Algebraic field theory operads and linear quantization

We generalize the operadic approach to algebraic quantum field theory [arXiv:1709.08657] to a broader class of field theories whose observables on a spacetime are algebras over any single-colored operad. A novel feature of our framework is that it gives rise to adjunctions between different types of field theories. As an interesting example, we study an adjunction whose left adjoint describes the quantization of linear field theories. We also develop a derived version of the linear quantization adjunction for chain complex valued field theories, which in particular defines a homotopically meaningful quantization prescription for linear gauge theories.


Introduction and summary
The aim of this paper is to generalize and extend the operadic approach to algebraic quantum field theory initiated in [BSW17]. The structures we shall formalize and investigate are functors A : C → Alg P from a small category C to the category of algebras over a single-colored operad P, which are required to satisfy a suitable generalization of the Einstein causality axiom. One should interpret C as a category of spacetimes and P as the operad controlling the algebraic structure of the observables on a fixed spacetime. Choosing the associative operad P = As, we recover the framework for quantum field theory developed in [BSW17]. There however exist also other interesting choices which are not covered by the latter work. For example, (1) classical field theories may be obtained by choosing the Poisson operad P = Pois in our construction, and (2) linear field theories, which we describe in terms of Heisenberg Lie algebras of presymplectic vector spaces, by choosing the unital Lie operad P = uLie. Therefore, our constructions provide a flexible framework to formalize and investigate a broad range of different types of field theories (linear, classical, quantum, etc.) from a common algebraic perspective based on operad theory.
One of the key observations in the present paper is that to each of such types of field theories there corresponds a colored operad which controls their algebraic structures, i.e. field theories are precisely the algebras over this colored operad. We denote the relevant colored operad by P (r 1 ,r 2 ) C and note that it depends on two different kinds of input data, which control the spacetime category of interest and the type of field theory. More precisely, the first datum is an orthogonal category C = (C, ⊥) (cf. Definition 3.1) and and the second is a bipointed single-colored operad P (r 1 ,r 2 ) = (P, r 1 , r 2 : I[2] ⇒ P) (cf. Definition 3.14). As in the previous paragraph, we interpret C as a category of spacetimes and P as the operad controlling the algebraic structure of the observables on a fixed spacetime. The data ⊥ and r 1 , r 2 are used to formalize a suitable generalization of the Einstein causality axiom of quantum field theory, see Definition 3.3. We shall provide a precise description of the colored operads P (r 1 ,r 2 ) C in Definitions 3.9 and 3.11.
We will show that the results for quantum field theories obtained in [BSW17] extend to our more general and flexible framework. In particular, from the functoriality of the assignment C → P (r 1 ,r 2 ) C of our colored operads to orthogonal categories, we obtain adjunctions between the categories of field theories corresponding to different spacetime categories. This includes generalizations of the time-slicification and local-to-global adjunctions from [BSW17], which have already found interesting applications to quantum field theory on spacetimes with boundaries [BDS18]. A novel feature of our framework, which is not captured by [BSW17], is a second kind of functorial assignment P (r 1 ,r 2 ) → P (r 1 ,r 2 ) C of our colored operads to bipointed single-colored operads. This results in adjunctions between the categories of field theories of different types. An interesting example, which we shall study in detail in this paper, is given by an adjunction whose left adjoint describes the quantization of linear field theories. This is induced by a canonical single-colored operad morphism uLie → As from the unital Lie operad to the associative operad.
Reformulating the usual quantization of linear field theories in terms of (the left adjoint of) an adjunction has profound technical advantages for studying gauge theories. Recall that the observables of gauge theories form chain complexes rather than vector spaces, which are obtained by e.g. the BRST/BV formalism. Because the category Ch(K) of chain complexes of vector spaces over a field K carries a canonical model structure with weak equivalences given by quasi-isomorphisms (see e.g. [Hov99]), it is natural ask whether field theories with values in Ch(K) form a model category as well. Based on [BSW18,Hin97,Hin15], we shall show that this is the case and moreover that the linear quantization adjunction is a Quillen adjunction. Employing techniques from the theory of derived functors (see e.g. [DS95,Hov99]), we obtain a derived linear quantization functor which provides a homotopically meaningful quantization prescription for linear gauge theories in the sense that it maps weakly equivalent linear gauge theories to weakly equivalent quantum gauge theories. A similar construction in the framework of factorization algebras [CG17] has been recently investigated in [GH18]. We shall also analyze in some detail the interplay of our derived linear quantization functor with (a homotopy theoretical generalization of) the time-slice axiom and local-to-global property of gauge theories. In a future work, we attempt to develop concrete constructions and examples of linear gauge theories from suitable geometric data such as stacks of fields and Lagrangian densities. The derived linear quantization functor from the present work then will be relevant to quantize these linear gauge theories.
The outline of the remainder of this paper is as follows: In Section 2 we shall fix our notations and briefly recall some background material on colored operads and their algebras. In Section 3 we introduce our rather broad concept of field theories which we study in this paper and show that the corresponding categories admit a description in terms of algebras over a suitable colored operad P (r 1 ,r 2 ) C . We also show that the assignment (C, P (r 1 ,r 2 ) ) → P (r 1 ,r 2 ) C of these operads to orthogonal categories C and bipointed single-colored operads P (r 1 ,r 2 ) is functorial. In Section 4 we harness this functorial behavior in order to study adjunctions between the categories of field theories corresponding to different C and P (r 1 ,r 2 ) . We first show that the results from [BSW17] on specific properties of the adjunctions corresponding to changing C easily generalize to our more flexible framework. After this we focus on our novel class of adjunctions corresponding to changing P (r 1 ,r 2 ) and investigate their interplay with the time-slice axiom and local-to-global property of field theories. A particularly relevant example of such an adjunction is the linear quantization adjunction which we study in detail in Section 5. In Section 6 we extend the results of the previous sections to the case of Ch(K)-valued field theories (which includes gauge theories) by using techniques from model category theory.

Closed symmetric monoidal categories
Throughout this paper we fix a closed symmetric monoidal category M, which we further assume to be complete and cocomplete, i.e. all small limits and colimits exist in M. The monoidal product is denoted by ⊗ : M × M → M, the monoidal unit by I ∈ M and the internal hom by [−, −] : M op × M → M, where (−) op denotes the opposite category. The symmetric braiding is denoted by τ : m ⊗ m ′ → m ′ ⊗ m, for all m, m ′ ∈ M. We shall always suppress the associator and unitors and in particular simply write m 1 ⊗ · · · ⊗ m n for multiple tensor products of objects m 1 , . . . , m n ∈ M. Because M is by assumption cocomplete, there exists a Set-tensoring ⊗ : Set × M → M, which we denote with abuse of notation by the same symbol as the monoidal product. Explicitly, for any set S ∈ Set and m ∈ M, we define where is the coproduct in M.
Example 2.1. A simple example of a bicomplete closed symmetric monoidal category is the Cartesian closed category Set of sets. Here ⊗ = × is the Cartesian product, I = { * } is any singleton set and [S, T ] = Map(S, T ) is the set of maps from S to T . The symmetric braiding is given by the flip map τ : S × T → T × S , (s, t) → (t, s). ▽ Example 2.2. Another standard example of a bicomplete closed symmetric monoidal category is the category Vec K of vector spaces over a field K. Here ⊗ is the usual tensor product of vector spaces, I = K is the 1-dimensional vector space and [V, W ] = Hom K (V, W ) is the vector space of linear maps from V to W . The symmetric braiding is given by the flip map τ :

Colored operads
We provide a brief review of those aspects of the theory of colored operads that are relevant for this work. We refer to e.g. [Yau16], [BM07] and [BSW17] for a more detailed presentation.
Let C ∈ Set be a non-empty set, which we shall call the 'set of colors'. We will use the notation c := (c 1 , . . . , c n ) ∈ C n for elements of the n-fold product set.
Definition 2.3. A C-colored operad O with values in M is given by the following data: • for each n ≥ 0 and (c, t) ∈ C n+1 , an object O t c ∈ M (called the object of operations from c to t); • for each n ≥ 0, (c, t) ∈ C n+1 and permutation σ ∈ Σ n , an cσ (called the permutation action), where cσ := (c σ(1) , . . . , c σ(n) ); • for each n > 0, k 1 , . . . , k n ≥ 0, (a, t) ∈ C n+1 and (b i , a i ) ∈ C k i +1 , for i = 1, . . . , n, an This data is required to satisfy the standard permutation action, equivariance, associativity and unitality axioms, see e.g. [Yau16, Definition 11.2.1]. A morphism φ : O → P between two C-colored operads O and P with values in M is a family of M-morphisms Colored operads generalize the concept of (enriched) categories in the following sense. In contrast to allowing only for 1-to-1 operations, such as the morphisms C(c, c ′ ) in a category C, colored operads also describe n-to-1 operations in terms of the objects of operations O t c . The operadic composition generalizes the usual categorical composition to operations of higher arity and the operadic unit is analogous to the identity morphisms in a category. Permutation actions are a new feature for operations of arity ≥ 2 and they have no analog in ordinary category theory. The following example clarifies how every category defines a colored operad with only 1-ary operations.
Example 2.4. Let C be a small category and denote its set of objects by C 0 . The following construction defines a C 0 -colored operad Diag C ∈ Op C 0 (Set) with values in M = Set, which is called the diagram operad of C, see e.g. [BM07]. For (c, t) ∈ C n+1 0 , one defines the set of operations by The permutation action is uniquely fixed because Σ 1 = {e} is the trivial group. The only nontrivial operadic compositions are γ : Diag C is given by the identity morphisms in the category C. One confirms that this defines a colored operad in the sense of Definition 2.3. ▽ Many interesting examples of (colored) operads can be conveniently defined in terms of generators and relations, see e.g. the examples below. Let us briefly explain how this construction works. We denote by Seq C (M) the category of C-colored (non-symmetric) sequences with values in M. An object X ∈ Seq C (M) is a family of objects X t c ∈ M, for all n ≥ 0 and (c, t) ∈ C n+1 , and a Seq C (M)-morphism f : X → Y is a family of M-morphisms f : X t c → Y t c , for all n ≥ 0 and (c, t) ∈ C n+1 . There exists a forgetful functor U : Op C (M) → Seq C (M) that forgets the permutation action, operadic composition and operadic unit of a C-colored operad. This functor has a left adjoint which is called the free C-colored operad functor, i.e. we have an adjunction (2.4) Given any choice of generators G ∈ Seq C (M), we consider the corresponding free C-colored operad F (G) ∈ Op C (M). In order to implement relations, we consider R ∈ Seq C (M) together with two parallel Seq C (M)-morphisms r 1 , r 2 : R ⇒ U F (G). Note that because (2.4) is an adjunction, the latter is equivalent to two parallel Op C (M)-morphisms r 1 , r 2 : F (R) ⇒ F (G), which we denote with abuse of notation by the same symbols. Because the category Op C (M) is cocomplete, the following construction defines a C-colored operad.
Definition 2.5. The C-colored operad presented by the generators G ∈ Seq C (M) and relations r 1 , r 2 : R ⇒ U F (G) is defined as the coequalizer Example 2.6. Consider for the moment M = Set. The associative operad As ∈ Op { * } (Set) is the single-colored operad (i.e C = { * } is a singleton) presented by the following generators and relations: We define the set of generators of arity n by (2.6) for all n ≥ 0. The generator µ in arity 2 is interpreted as a multiplication operation and the generator η in arity 0 as a unit element. To implement associativity and left/right unitality of these operations, we consider for all n ≥ 0, together with the two Seq { * } (Set)-morphisms r 1 , r 2 : R → U F (G) defined by where the operadic composition and unit are those of the free operad F (G). The associative operad As := F (G)/(r 1 = r 2 ) ∈ Op { * } (Set) is defined as the corresponding coequalizer.
It is instructive and useful to visualize the generators and relations in terms of rooted trees. The generators are depicted by (2.9a) and the relations (in the order they appear in (2.8)) then read as (2.9b) Let us note that the associative operad can be defined in any bicomplete closed symmetric monoidal category M. Using the Set-tensoring (2.1) and the unit object I ∈ M, we define generators G ⊗ I ∈ Seq { * } (M) and relations r 1 ⊗ I, r 2 ⊗ I : (2.10a) The relations are given by antisymmetry and the Jacobi identity where the numbers below the trees indicate input permutations.
Note that for defining the Lie relations we had to use the natural Abelian group structure on the Hom-sets of Vec K , i.e. addition of linear maps between vector spaces. Hence, the Lie operad can not be defined in a generic bicomplete closed symmetric monoidal category M. If however M is an additive category, then one can define the Lie operad Lie ∈ Op { * } (M) with values in M along the same lines as above. (2.11b) which express that µ is commutative and that {·, ·} is a derivation in the right entry (and hence by antisymmetry also a derivation in the left entry). Computing the operadic composition of the derivation relation with ½ ⊗ η ⊗ η implies that In addition to the antisymmetry and Jacobi identity relations (cf. (2.10)) for [·, ·], we demand the compatibility relation = 0 (2.14) between the Lie bracket and the unit. ▽ We shall often require a generalization of the concept of colored operad morphisms from Definition 2.3 to morphisms that do not necessarily preserve the underlying sets of colors. As a preparation for the relevant definition, note that for every D-colored operad P ∈ Op D (M) and every map of sets f : C → D, one may define the pullback C-colored operad f * (P) ∈ Op C (M). Concretely, it is defined by setting f * (P) t c := P f (t) f (c) , for all n ≥ 0 and (c, t) ∈ C n+1 , and restricting the permutation action, operadic composition and operadic unit in the evident way.

Algebras over colored operads
We have seen above that a colored operad O describes abstract n-to-1 operations, for all n ≥ 0, together with a composition law γ, specified identities ½ and a permutation action O(σ) that allows us to permute the inputs of operations. Forming concrete realizations/representations of these abstract operations leads to the concept of algebras over colored operads.
Definition 2.11. An algebra A over a C-colored operad O ∈ Op C (M), or shorter an O-algebra, is given by the following data: • for each c ∈ C, an object A c ∈ M; • for each n ≥ 0 and (c, t) ∈ C n+1 , an M-morphism α : A c i with the convention that A ∅ = I for n = 0. This data is required to satisfy the standard associativity, unity and equivariance axioms, see e.g. [Yau16, Definition 13.2.3]. A morphism κ : A → B between two O-algebras A and B is a family of M-morphisms κ : We denote the category of O-algebras by Alg O .
Example 2.12. Consider the diagram operad Diag C ∈ Op C 0 (Set) from Example 2.4. A Diag Calgebra is a family of sets A c ∈ Set, for all objects c ∈ C 0 in the category C, together with maps α : for all c, t ∈ C 0 . (Here we already used that Diag C only contains 1-ary operations.) Because Diag C t c = C(c, t) is the Hom-set, the latter data is equivalent to specifying for each C-morphism f : c → t a map of sets A(f ) := α(f, −) : A c → A t . The axioms for O-algebras imply that A(g f ) = A(g) A(f ), for all composable C-morphism, and A(id) = id for the identities. Hence, a Diag C -algebra is precisely a functor C → Set, i.e. a diagram of shape C. One observes that morphisms between Diag C -algebras are precisely natural transformations between the corresponding functors. ▽ Example 2.13. Consider for the moment M = Set and the associative operad As ∈ Op { * } (Set) from Example 2.6. An As-algebra is a single set A = A * ∈ Set together with an As-action. The latter is equivalent to providing a family of maps α : As(n) → Map(A ×n , A), for all n ≥ 0, which define an Op { * } (Set)-morphism to the endomorphism operad End(A), see e.g. [Yau16, Definition 13.8.1]. Because As is presented by generators and relations (cf. Example 2.6), this is equivalent to defining α on the generators such that the relations hold true. This yields two maps µ A := α(µ) : A × A → A and η A := α(η) : { * } → A, which because of the relations have to satisfy the axioms of an associative and unital algebra in Set. One finds that morphisms of As-algebras are precisely morphisms of associative and unital algebras.
For a general bicomplete closed symmetric monoidal category M, one obtains that the category Alg As of algebras over As ∈ Op { * } (M) is the category of associative and unital algebras in M. In particular, for M = Vec K this is the category of associative and unital K-algebras.
, the pullback functor (f, φ) * : Alg P → Alg O has a left adjoint, which is called operadic left Kan extension. We denote the corresponding adjunction by (2.16) Example 2.16. Every functor F : C → D defines an evident Op(Set)-morphism (F 0 , F ) : between the corresponding diagram operads, cf. Example 2.4. Recalling from Example 2.12 that Alg Diag C ∼ = Set C is the category of functors from C to Set (and similarly that Alg Diag D ∼ = Set D ), one shows that the pullback functor (F 0 , F ) * is the usual pullback functor F * := (−) • F : Set D → Set C for functor categories. Its left adjoint (F 0 , F ) ! is therefore the ordinary categorical left Kan extension Lan F : Set C → Set D . ▽ 3 Field theory operads

Orthogonal categories and field theories
Let us briefly recall the basic idea of algebraic quantum field theory, see e.g. [HK64, BFV03, BDFY15, FV15] for more details. Broadly speaking, a field theory in this setting is a functor from a suitable category of spacetimes to a category of algebraic structures of interest, that satisfies a list of physically motivated axioms. The prime example is given by functors A : Loc → Alg As from the category Loc of globally hyperbolic Lorentzian manifolds to the category of associative and unital algebras that satisfy the Einstein causality axiom. The latter is a property of the functor A : Loc → Alg As which demands that for every pair (f 1 : Loc-morphisms whose images are causally disjoint in M the diagram M denotes the (opposite) multiplication on A(M ). For the purpose of this paper, we consider the following generalization of the scenario sketched above. (Examples which justify this generalization are presented at the end of this subsection.) Let C be a small category which we interpret as a category of spacetimes. Instead of associative and unital algebras, let us take any single-colored operad P ∈ Op { * } (M) and consider the functor category Alg P C . An object in this category is a functor A : C → Alg P , i.e. an assignment of P-algebras to spacetimes, and the morphisms are natural transformations between such functors. To encode physical axioms which generalize the Einstein causality axiom above, we recall the concept of orthogonal categories from [BSW17].
Definition 3.1. An orthogonal category is a pair C := (C, ⊥) consisting of a small category C and a subset ⊥⊆ Mor C t × t Mor C of the set of pairs of morphisms with a common target, which satisfies the following properties: We shall also write We denote by OrthCat the category of orthogonal categories and orthogonal functors.
for all n ≥ 0. This means that each r i picks out an operation of arity 2 in P. For simplifying notation, we shall write and we call P (r 1 ,r 2 ) an (arity 2) bipointed single-colored operad.
Definition 3.3. A field theory of type P (r 1 ,r 2 ) on C is a functor A : C → Alg P that satisfies the following property: For all (f 1 : in M commutes, where α P c denotes the P-action on A(c) ∈ Alg P , cf. Definition 2.11. The category of field theories of type P (r 1 ,r 2 ) on C is defined as the full subcategory FT C, P (r 1 ,r 2 ) ⊆ Alg P C , (3.5) whose objects are all functors A : C → Alg P satisfying (3.4).
Remark 3.4. Our concept of field theories in Definition 3.3 is based on the idea that there exist two distinguished arity 2 operations in P, which act in the same way when pre-composed with an orthogonal pair f 1 ⊥ f 2 of C-morphisms. There exists an obvious generalization of this scenario to n-ary operations in P and orthogonal n-tuples of C-morphisms. We however decided not to introduce this more general framework for field theories, because all examples of interest to us are field theories in the sense of Definition 3.3. ▽ Example 3.5 (Quantum field theories). Consider the associative operad As ∈ Op { * } (M) from Example 2.6 and the two Seq { * } (M)-morphisms µ, µ op : I[2] ⇒ U (As) which select the multiplication and opposite multiplication operations. A field theory of type As (µ,µ op ) on C is a functor A : C → Alg As to the category of associative and unital algebras which satisfies the analog of (3.1). For C = Loc (cf. Example 3.2), this is a locally covariant quantum field theory [BFV03,FV15] that satisfies the Einstein causality axiom but not necessarily the time-slice axiom. As explained in [BSW17], the latter can be encoded by . A field theory of type As ([·,·],0) on C is a functor A : C → Alg As to the category of associative and unital algebras which satisfies the property that is the zero-map, for all (f 1 : .) This is equivalent to our description in Example 3.5, i.e.
FT C, As ([·,·],0) ∼ = FT C, As (µ,µ op ) . (3.7) This observation will be useful in Section 5 when we study the linear quantization adjunction. ▽  Example 3.8 (Linear field theories). In the usual construction of linear quantum field theories, see e.g. [BGP07,BDH13] for reviews, one first defines a functor L : Loc → PSymp to the category of presymplectic vector spaces, which is then quantized by forming CCR-algebras (CCR stands for canonical commutation relations). Recall that a presymplectic vector space (V, ω) is a pair consisting of a vector space V and an antisymmetric linear map ω : V ⊗ V → K. Notice that this is not an operation of arity 2 in the sense of operads because the target is the ground field and not V . Hence, PSymp is not the category of algebras over an operad and, as a consequence, functors L : Loc → PSymp do not define field theories in the sense of Definition 3.3.
However, there exists a canonical upgrade of every functor L : Loc → PSymp to a field theory in the sense of Definition 3.3. Given any presymplectic vector space (V, ω), one can define its Heisenberg Lie algebra H(V, ω). The underlying vector space of H(V, ω) is given by V ⊕ K and the Lie bracket There exists a canonical unit map η : Hence, Heisenberg Lie algebras are algebras over the unital Lie operad uLie ∈ Op { * } (M) given in Example 2.9. Because forming Heisenberg Lie algebras is functorial, we can define for every L : Loc → PSymp the composite functor H L : Loc → Alg uLie .
Consider now the two Seq { * } (M)-morphisms [·, ·], 0 : I[2] → U (uLie) which select the Lie bracket and the zero-operation. A field theory of type uLie ([·,·],0) on C is a functor A : C → Alg uLie to the category of unital Lie algebras which satisfies the property that is the zero-map, for all (f 1 :

Operadic description
In this section we show that the category of field theories from Definition 3.3 is the category of algebras over a suitable colored operad. This generalizes previous results in [BSW17] and it is the key insight that allows us to study a large family of universal constructions for field theories in Section 4. As a preparation for the relevant definition, we define an auxiliary colored operad that describes functors from a small category C to the category of P-algebras.
Definition 3.9. Let C be a small category with set of objects C 0 and let P ∈ Op { * } (M) be a single-colored operad. The C 0 -colored operad P C ∈ Op C 0 (M) is defined by the following data: • for n ≥ 0 and (c, t) ∈ C n+1 0 , the object of operations is where ⊗ is the Set-tensoring (2.1) and C(c, t) := n i=1 C(c i , t) is the product of Hom-sets; • for n ≥ 0, (c, t) ∈ C n+1 0 and σ ∈ Σ n , the permutation action P C (σ) is defined by for all f : = (f 1 , . . . , f n ) ∈ C(c, t), where ι f : P(n) → P C t c = C(c, t) ⊗ P(n) are the inclusion morphisms into the coproduct (cf. (2.1)) and f σ := (f σ(1) , . . . , f σ(n) ); • for n > 0, k 1 , . . . , k n ≥ 0, (a, t) ∈ C n+1 0 and (b i , a i ) ∈ C k i +1 0 , for i = 1, . . . , n, the operadic composition γ P C is defined by . . , f n g nkn ) ∈ C(b, t) is defined by composition in the category C; • for c ∈ C 0 , the operadic unit ½ P C is where id c : c → c is the identity morphism of c in the category C.
A straightforward check shows that this data defines a colored operad, cf. Definition 2.3.
Lemma 3.10. There exists a canonical isomorphism between the category of algebras over the colored operad P C ∈ Op C 0 (M) from Definition 3.9 and the category of functors from C to Alg P .
Proof. A P C -algebra is a family of objects A c ∈ M, for all c ∈ C 0 , together with a P C -action α : P C t c ⊗A c → A t . Because (3.11) is a coproduct, this is equivalent to a family of M-morphisms α f : P(n) ⊗ A c → A t , for all n ≥ 0, (c, t) ∈ C n+1 0 and f ∈ C(c, t), which satisfies the following compatibility conditions resulting from the axioms for algebras over colored operads Using that any f = (f 1 , . . . , f n ) ∈ C(c, t) can be written as f = id t n (f 1 , . . . , f n ), where id t n = (id t , . . . , id t ) is of length n, the diagram (3.16a) implies that α f factorizes as Hence, the P C -action α is uniquely specified by the following two types of M-morphisms: (1) α t := α idt n : P(n) ⊗ A ⊗n t → A t , for all t ∈ C 0 and n ≥ 0, and (2) The remaining conditions in (3.16) are equivalent to α t defining a P-action on A t , for all t ∈ C 0 , and A(f ) : A c → A t defining a functor C → Alg P to P-algebras. From this perspective, P C -algebra morphisms correspond precisely to natural transformations between functors from C to Alg P .
Definition 3.11. The operad of field theories of type P (r 1 ,r 2 ) on C is defined as the coequalizer The importance of this operad is evidenced by the following theorem. between the category of algebras over the colored operad P (r 1 ,r 2 ) C ∈ Op C 0 (M) from Definition 3.11 and the category of field theories of type P (r 1 ,r 2 ) on C from Definition 3.3.
Proof. Because P (r 1 ,r 2 ) C is defined as a coequalizer (3.20), its algebras are precisely those P Calgebras A ∈ Alg P C that satisfy the relations encoded by r 1,C , r 2,C : R ⊥ ⇒ U (P C ), cf. (3.19). Using the notations from the proof of Lemma 3.10, this concretely means that the diagram in M commutes, for all (f 1 : c 1 → t, f 2 : c 2 → t) ∈⊥. Using the isomorphism of Lemma 3.10, one easily translates this diagram to the diagram (3.4) for the functor A : C → Alg P corresponding to A ∈ Alg P C , which completes the proof.
Example 3.13. Recalling Examples 3.5, 3.7 and 3.8, our construction defines colored operads for quantum field theory As

Functoriality
Note that the field theory operad P (r 1 ,r 2 ) C ∈ Op C 0 (M) from Definition 3.11 depends on the choice of two kinds of data: (1) An orthogonal category C = (C, ⊥) and (2) a bipointed single-colored operad P (r 1 ,r 2 ) = (P, r 1 , r 2 : I[2] ⇒ U (P)). We will see that both of these dependencies are functorial. Recall from Definition 3.1 that orthogonal categories are the objects of the category OrthCat. The second kind of data may be arranged in terms of a category as follows.
Definition 3.14. The category of (arity 2) bipointed single-colored operads Op 2pt { * } (M) has the following objects and morphisms: An object is a pair P (r 1 ,r 2 ) = (P, r 1 , r 2 : I[2] ⇒ U (P)) consisting of a single-colored operad P ∈ Op { * } (M) and a parallel pair of Seq { * } (M)-morphisms r 1 , r 2 : I[2] ⇒ U (P) (cf.  Proof. For every morphism (F, φ) : (C, P (r 1 ,r 2 ) ) → (D, Q (s 1 ,s 2 ) ) in OrthCat × Op 2pt { * } (M) one can define an Op(M)-morphism φ F : P C → Q D between the corresponding auxiliary operads from Definition 3.9. Concretely, this morphism is specified by the components We now show that the assignment of the field theory operads is functorial too. For this we first note that one can define analogously to above a morphism R ⊥ C → R ⊥ D of colored sequences and one easily checks that this defines a morphism of parallel pairs in (3.19). (For this step one uses that F is an orthogonal functor and that φ preserves the points.) Because forming colimits is functorial, this defines an to the coequalizer of the corresponding pullback operads. (With an abuse of notation, we denoted by F both the free D 0 -colored operad functor (2.4) and the orthogonal functor F : C → D.) Notice that pullback operads arise at this point because Definition 3.11 considers colimits in the categories of operads with a fixed set of colors and not in the category Op(M). From the universal property of colimits one obtains a canonical ) to the pullback of field theory operad. The composition of the latter two morphisms defines our desired Op(M)-morphism, which we denote with abuse of notation by the same symbol φ F : P (r 1 ,r 2 ) C → Q (s 1 ,s 2 ) D as the one for the auxiliary operads.
As a consequence of this proposition, we obtain for every morphism (F, φ) : (C, P (r 1 ,r 2 ) ) → between the corresponding categories of field theories. From the concrete definition of φ F given in the proof of Proposition 3.15 and the identification in Theorem 3.12, one observes that the right adjoint (φ F ) * admits a very explicit description in terms of either of the two compositions in the commutative diagram In this diagram F * is the restriction to the categories of field theories of the pullback functor for functor categories for O = P and O = Q, and (φ * ) * is the restriction to the categories of field theories of the pushforward functor for functor categories for E = C and E = D, where φ * : Alg Q → Alg P is the pullback functor corresponding to the single-colored operad morphism φ : P → Q.

Universal constructions for field theories 4.1 Generalities
This section is concerned with analyzing in more depth the adjunctions in (3.25) and their relevance for universal constructions in field theory. Because of (3.26), this problem may be decomposed into three smaller building blocks: 3. the interplay between these two cases via the diagram of categories and functors in which the square formed by the right adjoints commutes by (3.26) and, as a consequence of the uniqueness (up to a unique natural isomorphism) of left adjoint functors, the square formed by the left adjoints commutes up to a unique natural isomorphism.
In the following subsections we study particular classes of examples of such adjunctions, all of which are motivated by concrete problems and constructions in field theory, and discuss their interplay. A particularly interesting example, which we will discuss later in Section 5, is given by an adjunction that describes the quantization of linear field theories. exhibits FT C, P (r 1 ,r 2 ) as a full coreflective subcategory of FT D, P (r 1 ,r 2 ) , i.e. the unit η : id → j * j ! of this adjunction is a natural isomorphism.

Full orthogonal subcategories
Proof. The proof is analogous to the corresponding one in [BSW17] and will not be repeated. (4.5) The right adjoint j * is the restriction functor which restricts field theories that are defined on all of Loc to the full orthogonal subcategory Loc ⋄ of spacetimes diffeomorphic to R m . More interestingly, the left adjoint j ! is a universal extension functor which extends field theories that are only defined on Loc ⋄ to all of Loc. It was shown in [BSW17] that j ! is a generalization and refinement of Fredenhagen's universal algebra construction [Fre90,Fre93,FRS92,Lan14]. A nontrivial application of a similar construction to quantum field theories on spacetimes with boundaries has been studied in [BDS18]. ▽ Remark 4.3. The result in Proposition 4.1 that j ! exhibits FT C, P (r 1 ,r 2 ) as a full coreflective subcategory of FT D, P (r 1 ,r 2 ) is crucial for a proper interpretation of j ! as a universal extension functor and j * as a restriction functor in the spirit of Example 4.2. Given any field theory B ∈ FT C, P (r 1 ,r 2 ) on the full orthogonal subcategory C ⊆ D, one may apply the left and then the right adjoint functor in (4.4) to obtain another field theory j * j ! (B) ∈ FT C, P (r 1 ,r 2 ) on C ⊆ D. The latter is interpreted as the restriction of the universal extension of B. By Proposition 4.1, the unit η B : B → j * j ! (B) defines an isomorphism between these two theories, which means that j ! extends field theories from C ⊆ D to all of D without altering their values on the subcategory C. ▽ An interesting application of the class of adjunctions in (4.4) is that they allow us to formalize a kind of local-to-global (i.e. descent) condition for field theories. Given a field theory A ∈ FT D, P (r 1 ,r 2 ) on the bigger category D, one may ask whether it is already completely determined by its values on the full orthogonal subcategory C ⊆ D. In the context of Example 4.2, the question is whether the value of a field theory on a general spacetime M ∈ Loc is already completely determined by its values on spacetimes diffeomorphic to R m , which is a typical question of descent. The following definition provides a formalization of this idea.
Definition 4.4. A field theory A ∈ FT D, P (r 1 ,r 2 ) on D is called j-local if the corresponding component of the counit ǫ A : j ! j * (A) → A is an isomorphism. The full subcategory of j-local field theories is denoted by FT D, P (r 1 ,r 2 ) j−loc ⊆ FT D, P (r 1 ,r 2 ) .
The following result, which extends eariler results from [BSW17] to our more general framework, shows that j-local field theories on the bigger category D may be equivalently described by field theories on the full orthogonal subcategory C ⊆ D.
Corollary 4.5. The adjunction (4.4) restricts to an adjoint equivalence (4.6) Proof. This is an immediate consequence of Proposition 4.1. exhibits FT C[W −1 ], P (r 1 ,r 2 ) as a full reflective subcategory of FT C, P (r 1 ,r 2 ) , i.e. the counit ǫ : L ! L * → id of this adjunction is a natural isomorphism.

Orthogonal localizations
Proof. The proof is analogous to the corresponding one in [BSW17] and will not be repeated. (4.8) The right adjoint L * is the functor that forgets that a field theory B ∈ FT Loc[W −1 ], P (r 1 ,r 2 ) satisfies the time-slice axiom. More interestingly, the left adjoint L ! assigns to a field theory A ∈ FT Loc, P (r 1 ,r 2 ) that does not necessarily satisfy the time-slice axiom a theory that does. Hence, one may call the left adjoint functor L ! a 'time-slicification' functor. Notice that the result in Proposition 4.6 that L * exhibits FT Loc[W −1 ], P (r 1 ,r 2 ) as a full reflective subcategory of FT Loc, P (r 1 ,r 2 ) has a concrete meaning. The isomorphisms ǫ B : L ! L * (B) → B given by the counit say that time-slicification does not alter those field theories that already do satisfy the time-slice axiom, which is of course a very reasonable property. ▽ An interesting application of the class of adjunctions in (4.7) is that they allow us to formulate a suitable criterion to test whether a theory A ∈ FT C, P (r 1 ,r 2 ) satisfies the 'time-slice axiom'.
Definition 4.8. A field theory A ∈ FT C, P (r 1 ,r 2 ) is called W -constant if the corresponding component of the unit η A : A → L * L ! (A) is an isomorphism. The full subcategory of W -constant field theories is denoted by FT C, P (r 1 ,r 2 ) W −const ⊆ FT C, P (r 1 ,r 2 ) .
The following result shows that W -constant field theories may be equivalently described by field theories on the orthogonal localization C[W −1 ].
Corollary 4.9. The adjunction (4.7) restricts to an adjoint equivalence Proof. This is an immediate consequence of Proposition 4.6.

Change of bipointed single-colored operad
Our third class of examples are adjunctions that correspond to morphisms φ : P (r 1 ,r 2 ) → Q (s 1 ,s 2 ) of bipointed single-colored operads, i.e. (4.10) Let us stress that these adjunctions are conceptually very different to the ones we studied in the previous two subsections because they change the type of field theories and not the orthogonal category on which field theories are defined. In particular, such adjunctions can not be formulated within the original operadic framework for algebraic quantum field theory developed in [BSW17] as they crucially rely on our more flexible definition of field theory operads, cf. Definition 3.11. In Section 5 we study an interesting example given by an adjunction that describes the quantization of linear field theories.
We observe the following preservation results for j-local field theories (cf. Definition 4.4) and for W -constant field theories (cf. Definition 4.8) under the adjunctions (4.10).
Proof. Item a): Let A ∈ FT D, P (r 1 ,r 2 ) j−loc be any j-local field theory of type P (r 1 ,r 2 ) , i.e.
is an isomorphism. This follows from the commutative diagram where isomorphisms are indicated by ∼ =. In more detail, the top square commutes by naturality of the counit and the vertical arrows are isomorphisms because A is j-local. The middle square commutes because of (4.3). The bottom triangle is the triangle identity for the adjunction and the unit (vertical arrow) is an isomorphism because of Proposition 4.1.

Item b): Let
A ∈ FT C, Q (s 1 ,s 2 ) W −const be any W -constant field theory of type Q (s 1 ,s 2 ) , i.e. η A : A → L * L ! (A) is an isomorphism. The claim is that the field theory (φ * ) * (A) ∈ FT C, P (r 1 ,r 2 ) of type P (r 1 ,r 2 ) is W -constant as well, i.e. η (φ * ) * (A) : is an isomorphism. This follows from the commutative diagram In more detail, the top square commutes by naturality of the unit and the vertical arrows are isomorphisms because A is W -constant. The middle square commutes because of (4.3). The bottom triangle is the triangle identity for the adjunction and the counit (vertical arrow) is an isomorphism because of Proposition 4.6.
Let us emphasize that the result in Proposition 4.10 is asymmetric in the sense that j-local field theories are preserved by the left adjoints (φ * ) ! and W -constant field theories are preserved by the right adjoints (φ * ) * . The reason for this asymmetry is that the former property is formalized by a coreflective full subcategory (cf. Proposition 4.1 and Definition 4.4) while the latter property by a reflective full subcategory (cf. Proposition 4.6 and Definition 4.8). The opposite preservation properties do not hold true in general because in (4.3) only the square formed by the left adjoints and the square formed by the right adjoints commutes (up to a natural isomorphism). We however would like to note the following special case in which there exists a further preservation result. This will become relevant in Section 5 below.  (s 1 ,s 2 ) ) is (naturally isomorphic to) the restriction to the categories of field theories of the pushforward functor for functor categories where the adjunction φ ! : Alg P ⇄ Alg Q : φ * corresponds to the single-colored operad morphism φ : P → Q. Then the left adjoint functor (φ * ) ! : FT(C, P (r 1 ,r 2 ) ) → FT(C, Q (s 1 ,s 2 ) ) preserves W -constant field theories.
Proof. Consider the diagram (4.3) for F = L and observe that under our hypothesis the square form by the (φ * ) ! and L * commutes (up to a natural isomorphism), i.e.
where in the second step we used that pullback and pushforward functors for functor categories commute. Replacing (φ * ) * by (φ * ) ! in the proof of Proposition 4.10 b) then proves our claim.
We conclude this section with a technical lemma that provides a criterion to detect whether the hypotheses of Proposition 4.11 are fulfilled. Recall from Definition 3.11 that there exists a natural projection Op C 0 (M)-morphism π : P C → P (r 1 ,r 2 ) C from our auxiliary operads to the field theory operads. Given any Op 2pt { * } (M)-morphism φ : P (r 1 ,r 2 ) → Q (s 1 ,s 2 ) , this yields the square of adjunctions in which the square formed by the right adjoints commutes, i.e. (φ * ) * π * = π * (φ * ) * , and hence the square formed by the left adjoints commutes (up to a unique natural isomorphism), i.e. (φ * ) ! π ! ∼ = π ! (φ ! ) * . Notice that the vertical adjunctions exhibit the field theory categories as full reflective subcategories of the functor categories. An immediate consequence is the following Lemma 4.12. If the functor (φ ! ) * π * : FT C, P (r 1 ,r 2 ) → Alg Q C factors through the full reflective subcategory FT C, Q (s 1 ,s 2 ) ⊆ Alg Q C , then the left adjoint (φ * ) ! : FT C, P (r 1 ,r 2 ) → FT C, Q (s 1 ,s 2 ) is (naturally isomorphic to) the restriction to the categories of field theories of the pushforward functor (φ ! ) * : Alg P C → Alg Q C .

Linear quantization adjunction
Throughout this section we assume that the underlying bicomplete closed symmetric monoidal category M is additive. Recalling Example 3.5 and also Remark 3.6, we define the category of quantum field theories on an orthogonal category C by which one easily confirms to be well-defined by using the relations of the associative operad (cf. Example 2.6) and the ones of the unital Lie operad (cf. Example 2.9). It is evident that φ : uLie ([·,·],0) → As ([·,·],0) defines an Op 2pt { * } (M)-morphism in the sense of Definition 3.14. By (4.10) this induces an adjunction between the category of linear field theories and the category of quantum field theories, which we shall denote by The aim of this section is to study this adjunction in detail and in particular to show that the left adjoint Q lin admits an interpretation as a linear quantization functor.
Let us first provide an explicit description of the right adjoint functor U lin = (φ * ) * . Note that the functor φ * : Alg As → Alg uLie from associative and unital algebras to unital Lie algebras is very explicit. It assigns to any (A, µ A , η A ) ∈ Alg As the unital Lie algebra φ * (A, µ A , η A ) = (A, µ A −µ op A , η A ) ∈ Alg uLie , where the Lie bracket is given by the commutator. The corresponding pushforward functor U lin = (φ * ) * : QFT(C) → LFT(C) carries out this construction object-wise on C. Concretely, for A : C → Alg As ∈ QFT(C), the functor underlying U lin (A) ∈ LFT(C) is given by U lin (A)(c) = φ * A(c) ∈ Alg uLie , for all c ∈ C.
We now provide an explicit description of the left adjoint functor Q lin in (5.4). Our strategy is to analyze the pushforward functor (φ ! ) * : Alg uLie C → Alg As C for the functor categories and to prove that it satisfies the criterion of Lemma 4.12. As a consequence of this lemma, the restriction to the categories of field theories of the pushforward functor (φ ! ) * defines a model for the left adjoint functor Q lin .
Let us describe first the left adjoint functor of the adjunction φ ! : Alg uLie ⇄ Alg As : φ * between algebras over single-colored operads. The following construction, which we will call the unital universal enveloping algebra construction, defines a model for the left adjoint φ ! . Let V ∈ Alg uLie be any unital Lie algebra, with Lie bracket [·, ·] : V ⊗ V → V and unit η : I → V . As the first step, we form the usual tensor algebra T ⊗ V := ∞ n=0 V ⊗n ∈ Alg As , i.e. the free As-algebra of the underlying object V ∈ M, with multiplication µ ⊗ : T ⊗ V ⊗ T ⊗ V → T ⊗ V and unit η ⊗ : I → T ⊗ V . We then consider the two parallel M-morphisms where ι 1 : V → T ⊗ V is the inclusion into the coproduct, which compare the commutator of T ⊗ V with the Lie bracket of V . We form the corresponding coequalizer in Alg As and notice that U ⊗ V is the universal enveloping algebra of the underlying Lie algebra (V, [·, ·]) ∈ Alg Lie . As the final step, we consider the two parallel M-morphisms which compare the unit of V with the unit of T ⊗ V , and form the corresponding coequalizer in Alg As . All of these constructions are clearly functorial.
Proof. It is easy to construct a natural bijection Hom Alg As (φ ! (V ), A) ∼ = Hom Alg uLie (V, φ * (A)), for all V ∈ Alg uLie and A ∈ Alg As . Concretely, given κ : φ ! (V ) → A in Alg As , then κ π ′ π ι 1 : V → φ * (A) defines an Alg uLie -morphism. On the other hand, given ρ : V → φ * (A) in Alg uLie , then the canonical extension to an Alg As -morphism ρ : T ⊗ V → A on the tensor algebra descends to the quotients in (5.5) and (5.6). Proof. By hypothesis, given any orthogonal pair (f 1 : As is the zero map too. This is an immediate consequence of our definition of the unital universal enveloping algebra (cf. (5.5) and (5.6)) and the fact that the commutator bracket satisfies the Leibniz rule in both entries. (Hint: The latter property is used to expand the commutator of polynomials to a sum of terms containing as a factor the commutator of generators, which is identified via (5.5) with the Lie bracket.) As a consequence of Lemma 4.12, we obtain PSymp → Alg uLie , cf. Example 3.8. It is easy to check that the composition φ ! H : PSymp → Alg As of the Heisenberg Lie algebra functor and the unital universal enveloping algebra functor is naturally isomorphic to the usual (polynomial) CCR-algebra functor CCR : PSymp → Alg As that is used in the quantization of linear field theories, cf. [BGP07,BDH13]. In particular, we obtain a natural isomorphism Q lin H L ∼ = CCR L : C → Alg As , which means that our quantization prescription via Q lin is in this case equivalent to the ordinary CCR-algebra quantization of linear field theories. ▽ We would like to emphasize that our linear quantization functor preserves both j-locality and W -constancy, i.e. it preserves descent and the time-slice axiom of field theories.

Towards derived quantization of linear gauge theories
The techniques we developed in this paper can be refined to the case where M is a suitable symmetric monoidal model category. Let us recall that a model category is a category that comes equipped with three distinguished classes of morphisms -called weak equivalences, fibrations and cofibrations -that satisfy a list of axioms going back to Quillen, see e.g. [DS95] for a concise introduction. The main role is played by the weak equivalences, which introduce a consistent concept of "two things being the same" that is weaker than the usual concept of categorical isomorphism. For example, the category M = Ch(K) of (possibly unbounded) chain complexes of vector spaces over a field K may be endowed with a symmetric monoidal model category structure in which the weak equivalences are quasi-isomorphisms, see e.g. [Hov99].
Model category theory plays an important role in the mathematical formulation of (quantum) gauge theories. In particular, the 'spaces' of fields in a gauge theory are actually higher spaces called stacks, which may be formalized within model category theory. We refer to e.g. [Sch13] for the general framework and also to [BSS18] for the example of Yang-Mills theory. Consequently, the observable 'algebras' in a quantum gauge theory are actually higher algebras, e.g. the differential graded algebras arising in the BRST/BV formalism. We refer to [Hol08,FR12,FR13] for concrete constructions within the BRST/BV formalism in algebraic quantum field theory and also to [BSW18] for the relevant model categorical perspective.
The aim of this last section is to refine the linear quantization adjunction from Section 5 to the framework of model category theory. In particular, we will construct a derived linear quantization functor, which is an essential ingredient to quantize linear gauge theories to quantum gauge theories in a way that is consistent with the concept of weak equivalences. In order to simplify our presentation, we restrict ourselves to the case where M = Ch(K) is the symmetric monoidal model category of chain complexes of vector spaces over a field K of characteristic zero, e.g. K = C. In this section we shall freely use terminology and results from general model category theory [DS95,Hov99] and more specifically the model structures for colored operads and their algebras [Hin97,Hin15]. We refer to [BSW18] for a more gentle presentation of how these techniques can be applied to Ch(K)-valued algebraic quantum field theory.
Our first (immediate) result is that the categories FT(C, P (r 1 ,r 2 ) ) of field theories with values in M = Ch(K) from Definition 3.3 are model categories, i.e. there exists a consistent concept of weak equivalences for Ch(K)-valued field theories. Furthermore, the adjunctions in (3.25) are compatible with these model category structures in the sense that they are Quillen adjunctions.
Proof. This is a consequence of Theorem 3.12 and Hinich's results [Hin97,Hin15], which show that all colored operads in Ch(K) are admissible for K a field a characteristic zero.
As a specific instance of these general results, we obtain that both the category of Ch(K)valued linear field theories LFT(C) and the category of Ch(K)-valued quantum field theories QFT(C) carry a canonical model structure. Moreover, the linear quantization adjunction (5.4) is a Quillen adjunction. Using the general technique of derived functors (see e.g. [DS95,Hov99]), this means that we can construct a left derived linear quantization functor LQ lin and a right derivation of its right adjoint RU lin . Because all objects in the model categories of field theories from Proposition 6.1 are fibrant objects, there is no need to introduce a fibrant replacement functor and we can simply define the right derived functor by By Ken Brown's lemma, it follows that this functor preserves weak equivalences. It is important to stress that it is the derived functor LQ lin and not the non-derived functor Q lin which provides a meaningful quantization prescription for linear gauge theories that is consistent with the crucial concept of weak equivalences.
Remark 6.3. Regarding applications to the construction of quantum gauge theories, we would like to note that the derived linear quantization functor LQ lin always exists, because cofibrant replacements always exist in a model category. However, finding explicit models to compute cofibrant replacements is generically quite hard, which means that making these constructions practically applicable requires further work. We hope to come back to this issue in another paper, where we also plan to discuss concrete examples of linear quantum gauge theories. ▽ We would like to conclude this section by presenting a first attempt towards a natural homotopical generalization of the j-locality property (cf. Definition 4.4) and the W -constancy property (cf. Definition 4.8) in the context of model category theory. To motivate the definitions below, let us recall that given a Quillen adjunction F ⊣ G between model categories, the ordinary unit η : id → G F and counit ǫ : F G → id are built from the non-derived functors and hence they are in general not homotopically meaningful. As a consequence, our previous concepts of j-locality and W -constancy are in general not preserved under weak equivalences. To address this issue, we will consider the derived functors LF = F Q and RG = G R and formalize j-locality and W -constancy in terms of the derived unit and counit of the Quillen adjunction where q : Q → id (respectively r : id → R) is the natural weak equivalence corresponding to the cofibrant replacement functor Q (respectively the fibrant replacement functor R). Because all objects are fibrant objects in the model categories of field theories from Proposition 6.1, we can choose in what follows R = id.
We focus first on a homotopical generalization of the j-locality property from Definition 4.4.
Definition 6.4. Let j : C → D be a full orthogonal subcategory and P (r 1 ,r 2 ) any bipointed singlecolored operad. A field theory A ∈ FT D, P (r 1 ,r 2 ) is called homotopy j-local if the corresponding component of the derived counit weak equivalences and the commutative diagram The vertical arrows in the top square are weak equivalences because A is by hypothesis homotopy j-local. The vertical arrows in the two other squares are weak equivalences because left Quillen functors preserve cofibrant objects and weak equivalences between cofibrant objects. The bottom horizontal arrow is a weak equivalence because of Lemma 6.5. Finally, the natural isomorphism in the underbraces is due to (4.3).
Corollary 6.7. The derived linear quantization functor (6.2) maps homotopy j-local linear field theories to homotopy j-local quantum field theories.
We propose a homotopical generalization of the W -constancy property from Definition 4.8.
Definition 6.8. Let L : C → C[W −1 ] be an orthogonal localization and P (r 1 ,r 2 ) any bipointed single-colored operad. A field theory A ∈ FT C, P (r 1 ,r 2 ) is called homotopy W -constant if the corresponding component of the derived unit is a weak equivalence in FT C, P (r 1 ,r 2 ) .
It turns out that analyzing the homotopy W -constancy property is in general more subtle than our study of the homotopy j-locality property above. The reason for this is the following observation, which implies that L ! ⊣ L * is only under certain conditions a Quillen reflection. Proof. This follows from (6.3), the 2-of-3 property of weak equivalences and Proposition 4.6.
In order to prove some desirable properties of homotopy W -constant field theories, we have to introduce extra assumptions on the orthogonal localization functor L. It is an interesting and relevant question whether these assumptions are satisfied for the orthogonal localization L : Loc → Loc[W −1 ] appearing in locally covariant field theory, cf. Example 3.5. To answer this question, one has to perform an explicit study of cofibrant objects in the model categories of field theories, which is very technical and beyond the scope of this work.
Assumption A. The derived counit of the Quillen adjunction L ! ⊣ L * is a natural weak equivalence, i.e. L ! ⊣ L * is a Quillen reflection.
Assumption B. The functor L * L ! Q maps to the full subcategory of cofibrant objects.
Under Assumption A, it is easy to prove that the (derived) right adjoint functor RL * = L * constructs examples of homotopy W -constant field theories.
Proof. This follows from the commutative diagram where for the right vertical arrows we used Lemma 6.9 and Proposition 4.6.
Under both Assumptions A and B, we can prove that Proposition 4.11 generalizes to our model categorical setting.
Proof. Let A ∈ FT(C, P (r 1 ,r 2 ) ) be any homotopy W -constant field theory. We have to prove that the derived unit η (φ * ) ! Q(A) : Q (φ * ) ! Q(A) → L * L ! Q (φ * ) ! Q(A) corresponding to the field theory (φ * ) ! Q(A) ∈ FT(C, Q (s 1 ,s 2 ) ) is a weak equivalence. This follows from the 2-of-3 property of weak equivalences and the commutative diagram The vertical arrows in the top square are weak equivalences because (φ * ) ! is a left Quillen functor and η A is a weak equivalence between cofibrant objects by hypothesis and Assumption B. The vertical arrows in the bottom square are the isomorphisms explained in the proof of Proposition 4.11. Finally, the bottom horizontal arrow is a weak equivalence because of Lemma 6.10, which uses Assumption A.
Corollary 6.12. Suppose that both Assumptions A and B hold. Then the derived linear quantization functor (6.2) maps homotopy W -constant linear field theories to homotopy W -constant quantum field theories.