical axiom. In fact, below Theorem 3.15 we expand on this to answer [7, Open Problem 4.14]. The local-to-global principle (Definition 5.1) is incorporated in a new model structure that is an instance of a general construction explained in Section 4 which culminates in Theorem 4.9.

⊥ ⊥ Theorem B (Theorem 5.3). Let C⋄ ֒→ C be an inclusion of a full orthogonal subcategory. Then, the homotopy theory of algebraic quantum ﬁeld theories over ⊥ C⋄ ⊥ C which satisfy the C⋄-local-to-global principle is modelled by QFT (C ). Finally, for the combination of both axioms, time-slice plus local-to-global, we ﬁnd again two model structures presenting the associated homotopy theory which are equivalent as expected.

⊥ ⊥ Theorem C (Theorems 5.9, 5.11, 5.13). Let C⋄ ֒→ C be an inclusion of a full orthogonal subcategory and S a suitable set of maps in C. Then, the homotopythe- ory of algebraic quantum ﬁeld theories over C⊥ satisfying the weak S-time-slice

3 axiom and the C⋄-local-to-global principle is presented by the Quillen equivalent C⋄ ⊥ C⋄ ⊥ model structures QFT w S(C ) and LS QFT (C ). Coming back to the general construction of Section 4, it is mandatory to com- ment that it has its own importance in abstract homotopy theory and so several applications are expected. Indeed, in [16], the homotopy theory of factorization algebras is investigated by these means. It is worth noting that most of the results in this work are valid for a closed symmetric monoidal model category V satisfying some general requirements con- sisting on a combination of hypothesis appearing in Section 4 and in [15] (or [16, Section 6]). However, we have chosen to state them for chain-complex valued algebraic ﬁeld theories over a (commutative) ring R which contains the rationals for simplicity and due to connections with the existing literature. This restriction is not made explicit in order to enforce the cited generality. For instance, Sections 2 and 4 are written for a general V, whereas the rest of the sections should be interpreted under the assumption V = Ch(R).

Outline: We include a brief summary of the contents of this work. Section 2 presents basic deﬁnitions in the theory of AQFTs and a straightforward construction of the canonical operad developed in [10] and [12]. The rest of the paper is organized depending on which further axiom of AQFTs is under study. Section 3 is devoted to the analysis and discussion of the time-slice axiom and its associated homotopy theories, going from the strict version to the homotopical one, which is seen as part of structure or as a property of AQFTs. The local-to-global principle introduced in the work of Benini and collaborators, and its interaction with the time-slice axiom, is the focus on Section 5. To fully understand the constructions in this last section, the content of Section 4 is necessary. It yields a machine to construct cellularizations of model categories of operadic algebras. Despite this fact, this homotopical discussion can be used as a necessary black box, since it re- produces the local-to-global principle model categorically. At the end, we include a table and a diagram (Section 6) condensing the ﬁeld theory related material obtained in this paper.

2 Algebraic quantum ﬁeld theories

2.1 Deﬁnition and examples Algebraic quantum ﬁeld theory is an axiomatic approach to quantum physics which focuses on the assignation of algebras of observables to spacetime regions.

4 The fundamental idea of such objects is that the algebraic structure on the observables is supposed to capture quantum information, so it is not commutative in general, although observables coming from “disjoint spacetime regions" do com- mute. The notion of orthogonal category formalizes the concept of spacetime regions and the relation of being disjoint.

Deﬁnition 2.1. [33, Deﬁnition 8.2.1] An orthogonal category C⊥ is a pair (C, ⊥) given by a category C with choices of sets of morphisms

⊥ = ⊥ ({U, W}; V) ⊆ C(U, V) × C(W, V) U,W,V∈ob C which are stable by composition in C and are symmetric in (U, W). It is said that two C-morphisms f and g are orthogonal, denoted f ⊥ g, if they have the same codomain and (f, g) belongs to ⊥.

Notation 2.2. In order to maintain the physical motivation, we call spacetime regions to the objects of an orthogonal category.

In turn, the assignation of algebras of observables to spacetime regions is cap- tured by the following definition. Let us fix for this section a (bicomplete and closed) symmetric monoidal category V as a target for field theories.

Deﬁnition 2.3. Let C⊥ be an orthogonal category. An algebraic quantum ﬁeld theory over C⊥ is a C-diagram of monoids A: C → Mon(V) such that for any pair of orthogonal maps (f: U → V) ⊥ (g: W → V) in C, the following square commutes

A(U) ⊗ A(W) A(f)⊗A(g) A(f)⊗A(g)

A(V) ⊗ A(V) A(V) ⊗ A(V). op µV µV

A(V)

The full subcategory of algebraic quantum ﬁeld theories within [C, Mon(V)] is denoted QFT(C⊥).

Benini and his collaborators proposed this formalization of algebraic quantum ﬁeld theories in [10] in order to cover several proposals in the literature. We collect

5 some examples on the following table (see also [33] and references therein).

Underlying category C Orthogonality relation ⊥ Algebraic quantum field theory bounded lattice ∧-divisors of 0, Quantum field theories on i.e. u ⊥ v ⇔ u ∧ v = 0 bounded lattice oriented n-manifolds with disjoint image, Chiral conformal quantum field orientation-preserving open i.e. f ⊥ g ⇔ Im f ∩ Im g = o✁ theories embeddings oriented riemannian disjoint image Euclidean quantum field n-manifolds with theories orientation-preserving isometric open embeddings oriented, time-oriented, globally causally disjoint images, Locally covariant quantum field hyperbolic Lorentzian i.e. f ⊥ g if and only if theories manifolds with isometric open there is no causal curve joining embeddings preserving Im f and Im g orientation and time-orientation whose image is causally compatible category of regions for a causally disjoint regions Algebraic quantum field spacetime with timelike theories on spacetime with boundary timelike boundary

2.2 Operadic presentation in a nutshell

Given an orthogonal category C⊥, there is a functorial construction of an operad O⊥ ⊥ C whose algebras are algebraic quantum ﬁeld theories over C . More concretely, there is an equivalence of categories

O⊥ ⊥ C - Alg ≃ QFT(C ). O⊥ In this section, we recall the definition of C , given originally in [10], with some additional observations which makes such definition more transparent in our view. First, observe that QFT(C⊥) is a full subcategory of [C, Mon(V)], which is itself the category of algebras over a certain operad. To find this first operad, note that the category of monoids in V is equivalent to the category of uAss-algebras in V, where uAss the operad of associative algebras with unit. Looking at C as an operad as well, we have the identification

[C, Mon(V)] ∼= C - Alg uAss - Alg(V) . The Boardman-Vogt tensor product (see [31] or the original source [11, Section II.3]), denoted ⊗BV, combines C and uAss to obtain another operad, C ⊗BV uAss,

6 which comes with the identiﬁcation

C ⊗ uAss - Alg(V) ∼= C - Alg uAss - Alg(V) . BV In other words, the operad governing C-diagrams of monoids in V is just C ⊗BV uAss. Within such diagrams, algebraic ﬁeld theories are those for which certain operations coincide. More precisely, looking at A ∈ QFT(C⊥) as an algebra over the Boardman-Vogt tensor product, it must satisfy

U W U W

f g f g A = A µ op V µV V V for all f ⊥ g in C. Therefore, one can present this algebraic structure by con- structing a suitable quotient of C ⊗BV uAss which identiﬁes those operations. This is done by introducing a free operad F(R⊥) that select these operations, and co- equalizing the corresponding maps

µ⊥ coeq O⊥ F(R⊥) C ⊗ uAss C . op BV µ⊥

The underlying symmetric sequence of the free operad on the left is simply

I if n =2 ⊥({U1,U2};V) R⊥({U1 ... Un}; V)= ` ,  ✓o otherwise  and by freeness, the map µ⊥ corresponds to choosing the set of operations

U W

f g where µV is associated to µ2 ∈ uAss(2) µV and f ⊥ g are orthogonal in C V

op op in C ⊗BV uAss (and analogously for µ⊥ , just replace µV by µV above). The recognition of the Boardman-Vogt tensor product in the construction of O⊥ C is not vacuous. It leads to a broad generalization of the functorial construction

OrthCat Operad2 pt V Operad V , C, P P(r1,r2) × {∗} ( ) −→ ( ) ( ) 7→ C discussed at [12, 13] . Now, it can be seen as the composition of:

7 ×2 • the Boardman-Vogt tensor ⊗BV : Operad(V) → Operad(V), to construct the operad PC =∼ C ⊗BV P encoding C-diagrams of P-algebras (see [31]);

• a colimit construction to identify operations in PC ∼= C ⊗BV P as ruled out by the orthogonal relation in C.

Hence, it is admissible to consider orthogonal categories enriched in V and more general quotients than those coming from a monochromatic bipointed operad (following the notation on [13]). For example, the category Loc of globally hyperbolic lorentzian manifolds [20] could be endowed with some topological structure and the bipointed operad could have multiple colors. We expect to exploit this flexi- bility in subsequent work. Remark 2.4. The recognition of the Boardman-Vogt tensor product provides a canonical presentation by generators and relations of the operad (see [31]). As O⊥ expected, it coincides with the presentation by generators and relations of C given in [10]. From now on, operads and algebras are consider in V = Ch(R), with Q ⊆ R, unless otherwise specified. This choice also affects our notation, e.g. Set-operads V V are consider in via the strong monoidal functor Set → , X 7→ X I, where I is the monoidal unit of V. ` Remark 2.5. We decided to work on chain complexes over a ring R containing the rationals mainly for its connection with the foundational work of Benini et al., and because of the simplicity of the homotopy theory of operads and algebras in this context. For instance, the strictification theorem for operadic algebras in Ch(R) is one of the principal advantages of working within this setting. This theorem asserts that operadic algebras over a colorwise cofibrant operad presents the homotopy theory of their homotopy coherent version, because such operads are automatically Σ-cofibrant by [9, Proposition 2.11] and because weak equivalences of Σ-cofibrant operads induce Quillen equivalences between the corresponding model categories of operadic algebras [9, Theorem 2.10]. O⊥ Suitable replacements of the operad C worked out by Yau [33] and Benini- Schenkel-Woike [9] lead to homotopy coherent variants of algebraic quantum field theories, which may have a priori a richer homotopy theory than the strict version explored in this work. Their contribution consists on choosing a weak equivalent O⊥ ∼ O⊥ operad C,∞ −→ C , which has more cofibrancy than the original one in some ⊥ O⊥ precise sense, an looking at QFT∞(C )= C,∞ - Alg. However, the strictification theorem states that no more homotopical information is gained if we work within Ch(R), since we have the Quillen equivalence

O⊥ O⊥ C,∞ - Alg ≃ C - Alg .

8 Nevertheless, as briefly commented in the introduction, our techniques could be adapted for homotopy coherent algebraic quantum field theories as well, lead- ing to parallel results. The importance of this generalization is that it is possible to replicate our homotopical results in more general contexts, for instance allowing chain-complexes over a general base ring R, some flavour of spectra, etc. For the record, we highlight the choice commented above.

Assumption 2.6. V represents the (projective) model category of chain-complexes over a commutative ring R containing the rationals.

This operadic reformulation serves to find a candidate of Quillen model structure to present the homotopy theory of algebraic quantum field theories [9]. The following notation fixes that choice. ⊥ O⊥ Notation 2.7. QFT (C ) denotes the projective model structure on C - Alg.

3 Time-slice localization

Given an algebraic quantum field theory, a way to introduce dynamics into the picture is via the so called time-slice axiom on the theory (see [7]or[20] for motivation of the concept). Formally, it consists on a choice of maps S in C between space-time regions such that A ∈ QFT(C⊥) satisfies that A(f) is an isomorphism for any f ∈ S. Such field theories are said to satisfy the S-time-slice axiom. In this section, we propose two methods to study the homotopy theory of algebraic quantum field theories satisfying the time-slice axiom (or its relaxed version, see Definition 3.6). For internal references, we adopt the following alternative notation.

Definition 3.1. A field theory A ∈ QFT(C) satisfying the S-time-slice axiom is said to be S-constant. The full subcategory of QFT(C⊥) spanned by S-constant ⊥ field theories is denoted QFTS(C ). Remark 3.2. In the sequel, we will make an extensive use of localization for operads and model categories, and hence of its associated terminology. For an introductory survey which is full of ideas see [18] and for a condensed model categorical treatment see [2].

3.1 Strict time-slice axiom Let us begin with a brief remainder of the time-slice axiom discussion in [7, 9, 10]. There, it is noticed that looking at an algebraic ﬁeld theory as a diagram

9 A: C → Mon(V), S-constancy can be expressed by saying that the diagram fac- tors uniquely through the localization of C at S, denoted C → CS. Indeed, the localization CS can be endowed with a canonical orthogonal relation, the so called pushforward orthogonal relation ⊥S; it is constructed in [10, Lemma 3.19]. The ⊥S importance of CS is that it encodes the algebraic structure of S-constant field theories. More precisely, in [10, Lemma 3.20, Proposition 4.4], it is showed that the induced morphism ℓ O⊥ O⊥S yields an equivalence of categories of algebraic : C → CS field theories ∗ ⊥S ∼ ⊥ ℓ : QFT(CS ) −→ QFTS(C ). Thus, a change of orthogonal category permits the recognition of S-constancy as structure and not as a property on field theories. Consequently, the homotopy theory of S-constant field theories may be presented by the projective model structure over O⊥S - Alg QFT C⊥S (see [9, Theorems 3.8 and 3.10]). CS ≃ ( S ) Notation 3.3. C⊥ denotes the projective model structure on O⊥S - Alg. QFT S( ) CS Observe that this homotopy theory only deals with field theories satisfying the S-time-slice axiom strictly, as introduced above, and this fact excludes algebraic field theories coming from gauge theory (see [7, Section 3.4]). The way to include such examples will be a relaxation of the strict S-time-slice axiom. In order to do so, we will need a more canonical description of O⊥S , motivated by the algebraic CS field theories that it classifies.

⊥ ⊥S Proposition 3.4. Let C be an orthogonal category and CS its localization at S. Then, the canonical morphism O⊥ O⊥S induces an equivalence of operads C → CS

O⊥ −→∼ O⊥S , C S CS O⊥ where the left-hand side denotes the Set-localization of the operad C at S. Proof. The claim follows from the following chain of isomorphisms:

(i) ⊥S ∼ Operad O , P = equalizer Operad(CS ⊗ uAss, P) ⇒ Operad(F(R⊥ ), P) CS BV S h i (ii) ∼ PuAss ⇒ P = equalizer Operad(CS, ) Operad(F(R⊥S ), ) h i (iii) ∼ PuAss P = equalizer OperadS(C, ) ⇒ Operad(F(R⊥), ) h i (iv) ∼ P P = equalizer OperadS(C ⊗ uAss, ) ⇒ Operad(F(R⊥), ) h BV i (v) ∼ O⊥ P = OperadS( C , ).

10 O⊥ We have used: (i) description of the operad C as a quotient of the Boardman- Vogt tensor product; (ii) Boardman-Vogt tensor product is left-adjoint to internal hom in Operad; (iii) universal property of the localization CS on the ﬁrst factor of the equalizer. On the second factor, we apply that the orthogonal relation ⊥S is generated by ⊥. That is, by the explicit description for the pushforward orthogonal relation of [10, Lemmas 3.19] and the proof of [10, Lemma 3.20], we see that the equalizer requiring ⊥-commutativity is the same as the one that requires ⊥S-commutativity; (iv) same adjunction than (ii); (v) universal property of localization together with (i). Remark 3.5. Proposition 3.4 reproves that the morphism ℓ O⊥ O⊥S induces : C → CS ∗ ⊥S ∼ ⊥ an equivalence of categories ℓ : QFT(CS ) −→ QFTS(C ). See the beginning of this section.

3.2 Weak time-slice axiom Motivated by toy models explained in [7], it is natural to relax the S-time-slice axiom to a homotopical variant. Definition 3.6. An algebraic quantum field theory A ∈ QFT(C⊥) satisfies the weak S-time-slice axiom, or equivalently it is weakly S-constant, if A(f) is a weak ⊥ equivalence whenever f ∈ S. We denote by QFTw S(C ) the full subcategory of QFT(C⊥) spanned by weakly S-constant field theories. We propose two approaches to study the homotopy theory of these algebraic quantum field theories and we show that these two approaches are equivalent in a precise sense. O⊥ Structural approach: It consists on enhancing the operad C to encode the structure that turns the maps in S to weak equivalences. Due to Proposition 3.4, the idea is similar to the one used in [10] to study S-constant field theories. We will use the following operad, whose construction is explained in [15]. O⊥ O⊥ Notation 3.7. The operad LS C denotes the homotopical localization of C at the set of unary operations S. Its fundamental property is given by the following result. Proposition 3.8. There is a canonical equivalence ⊥ O⊥ Ho QFTw S(C ) ≃ Ho LS C - Alg . O⊥ The operad LS C satisfies a homotopical universal property, so it is unique up to equivalence. Several candidates of homotopical localizations for operads can be given, but since we do not need any particular presentation, we prefer to state O⊥ our results making no reference to specific models for LS C .

11 O⊥ Remark 3.9. The existence of LS C can be also proved with a good deal of homotopy theory of operads in spaces, i.e. by the use of a formal procedure to invert unary operations of ∞-operads in the sense of Lurie. O⊥ Remark 3.10. The localization LS C is performed directly at the level of operads, in contrast to the strict case, where the localization is done for the orthogonal ⊥S category CS , see Subsection 3.1. With this observation, we want to stress that the orthogonal relation on C does not need to be pushed forward, and that a generalization of orthogonal relations to enriched categories is not yet required. Since weakly S-constant ﬁeld theories are presented as algebras over the op- O⊥ erad LS C , the projective model structure on these algebras, which we denote ⊥ appealingly by QFT w S(C ), presents the homotopy theory of those ﬁeld theories. We collect these facts in the following result.

⊥ Proposition 3.11. The homotopy theory of QFTw S(C ) is presented by the Quillen ⊥ model structure QFT w S(C ), characterized by having weak equivalences and ﬁ- brations deﬁned pointwise.

Remark 3.12. Recall that a map of algebraic quantum field theories A → B is said to be pointwise in some class of maps if A(U) → B(U) belongs to this class for any space-time region U ∈ C. Let us now explain the second alternative. ⊥ Property-based approach: Field theories in QFTw S(C ) have underlying al- O⊥ gebraic structure encoded by LS C . Now we want to relax the algebraic structure by providing a model structure on QFT(C⊥) which also presents the homotopy theory of weakly S-constant field theories. For this purpose, we use left Bousfield localization [2] for a suitable set of maps. A suitable choice of a localizing set of maps in QFT(C⊥) is given in [15]. A summary of the construction can be also found in [16, Section 6]. Using the same methods, it is possible to construct a set of maps S in QFT(C⊥) representing S ⊆ C in such a way that the following result holds by the same arguments.

Proposition 3.13. An algebraic ﬁeld theory A ∈ QFT(C⊥) is S-local if and only if it is weakly S-constant.

Proof. See [15]or[16, Theorem 6.5]. We are now ready to give the second model structure to present the homotopy theory of weakly S-constant ﬁeld theories.

Proposition 3.14. The model structure QFT (C⊥) admits the left Bousfield local- ⊥ ization at S, denoted LS QFT (C ), whose fibrant objects are fibrant objects in QFT (C⊥) that satisfy the weak S-time-slice axiom.

12 Proof. The existence of the left Bousﬁeld localization is ensured by [2, Theorem 4.7] since the projective model on operadic algebras valued in Ch(R) is left proper; this fact is proved in Proposition 4.13. The characterization of ﬁbrant objects is the content of Proposition 3.13.

⊥ A comparison with the model QFT w S(C ) (Proposition 3.11) is now in order. O⊥ O⊥ Theorem 3.15. The localization morphism ℓ: C → LS C induces a Quillen ⊥ ⊥ equivalence LS QFT (C ) ⇄ QFT w S(C ).

⊥ ⊥ Proof. We have to show that the Quillen pair QFT (C ) ⇄ QFT w S(C ) induced by ℓ descends to the localization and that such descent gives rise to an equivalence at the level of homotopy categories. The universal property of the localization proves the ﬁrst claim since S goes O⊥ to equivalences in LS C via ℓ. To check that the Quillen pair establishes an equivalence for homotopy categories, use Propositions 3.11 and 3.14 to obtain the chain of equivalences

⊥ ⊥ ⊥ Ho LS QFT (C ) ≃ Ho QFTw S(C ) ≃ Ho QFT w S(C ) .

We close this section by explaining how Theorem 3.15 answers [7, Open Prob- lem 4.14]. First, let us recollect the ingredients and the content of this problem. In [7, Example 4.13], the authors introduce a new concept in the theory of algebraic ﬁeld theories, homotopy S-constancy. To recall their notion, let us denote O⊥ O⊥S the localization map and L: C → CS

⊥ ⊥ ∗ LL♯ : QFT (C ) ⇄ QFT S(C ): L its associated derived adjoint pair.

Definition 3.16. A field theory A ∈ QFT(C⊥) is homotopy S-constant if the ∗ canonical map A →L LL♯ A is an equivalence. Benini and Schenkel wondered if homotopy S-constancy captures weak S- constancy in our sense, as discussed at the end of [7, Section 3.4]. However, it is not even clear if an strict S-constant field theory (Definition 3.1) is homotopy S- constant, whereas strict S-constancy clearly implies weak S-constancy. The main ∗ obstruction for this to happen is that the derived counit LL♯L ⇒ id is not known to be a natural equivalence. Hence, we can reformulate [7, Open Problem 4.14] as the following list of items:

(Q1) Is homotopy S-constancy equivalent to weak S-constancy?

13 (Q2) Does strict S-constancy imply homotopy S-constancy?

∗ (Q3) Is the derived counit LL♯L ⇒ id a natural equivalence? We solve these questions by considering weak S-constancy instead of homotopy S-constancy, by virtue of the results discussed in this section. Denoting by O⊥ O⊥ ℓ: C → LS C the homotopical localization map and

⊥ ⊥ ∗ Lℓ♯ : QFT (C ) ⇄ QFT w S(C ): ℓ its associated derived adjoint pair, we now explain how all the complications van- ∗ ∗ ish if one replaces LL♯ ⊣ L by Lℓ♯ ⊣ ℓ . Note that the homotopy category ⊥ ⊥ Ho QFTw S(C ) is by Theorem 3.15 a full reflective subcategory of Ho QFT(C ) ⊥ with reflector Lℓ♯, since Ho QFTw S(C ) is presented in Proposition 3.14 as a left Bousfield localization of QFT (C⊥). This fact may be reinterpreted as: (i) the de- ∗ ⊥ rived counit Lℓ♯ℓ ⇒ id is a natural equivalence and (ii) A ∈ QFT (C ) is weakly ∗ S-constant iff A → ℓ Lℓ♯ A is an equivalence. Moreover, strictly S-constant field theories are canonically examples of weakly S-constant field theories. Now we compare homotopy S-constancy and weak S-constancy. By the universal property of ℓ, there is a canonical factorization of L through ℓ which yields a factorization of the right Quillen functor L∗

∗ ⊥ ℓ ⊥ ⊥ ∗ QFT (C ) ←− QFT w S(C ) ←− QFT S(C ): L .

⊥ ⊥ The best scenario would be that the right map QFT w S(C ) ← QFT S(C ) were part of a Quillen equivalence, since this is equivalent to the statement (Q1): homotopy S-constancy is equivalent to weak S-constancy (because both are properties stable under equivalences for algebraic field theories in QFT(C⊥)). Thus, the original question (Q1) can be reformulated as an strictification problem from weak S-constancy to strict S-constancy. However, such result in general seems unlikely, as the following example shows. Example 3.17. Let C⊥0 be an orthogonal category with the initial orthogonal relation. Then, the category of algebraic quantum field theories QFT(C⊥0 ) is the category Fun(C, Mon(V)) of C-diagrams of monoids. Then, for a set of maps S ⊥0 ⊥0 in C, the Quillen functor QFT w S(C ) ← QFT S(C ) yields a Quillen equivalence if and only if the canonical restriction functor

category of functors category of functors Ho ←− Ho (1) LS C → Mon(V) CS → Mon(V) gives rise to an equivalence of categories. From this starting point, our goal is to produce a concrete example which shows that this fact is not always true. Indeed,

14 it is quite unlikely that the functor (1) is in general an equivalence since LS C and CS can be potentially very different. Consider that V is the category of simplicial sets Set∆ with the Kan-Quillen model structure. Then, on the one hand, the ordinary localization CS coincides with the category π0LS C, obtained from LS C by taking connected components of its mapping spaces [19, Corollary 4.2]. On the other hand, any small simplicial category (or if you prefer any small ∞-category) can be obtained, up to Dwyer- Kan equivalence, as LS C for a suitable choice of (C, S) due to [1]. Thus, one can take (C, S) in such a way that LS C is a model for the classifying ∞-groupoid BG of a simply connected topological group G. Under these conditions, the functor (1) becomes the diagonal functor category of functors Ho ←− Ho Mon(V), BG → Mon(V) which sends a monoid to the constant functor at that monoid. If we further consider that Set∆ is endowed with the cocartesian monoidal structure, the category of monoids is just Set∆ itself. The diagonal functor in this case can be identiﬁed with the composite

∼ category of functors Ho Set∆/BG ←− Ho ←− Ho Set∆, BG → Set∆ deﬁned by sending a space X to the second factor projection X ×BG → BG, as a consequence of [27, Section 8] (or the references cited therein for the simplicial case). It is clear that this functor cannot be an equivalence for a general G; that is, the answer to (Q1) is negative. For instance, the universal ﬁbration EG → BG does not belong to the essential image of this functor, since this would imply that H∗(BG) can be embedded into H∗(EG) ∼= H∗(pt). Moreover, this functor is not fully-faithful in general. One way to see it is by computing the (homotopy) mapping space between images of the functor in Set∆/BG to see that

MapBG(X ×BG, Y ×BG) ≃ Map(X ×BG, Y). Taking connected components, the map ∼ Ho Set∆(X, Y) −→ π0 Map(X ×BG, Y) = Ho Set∆/BG(X ×BG, Y ×BG) will not be a bijection in general. In fact, for X contractible, G the Eilenberg- MacLane space K(Z, 2) and Y = BG, the map between connected components ∼ 3 ∼ becomes 0 = π0(Y) −→ H (K(Z, 3), Z) = Z. Note that the failure of fully-faithfulness above states that

∗ ⊥ ⊥ ξ : Ho QFTS(C ) −→ Ho QFTw S(C )

15 ∗ is not fully-faithful, or equivalently that the derived counit Lξ♯ξ ⇒ id is not a natural equivalence, where ξ is the remaining factor in the factorization L = ξ · ℓ. Observe that one can write the derivedcounit in (Q3) as the following composition

∗ ∗ ∗ ∗ LL♯L = Lξ♯Lℓ♯ℓ ξ Lξ♯ξ id .

The natural transformation on the left is a natural equivalence (apply Lξ♯ to the ∗ natural equivalence in (i) above), and thus LL♯L ⇒ id is a natural equivalence ∗ iff Lξ♯ξ ⇒ id is so. Therefore, (Q3) is negative in general. Remark 3.18. Another source of examples which answer [7, Open Problem 4.14] can be obtained by the use of Sullivan models of (equivariant) rational homotopy theory. In that case, we have to choose an orthogonal category C⊥1 with the ﬁnal orthogonal relation and V = Ch(Q).

4 Extension model structure

Our goal is to produce the extension model structure for operadic algebras, which, aside its great intrinsic interest, will be crucial in the main results of next section. Given an inclusion of operads, the formal idea of the construction consists on finding a cellularization, also known as right Bousfield localization, of the projective model on algebras for the bigger operad, which is Quillen equivalent to the projective model for the smaller one. The discussion in this section is more technical, and related to the theory of Quillen model stuctures. Therefore, it could be skipped in a first lecture and con- sulted when referred in the sequel. Let us fix a closed symmetric model category V (see [21]). We think about such V as a sufficiently structured homotopical cosmos in which our constructions hold. Several hypotheses must be satisfied for the extension model to exist, and they will be stated in due time, when needed.

Notation 4.1. We say that a functor, between homotopical categories, is homotopical if it preserves equivalences.

Let ι: B → N be a morphism of V-operads. Then, the induced restriction functor ι∗ between algebras admits a left adjoint

∗ ι♯ : AlgB(V) ⇄ AlgN(V): ι .

We want to deal with homotopy theories on these categories, so we assume:

Hypothesis 4.2. V is coﬁbrantly generated and B, N are admissible operads.

16 ∗ Endowing both categories with the projective model, the adjunction ι♯ ⊣ ι is a Quillen pair with ι∗ being homotopical. Fixing a functorial coﬁbrant replacement (Q, q) on AlgB(V), we get the derived pair

∗ ι♯ Q: AlgB(V) ⇄ AlgN(V): ι ,

∗ and the composite endofunctor Q = ι♯ Q ι on AlgN(V) admits an augmentation that should be seen as the derived counit,

ι q ι∗ ∗ ♯ ∗ counit ǫ: Q = ι♯ Q ι ι♯ι id .

The construction of the extension model is an application of the Bousfield- Friedlander machinery (see [29]) to the augmented endofunctor (Q, ǫ). Note that the augmentation is reversed with respect to [29], as Q is a candidate for cofibrant replacement. This causes no trouble since [28, Theorem 1.1] is completely dualizable. The Bousfield-Friedlander theorem will produce a new model on AlgN(V) (in fact a right Bousfield localization of the projective model), where:

• Weak equivalences are those maps that Q sends to equivalences.

• Coﬁbrant objects are proj-coﬁbrant algebras for which the augmentation ǫ is an equivalence.

• Cofibrations between cofibrant objects are just the proj-cofibrations.

These classes deserve a notation for future reference.

Definition 4.3. We denote by WQ the class of maps that Q sends to equivalences. We say that an N-algebra is B-colocal if the augmentation ǫ on it is an equiv- c alence. We write CofQ for the class of proj-cofibrations between proj-cofibrant algebras which are B-colocal.

Using this notation, the cited result (see [28, Theorem 1.1]) works if we check the following:

1. Q is homotopical.

2. ǫQ and Q ǫ are equivalences. W c 3. Q is stable under cobase changes along the class CofQ . The statement (1) is immediate since Q is a composite of homotopical functors. For (2) to hold we need the additional hypothesis:

17 ∗ i Hypothesis 4.4. The unit id → ι ι♯ is an equivalence on proj-coﬁbrant algebras .

Under this additional constraint, let us see that ǫQ is an equivalence. We deduce this fact from the commutative diagram

ǫQ

ι q ι∗ι Q ι∗ counit·ι Q ι∗ ∗ ∗ ♯ ♯ ∗ ∗ ♯ ∗ ι♯ Q ι ι♯ Q ι ι♯ι ι♯ Q ι ι♯ Q ι .

∼ ∼ ∗ ∗ ι♯ Q ·unit·Q ι ι♯·unit·Q ι

∗ ∗ ι♯ QQ ι ∼ ∗ ι♯ Q ι ι♯ q Q ι

The commutativitycomes from the naturality of the unit and the triangular identity ∗ of the adjunction ι♯ ⊣ ι ; the conclusion follows from 2 out of 3 for equivalences. The claim for Q ǫ follows from a similar diagram chasing. The remaining statement (3) is harder and requires further assumptions. We provide its proof in Proposition 4.6 below, but ﬁrst we present an useful characterization of our candidates for weak equivalences.

Lemma 4.5. The class WQ is the preimage of colorwise-equivalences in AlgB(V) along ι∗, i.e.

∼ WQ = f: A → B such that fιb : A(ιb) −→ B(ιb) ∀ b ∈ col(B) . n o Proof. Let us assume that f belongs to the class on the right hand side. Then, Q f is an equivalence since it is obtained from the equivalence ι∗f via the homotopical functor ι♯ Q. Conversely, assume that f belongs to WQ, i.e. Q f is an equivalence. Then we have a commutative diagram

∗ ∗ ∗ unit ∗ q ∗ ι Q A ι ι♯ Q ι A Q ι A ι A

∗ ∗ ∗ ∗ ∗ ι Q f ι ι♯ Q ι f Q ι f ι f ι∗ Q B ι∗ι Q ι∗ B Q ι∗ B ι∗ B ♯ unit q which by 2 out of 3, Hypothesis 4.4 and the fact that ι∗ is homotopical allows us to conclude the result.

i We interprete this condition as ι♯ being homotopically fully-faithful. It is ensured if ι is fully- faithful in the ordinary operadic sense.

18 Proposition 4.6. Let ABf p g

A B, ef

e e c be a pushout in AlgN(V) where f ∈ WQ and g ∈ CofQ. Then f belongs to WQ if the following conditions on the model structure on V are fulﬁlled: e • it is left proper.

• sequential colimits of coﬁbrationsii are homotopical.

Proof. First note that, as WQ is closed under retracts, we can consider without loss of generality that g is a cellular proj-cofibration, i.e. g is a transfinite composite of pushouts of generating proj-cofibrations. Recalling that the projective model on AlgN is transferred through the pair

N ◦ ⋆:[col(N), V] ⇄ AlgN(V): U, it is clear that each pushout referred above is of the form

N ◦ s(j) Aα p N ◦ j gα

N ◦ t(j) Aα+1, where j is a generating cofibration of V concentrated in one color. Due to [32, N V Proposition 4.3.17], gα, when viewed in [col( ), ] via the forgetful functor, can be described as a ω-transfinite composite of colorwise cofibrations:

1 2 0 gα 1 gα ω gα : Aα = Aα Aα ··· Aα = Aα+1 .

Arranging the two filtrations, Ug is described as a transfinite composite of cofibrations in [col(N), V]. Using this, the initial pushout square is decomposed into

iipossibly starting at a non coﬁbrant object.

19 the following commutative diagram in [col(N), V]:

ABf p

1 1 f 1 A0 0 B0 . . . . ω ω ω f A0 0 B0

0 0 0 A1 f 1 B1 p

1 1 1 A1 f 1 B1 . . . . ω ω ω f A1 1 B1 ......

. . . .

......

A ef B. e e We conclude that f belongs to WQ by the following facts: WQ is the class of arrows which are colorwise equivalences on ι(col(B)) (Lemma 4.5) and the first e step belongs to this class. Iterative steps can be computed colorwise as pushouts in V, so by left-properness, the property is inherited in iterative steps. Transfi- nite steps are transfinite composites of cofibrations in V, so the assumption on sequential colimits ensures that the property is inherited in transfinite steps. In the next statement we collect the needed hypotheses for future reference. Hypothesis 4.7. The model structure on V is left-proper and sequential colimits of cofibrations are homotopical.

Remark 4.8. Hypothesis 4.7 is slighty more general than assuming that sequential colimits are homotopical since it is implied by V being tractable or if all objects in V are coﬁbrant. On the other hand, sequential colimits are homotopical in a broad range of examples because it sufﬁces to ensure that the model structure is

20 cofibrantly generated with generating cofibrations that have compact domain and codomain [25, Section 2]. Gathering all arguments above and assuming that Hypotheses 4.2, 4.4 and 4.7 hold, we finally obtain the main result of this section.

Theorem 4.9. The projective model on AlgN admits a right Bousﬁeld localization, called the extension model structure, which enjoys the following properties:

• WQ is the class of weak equivalences.

• A coﬁbrant object is a proj-coﬁbrant algebra which is B-colocal.

• A coﬁbration g is a proj-coﬁbration such that

QAǫ A

Q g g

Q A ǫ A,

is a homotopy pushout in the projectivee model.e

∗ Furthermore, the Quillen pair ι♯ ⊣ ι descends to a Quillen equivalence between the projective model on AlgB(V) and the extension model on AlgN(V).

Proof. The first part is a direct application of the enhancement of Bousfield- Friedlander theorem in [29] as explained before. ∗ On the other hand, the Quillen pair ι♯ ⊣ ι descends to a Quillen pair for the extension model since ι∗ preserves fibrations and weak equivalences. Moreover, by Hypothesis 4.4 and construction, the derived unit and counit are respectively equivalences, and so we have the desired Quillen equivalence. We close this section by showing two fundamental properties of the extension model that will be useful to localize it.

Proposition 4.10. The extension model on AlgN(V) is coﬁbrantly generated.

Proof. Since the fibrations of the extension model are the proj-fibrations, it suffices to take as generating acyclic cofibrations the set J of generating acyclic proj- cofibrations. In order to find a set of generating cofibrations, note that they should detect acyclic fibrations, which are proj-fibrations that belong to WQ. The proj-fibration condition is fulfilled if we simply add J to our generating cofibrations. Being ad- ditionally in WQ is detected by right lifting property against a set I. Such setIis

21 just the image along the composite functor ι♯(B ◦ ⋆) of the set of generating coﬁ- brations in [col(N), V], due to Lemma 4.5 and adjunction. Note that the composite functor ι♯(B ◦ ⋆) ﬁts into the commutative square of left adjoint functors

col(ι) [col(B), V][! col(N), V]

B ◦ ⋆ N ◦ ⋆

AlgB AlgN . ι♯

Summarizing, (I ⊔ J, J) are the candidates for generating (acyclic) cofibrations of the extension model. By the aboveargument, it suffices to check that I⊔J admits a small object argument, but the same reasons given to ensure that generating proj- cofibrations admit Quillen small object argument are valid for the set I ⊔ J. For the last property of the extension model, we need more conditions on V for it to hold.

Hypothesis 4.11. For any cofibration j in V and any Σr-object G in V, the map r r idG ⊗Σr j is an h-cofibration [3, Definition 1.1], where j denotes the iterated r pushout product. That is, equivalences are stable under pushout along idG ⊗Σr j . Remark 4.12. As we see in Proposition 4.13 below, Hypothesis 4.11 is closely related to left properness of operadic algebras, issue which has being addresed by several authors in the literature. Note that symmetric h-monoidality [25, Sec- tion 2] implies Hypothesis 4.11, so a great amount of symmetric monoidal model categories satisfy the cited hypothesis.

Proposition 4.13. The projective and the extension model on AlgN(V) are left proper whenever V satisﬁes Hypothesis 4.11 and if sequential colimits in V are homotopical.

Proof. Our proof uses the ω-transfinite colorwise decomposition of cobase changes of generating cofibrations. This filtration is further explained in [32, Proposition n 4.3.17]. The sequence (gn)n∈N there consists on cofibrations whenever id ⊗Σn j is a cofibration. This condition requires that A is proj-cofibrant to ensure that the enveloping operad is Σ-cofibrant in general, as explained in [32, Subsection 5.4].

However, under Hypothesis 4.11,gn is a h-cofibration when j is a cofibration and no further conditions are required on A. Using this, the proof of Proposition 4.6 works for g any proj-cofibration, A any N-algebra and f any col(N)-equivalence or any col(B)-equivalence, and we are done.

22 5 Local-to-global cellularization

This section is devoted to the presentation of a model category structure which encapsulates the homotopy theory of AQFTs that satisfy a natural local-to-global condition, and its interaction with the models deﬁned in Section 3 that deal with the time-slice axiom.

5.1 The extension model on AQFTs First, we recall this canonical local-to-global condition (see [9, 10]).

⊥ ⊥ Definition 5.1. Let C⋄ ֒→ C be the inclusion of a full orthogonal subcategory ⊥ ⊥ ∗ and ι♯ : QFT (C⋄ ) ⇄ QFT (C ): ι the induced Quillen adjunction. Then, a ⊥ field theory A ∈ QFT(C ) is said to satisfy the C⋄-local-to-global axiom if the canonical map ∗ Lι♯ι A → A is an equivalence, where Lι♯ denotes the derived functor of ι♯. In the sequel, QFTC⋄ (C⊥) denotes the full subcategory of QFT(C⊥) spanned by field theories which satisfy the C⋄-local-to-global axiom. Remark 5.2. In [10], the authors proposed the above local-to-global principle as a substitute for Fredenhagen’s universal construction [14], which is proven to fail in certain situations because of the violation of Einstein causality, or in other words ⊥-commutativity ([10, Section 5]). Letting aside the homotopical discussion for a moment, it is easy to see that O⊥ O⊥ the morphism of operads ι: C⋄ → C induces an equivalence of categories

⊥ ⊥ ∼ A ∈ QFT(C ) s.t. ι♯ : QFT(C⋄ ) −→ ∗ , ι♯ι A −→ A is an iso which shows that the algebraic quantum field theories that satisfy the strict version of the C⋄-local-to-global principle are completely determined by their restriction to spacetime regions in C⋄. This fact clearly justifies the use of the cellularization discussed in Section 4. O⊥ O⊥ -The operad inclusion ι: C⋄ ֒→ C fulfills the requirements to apply Theo rem 4.9. Hence, QFT (C⊥) admits a cellularization, which we denote QFT C⋄ (C⊥), and whose essential properties are collected in the following theorem.

⊥ ⊥ .Theorem 5.3. Let C⋄ ֒→ C be the inclusion of a full orthogonal subcategory Then, the model structure QFT C⋄ (C⊥) has:

23 • as weak equivalences, morphisms of algebraic quantum field theories which are equivalences when evaluated on spacetime regions within C⋄; • as cofibrant objects, those cofibrant field theories in QFT (C⊥) which satisfy the C⋄-local-to-global axiom; • as fibrations, the class of projective fibrations, i.e the fibrations in QFT (C⊥).

Moreover, QFT C⋄ (C⊥) presents the homotopy theory of QFTC⋄ (C⊥) and it is ⊥ Quillen equivalent to QFT (C⋄ ). Proof. All the statements are proven in Theorem 4.9 except that QFT C⋄ (C⊥) presents the homotopy theory of QFTC⋄ (C⊥). This fact follows from the characterization of coﬁbrant objects in QFT C⋄ (C⊥) by the following chain of equivalences of categories

C⋄ ⊥ C⋄ ⊥ C⋄ ⊥ Ho QFT (C ) ≃ Ho QFT (C )coﬁbrant ≃ Ho QFT (C ).

Remark 5.4. Somehow not very surprisingly, the Quillen equivalence in Theorem C⋄ ⊥ ⊥ 5.3 between QFT (C ) and QFT (C⋄ ) implies that, in a homotopical sense, ﬁeld theories satisfying the C⋄-local-to-global axiom are completely characterized by their restriction to spacetime regions in C⋄.

5.2 Mixing localization and cellularization We discuss how the time-slice axiom and the local-to-global principle interact in terms of homotopy theory of AQFTs. We will need to repeat the distinction between the structural and the property-based approach in Section 3. For these purposes, we start with some notation.

C⋄ ⊥ ⊥ Notation 5.5. We denote by QFTw S(C ) the full subcategory of QFT(C ) spanned by weakly S-constant ﬁeld theories that satisfy the C⋄-local-to-global principle.

Structural approach: Our goal is to introduce the C⋄-local-to-global principle ⊥ into the model structure QFT w S(C ) by means of a cellularization as in the pre- O⊥ ceding section. In this situation, we work with LS C as the operad that governs the underlying algebraic structure. We want to make use again of the results of Section 4, so we must select a suitable map of operads B → N. It is quite natural to consider the morphism

24 O⊥ O⊥ ιS : LS⋄ C⋄ → LS C , where S⋄ represents the intersection of S with C⋄. One reason is because it ﬁts into the following commutative square of operads

O⊥ ℓ O⊥ C LS C

ι φ ιS (2)

⊥ ⊥ O LS⋄ O , C⋄ ℓ⋄ C⋄ which yields the following motivational result.

⊥ Proposition 5.6. The following conditions are equivalent for A ∈ QFTw S(C ).

∗ • LιS ♯ιS A → A is an equivalence.

∗ • Lφ♯φ A → A is an equivalence. Proof. The commutative square (2) induces the commutative diagram of adjoints pairs between categories

⊥ ⊥ Ho QFT (C ) Ho QFT w S(C )

⊥ ⊥ Ho QFT (C⋄ ) Ho QFT w S⋄ (C⋄ ).

Therefore, with appropiate compositions of adjoints there, we obtain a commuta- ⊥ tive diagram in Ho QFTw S(C )

∗ Lφ♯φ A (i) ≃ (iii)

∗ ∗ LιS ♯ · Lℓ⋄♯ℓ⋄ · ιS A A .

≃ (iv) (ii) ∗ LιS ♯ιS A

On the one hand, (i) is an equivalence (iso in the homotopy category) by the commutativity of the above diagram. On the other hand, (ii) is an equivalence because, ⊥ ⊥ ∗ by Theorem 3.15, the adjunction Lℓ⋄♯ : Ho QFT(C⋄ ) ⇄ Ho QFTw S(C⋄ ): ℓ⋄ is a reﬂection. Thus, (iii) is an equivalence iff (iv) is so. O⊥ Remark 5.7. Proposition 5.6 should be understood as a check that LS⋄ C⋄ -colocal algebras (see Section 4) satisfy in some sense the C⋄-local-to-global principle, because they are reconstructed from its restriction to spacetime regions in C⋄.

25 However, regardless the canonicity of ιS, a technical assumption is required on S in order to make Theorem 4.9 work. Instead of ﬁnding general conditions for these purposes, we consider a quite restrictive hypothesis which makes all arguments work. For more information, see Remarks 5.10 and 5.15.

Hypothesis 5.8. The set of maps S belongs completely to the full subcategory C⋄.

⊥ ⊥ Theorem 5.9. Assume that C⋄ ֒→ C is a full orthogonal subcategory and S a set of morphisms in C satisfying Hypothesis 5.8. Then, there exists a Quillen model C⋄ ⊥ ⊥ structure QFT w S(C ) on the category of QFTw S(C ) that satisfies: • the weak equivalences are the maps that are equivalences when evaluated on spacetime regions within C⋄; • the fibrations are the projective fibrations;

⊥ • it is Quillen equivalent to QFT w S⋄ (C⋄ );

⊥ • the cofibrant objects are the cofibrant objects in QFT w S(C ) which satisfy the appropiate C⋄-local-to-global principle. O⊥ O⊥ Proof. Hypothesis 5.8 implies that the map LS⋄ C⋄ → LS C satisfies Hypoth- esis 4.4. Hence, Theorem 4.9 applies, giving the desired model structure. The characterization of the weak equivalences follows from Proposition 5.6 and a vari- ation of Lemma 4.5. The appropiate C⋄-local-to-global principle just asserts that ⊥ ∗ A ∈ QFTw S(C ) satisfies that the canonical map Lφ♯φ A → A is an equivalence, where φ is the map of operads appearing in the square (2). Thus, the recognition of cofibrant objects is a consequence of Proposition 5.6.

O⊥ O⊥ Remark 5.10. It is not true that LS⋄ C⋄ → LS C satisﬁes Hypothesis 4.4 in general, so Theorem 4.9 could not be applicable. We expect that conditions such as

LS⋄ C⋄(U, V) −→ LS C(U, V) being an equivalence ∀ U, V ∈ C⋄ (homotopical fully-faithfulness) are sufﬁcient for these purposes, although some more work will be needed. Furthermore, even a manageable condition on S that ensures this requirement seems to be hard to ﬁnd, so we prefer to avoid this problematic in here.

26 Property-based approach: Now, the underlying algebraic structure is consid- O⊥ ered to be parametrized by the operad C , i.e. the underlying category will be ⊥ O⊥ ⊥ O⊥ QFT(C ) ≃ C - Alg despite QFTw S(C ) ≃ LS C - Alg. In order to present the homotopy theory of weakly S-constant field theories that satisfy the local-to- global principle, we need to perform two Bousfield localizations of the model structure QFT (C⊥); each one deals with one axiom independently. We perform the cellularization described in Theorem 5.3 to get the model structure QFT C⋄ (C⊥). Because of Propositions 4.10 and 4.13, the model structure QFT C⋄ (C⊥) admits the localization at the set of maps S that appears in C⋄ ⊥ Proposition 3.14. Let us denote such localized model by LS QFT (C ). The fundamental result is the recognition of the bifibrant objects in this model.

C⋄ ⊥ Theorem 5.11. The biﬁbrant objects in LS QFT (C ) are those biﬁbrant ob- ⊥ jects in QFT (C ) which satisfy the weak S-time-slice axiom and the C⋄-local-to- global principle.

Proof. The cellularization does not change the class of fibrations, and hence of fibrant objects, whereas the localization does not change the class of cofibrations. Thus, the result follows from Proposition 3.14 and Theorem 5.3.

C⋄ ⊥ Corollary 5.12. The model structure LS QFT (C ) presents the homotopy the- C⋄ ⊥ ory of QFTw S(C ), i.e. algebraic ﬁeld theories which satisfy the weak S-time-slice axiom and the C⋄-local-to-global principle. Proof. This follows from the equivalences of categories

C⋄ ⊥ C⋄ ⊥ C⋄ ⊥ Ho LS QFT (C ) ≃ Ho LS QFT (C )biﬁbrant ≃ Ho QFTw S(C ).

We ﬁnish by comparing the two approaches. O⊥ O⊥ Theorem 5.13. The localization morphism of operads ℓ: C → LS C induces a Quillen equivalence of model categories

C⋄ ⊥ ⇄ C⋄ ⊥ ∗ ℓ♯ : LS QFT (C ) QFT w S(C ): ℓ .

⊥ Proof. By deﬁnition of QFT w S(C ), we already have a Quillen pair

⊥ ⊥ QFT (C ) ⇄ QFT w S(C ) induced by ℓ. The idea is to prove that this pair descends to the cellularizations and the localization.

27 ∗ ⊥ ⊥ By Lemma 4.5, the functor ℓ : QFT w S(C ) → QFT (C ) preserves the class of equivalences in Theorem 4.9. Hence, we have the corresponding Quillen C⋄ ⊥ C⋄ ⊥ pair between cellularizations, QFT (C ) ⇄ QFT w S(C ). Taking account of the universal property of the localization, it remains to check C⋄ ⊥ that the set S is sent to equivalences in QFT w S(C ). The same argument given in Theorem 3.15 shows that this is the case. Once constructed the Quillen pair of the statement, proving that it is a Quillen equivalence boils down to showing that it induces an equivalence of homotopy categories. This follows from the equivalences

(i) C⋄ ⊥ C⋄ ⊥ Ho LS QFT (C ) ≃ Ho LS⋄ QFT (C )

(ii) ⊥ ≃ Ho LS⋄ QFT (C⋄ )

(iii) ⊥ ≃ Ho QFT w S⋄ (C⋄ ) (iv) C⋄ ⊥ ≃ Ho QFT w S(C ).

(i) is ensured by Hypothesis 5.8; (ii) is immediate since we are localizing Quillen equivalent model structures at the same set of maps; (iii) is a particular instance of Theorem 3.15; and (iv) is the third item in Theorem 5.9.

C⋄ ⊥ Corollary 5.14. The model structure QFT w S(C ) presents the homotopy theory C⋄ ⊥ of QFTw S(C ), i.e. algebraic field theories which satisfy the weak S-time-slice axiom and the C⋄-local-to-global principle. Remark 5.15. As explained in the structural approach to the homotopy theory of weakly S-constant field theories satisfying C⋄-local-to-global principle, we restrict ourselves to sets S fulfilling Hypothesis 5.8. Observe that this restriction is fundamental for the comparison between the structural and the property-based approach C⋄ ⊥ C⋄ ⊥ and even to see that QFT w S(C ) presents the homotopy theory of QFTw S(C ). Nevertheless, we hope that those results hold under weaker hypothesis. Note that the essential step where we need the restriction is the equivalence of categories

C⋄ ⊥ ⊥ Ho LS QFT (C ) ≃ Ho LS⋄ QFT (C⋄ ).

28 6 Comparison of homotopy theories

This section contains a table summarizing all the model structures discussed in this work and a diagram that represents the relations between them.

Homotopy theory of Quillen model structure Underlying algebraic structure ⊥ O⊥ algebraic ﬁeld theories QFT (C ) (Notation 2.7) algebras over C

strictly S-constant ﬁeld theories C⊥ (Notation 3.3) algebras over O⊥S QFT S( ) CS ⊥ O⊥ weakly S-constant ﬁeld theories QFT w S(C ) (Proposition 3.11) algebras over LS C

⊥ O⊥ weakly S-constant ﬁeld theories LS QFT (C ) (Proposition 3.14) algebras over C

algebraic ﬁeld theories QFT C⋄ (C⊥) (Theorem 5.3) algebras over O⊥ satisfying local-to-global principle C

weakly S-constant ﬁeld theories QFT C⋄ (C⊥) (Theorem 5.9) algebras over L O⊥ satisfying local-to-global principle w S S C

weakly S-constant ﬁeld theories L QFT C⋄ (C⊥) (Theorem 5.11) algebras over O⊥ satisfying local-to-global principle S C

In the following diagram are displayed these Quillen model categories with

29 Quillen adjunctions between them, oriented by their left adjoints.

⊥ QFT S(C )

⊥ ⊥ QFT (C ) QFT w S(C )

localization Q-equiv

⊥ cellularization LS QFT (C ) cellularization

C⋄ ⊥ C⋄ ⊥ QFT (C ) QFT w S(C ) Q-equiv Q-equiv localization Q-equiv

⊥ C⋄ ⊥ ⊥ QFT (C⋄ ) LS QFT (C ) QFT w S⋄ (C⋄ )

Q-equiv localization Q-equiv

⊥ LS⋄ QFT (C⋄ ) (Comparison)

Acknowledgments The author would like to thank his advisors, Ramon Flo- res and Fernando Muro, for their support. Talking about contributions, Example 3.17 was the consequence of an enjoyable discussion with Fernando Muro. Car- los Maestro Pérez also deserves recognition; his help with LaTeX matters is of unquestionable value.

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UNIVERSIDAD DE SEVILLA, FACULTAD DE MATEMÁTICAS, DEPARTAMENTO DE ÁLGEBRA-IMUS, AVDA. REINA MERCEDES S/N, 41012 SEVILLA, SPAIN Email address: [email protected] url: http://personal.us.es/vcarmona1