Relative Cauchy evolution for linear homotopy AQFTs

This paper develops a concept of relative Cauchy evolution for the class of homotopy algebraic quantum field theories (AQFTs) that are obtained by canonical commutation relation quantization of Poisson chain complexes. The key element of the construction is a rectification theorem proving that the homotopy time-slice axiom, which is a higher categorical relaxation of the time-slice axiom of AQFT, can be strictified for theories in this class. The general concept is illustrated through a detailed study of the relative Cauchy evolution for the homotopy AQFT associated with linear Yang-Mills theory, for which the usual stress-energy tensor is recovered.


Introduction and summary
An algebraic quantum field theory (AQFT) on Lorentzian manifolds can be described by a functor from a category of spacetimes to a category of algebras, so that A(M ) is the algebra of observables assigned to the spacetime M by the theory. 1 An important concept in this framework is the relative Cauchy evolution (RCE) [BFV03,FV15] that allows one to study the response of a theory A to perturbations of the spacetime geometry. It is described by a coherent family of automorphisms RCE (M,h) : A(M ) → A(M ) of the algebras of quantum observables that is labeled by pairs (M, h) consisting of a spacetime M and a sufficiently small compactly supported metric perturbation h on M . The first derivatives d dǫ RCE (M,ǫh) ǫ=0 of the family of RCE automorphisms along the metric perturbation h determine the quantum stress-energy tensor of the theory A.
In the usual construction of the RCE automorphisms, see e.g. [BFV03,FV15], it is crucial that the theory A satisfies the time-slice axiom. This axiom requires that A assigns to every Cauchy morphism f : M → N (a morphism of spacetimes whose image f (M ) ⊆ N contains a Cauchy surface of N ) an isomorphism A(f ) : A(M ) → A(N ) of algebras. These isomorphisms introduce a concept of time evolution for the quantum observables, which is the key ingredient to define the RCE automorphisms. See Section 2 for more details and a concise review of this construction.
The main goal of this paper is to develop a generalization of the concept of relative Cauchy evolution to homotopy AQFTs [BSW19], which are higher categorical refinements of AQFTs that are relevant to describe quantum gauge theories. Examples of homotopy AQFTs arise from the BRST/BV formalism [FR12,FR13] and its mathematical incarnation in terms of derived geometry [BBS19]. See also [CG17] for similar developments in the context of factorization algebras.
The key difference between homotopy AQFTs and ordinary ones is that they assign differential graded algebras (in short, dg-algebras) of quantum observables which, in the terminology of the BRST/BV formalism, contain also observables for the ghost fields and the antifields of a gauge theory. As evidenced by the concrete examples studied in [FR12,FR13] and [BBS19], physically relevant models of homotopy AQFT usually satisfy only a higher categorical relaxation of the time-slice axiom, which was called the homotopy time-slice axiom. This axiom demands that the homotopy AQFT A assigns to every Cauchy morphism f : M → N a weak equivalence A(f ) : A(M ) → A(N ) of dg-algebras, i.e. a morphism of dg-algebras whose underlying chain map is a quasi-isomorphism (an isomorphism at the level of homologies). In contrast to the isomorphisms in ordinary AQFT, such weak equivalences do not admit (strict) inverses and hence the usual construction of the RCE can not be applied directly to homotopy AQFTs.
Our proposal to remedy this issue is to seek a rectification theorem for the homotopy time-slice axiom of homotopy AQFTs. Loosely speaking, this consists of a replacement of a given homotopy AQFT A that satisfies the homotopy time-slice axiom by a weakly equivalent homotopy AQFT A st that satisfies the strict time-slice axiom and hence admits the usual RCE automorphisms. The technical details of this proposal are explained in Section 3. The main result is Theorem 5.4 which proves that, when restricted to a category that is relevant for the relative Cauchy evolution associated with a fixed but arbitrary pair (M, h), every linear homotopy AQFT satisfying the homotopy time-slice axiom admits such a strictification. Let us recall that a linear homotopy AQFT is a theory A = CCR(L, τ ) that is obtained through canonical commutation relation (CCR) quantization of Poisson chain complexes. In physics terminology, these are non-interacting (i.e. "free") quantum gauge theories, such as e.g. the linear Yang-Mills model studied in [BBS19]. Our construction of the weakly equivalent strictified theory A st is rather explicit as it is determined by CCR-quantization of (a functor of) Poisson chain complexes (LL ! (L), τ L ) that we describe in detail in Sections 4 and 5. The relative Cauchy evolution for the weakly equivalent strictified theory A st takes a very simple form, given explicitly in (5.20) and (4.23).
Even though our concept of relative Cauchy evolution that is obtained by the Rectification Theorem 5.4 is mathematically sound and relatively simple to describe, its physical interpretation is a priori less clear. We shall address this issue in Section 6 by studying as a concrete example the linear Yang-Mills model from [BBS19]. The key result is Proposition 6.1 which proves that, at the level of linear quantum fields, our abstract concept of relative Cauchy evolution through rectification admits an equivalent, but physically more transparent and familiar, description in terms of a generalization of the usual RCE automorphism construction obtained by choosing quasi-inverses for the quasi-isomorphisms assigned to Cauchy morphisms. This equivalent model can be worked out in detail, see in particular Proposition 6.6 and Remark 6.7, leading to explicit formulas for the relative Cauchy evolution of linear Yang-Mills theory that generalize earlier nonhomotopical results in [FL16]. We shall further compute the associated stress-energy tensor and find that, up to exact terms that can be removed by a homotopy, it only receives contributions from the gauge field and not from the ghosts and antifields. More precisely, the stress-energy tensor that is obtained via our homotopical concept of relative Cauchy evolution agrees with the one of the standard Maxwell action S = − M 1 2 F ∧ * F . This is a very pleasing result because it substantiates mathematically the physical expectation that ghosts and antifields do not contribute to the energy content of a theory.
The outline of the remainder of this paper is as follows: In Section 2 we shall briefly recall the concept of relative Cauchy evolution in ordinary AQFT [BFV03,FV15] and provide a reformulation in terms of localization of categories that will be useful for generalizing to homotopy AQFT. Section 3 outlines our proposal for how to obtain a well-defined concept of relative Cauchy evolution for homotopy AQFTs satisfying the homotopy time-slice axiom. The key idea of our approach is to seek a rectification theorem that allows us to strictify the homotopy time-slice axiom and thereby make available the usual RCE automorphism construction in the context of homotopy AQFT. Restricting to linear homotopy AQFTs [BBS19] and the category (2.7) that is relevant for the relative Cauchy evolution associated with a fixed but arbitrary pair (M, h), we can successfully prove such a rectification theorem, see Theorem 5.4. The formulation and proof of this theorem is slightly abstract as it requires techniques from model category theory [Hov99,DHKS04,Rie14], in particular derived functors and their concrete models obtained via bar resolutions [Fre09]. These more technical aspects are discussed and worked out in detail in Sections 4 and 5. In Section 6 we apply our novel concept of relative Cauchy evolution for homotopy AQFTs to the linear Yang-Mills model from [BBS19]. We obtain explicit formulas involving Green operators, see in particular Proposition 6.6 and Remark 6.7, which generalize the earlier non-homotopical results in [FL16]. We also compute explicitly the stress-energy tensor for this example. This paper contains three appendices: Appendix A summarizes our conventions for bicomplexes, Appendix B recalls the bar construction, and Appendix C lists explicitly the homotopy coherence data that is needed to construct the Poisson structure in Section 5.

Relative Cauchy evolution and localization of categories
In this section we shall briefly review the concept of relative Cauchy evolution (RCE) for ordinary AQFTs [BFV03,FV15] from a perspective that will be useful for our generalization to homotopy AQFTs. We adopt the operadic formulation of AQFT [BSW21,BS19a] that automatically incorporates Einstein causality as an intrinsic part of the structure. Let us recall that an orthogonal category is a pair C := (C, ⊥) consisting of a category C and a subset ⊥ ⊆ Mor(C) t × t Mor(C) expressing which pairs of morphisms to a common target are considered to be orthogonal. A central example for AQFT is Loc := (Loc, ⊥ Loc ), where Loc is the usual category of all oriented and time-oriented globally hyperbolic Lorentzian manifolds M and morphisms f : M → N given by orientation and time-orientation preserving isometric embeddings with open and causally convex image. The orthogonality relation ⊥ Loc is determined by causal disjointness, i.e. (f 1 : M 1 → N ) ⊥ Loc (f 2 : M 2 → N ) if and only if the images f 1 (M 1 ) and f 2 (M 2 ) are causally disjoint open subsets of N . Associated to every orthogonal category C is a category AQFT(C) of AQFTs on C with values in the symmetric monoidal category Vec K of vector spaces over a (fixed) field K of characteristic 0. More precisely, the objects of AQFT(C) are given by Vec K -valued algebras over the AQFT operad O C associated with the orthogonal category C = (C, ⊥). By [BSW21, Theorem 3.16], each such object admits an equivalent description as a functor A : C → Alg K to the category of associative and unital K-algebras satisfying an abstract version of the Einstein causality axiom encoded by the orthogonality relation ⊥ on C. For Loc, the category AQFT(Loc) describes AQFTs on Loc in the sense of [BFV03], which satisfy the usual Einstein causality axiom but not necessarily the time-slice axiom.
Turning to the time-slice axiom, let us recall that a Loc-morphism f : M → N is called a Cauchy morphism if its image f (M ) ⊆ N contains a Cauchy surface of N . We denote by W ⊆ Mor(Loc) the subset of all Cauchy morphisms in Loc. There are two equivalent ways to implement the time-slice axiom. On the one hand, one can consider the full subcategory AQFT(Loc) W ⊆ AQFT(Loc) consisting of all AQFTs A ∈ AQFT(Loc) that assign to every Cauchy morphism (f : By construction, the localization functor L defines an orthogonal functor L : Loc → Loc[W −1 ], i.e. a functor that preserves the orthogonality relations. With this notation established, the time-slice axiom is automatically implemented in every theory belonging to the category AQFT(Loc[W −1 ]). These two perspectives are equivalent due to the following result, which was proven in [ The concept of relative Cauchy evolution is defined for each AQFT A that satisfies the timeslice axiom. It records the response of A to perturbations of the spacetime metric and thereby encodes information about the stress-energy tensor of the theory [BFV03,FV15]. In more detail, given a spacetime M ∈ Loc with metric denoted by g and a sufficiently small compactly supported metric perturbation h such that the spacetime M h with the perturbed metric g + h is also an object in Loc, one obtains a diagram in the category Loc. In this diagram M ± := M \ J ∓ M (supp h) ∈ Loc is the spacetime M with the causal past/future of the support of the perturbation h removed and i ± , j ± are the canonical inclusion morphisms. It is important to observe that all morphisms in the diagram (2.3) are Cauchy morphisms, but (except in the trivial case h = 0) not isomorphisms. Given now any A ∈ AQFT(Loc) W that satisfies the time-slice axiom, its application to the morphisms in the diagram (2.3) gives isomorphisms and one defines the relative Cauchy evolution associated with the pair (M, h) as the automorphism . Given now any B ∈ AQFT(Loc[W −1 ]), the relative Cauchy evolution is defined by applying B to the automorphism (2.5), i.e.
It is clear that the two formulations of the relative Cauchy evolution given in (2.4) and (2.6) agree when A ∈ AQFT(Loc) W and B ∈ AQFT(Loc[W −1 ]) define the same theory, i.e. A = L * B with L * the right adjoint in Proposition 2.1.
Summing up, for theories defined on the localized orthogonal category Loc[W −1 ], the relative Cauchy evolution is simply given by the application of a theory B ∈ AQFT(Loc[W −1 ]) to certain automorphisms (2.5) in Loc[W −1 ]. The compatibility conditions between RCE automorphisms and Loc-morphisms listed in [FV12] are then directly encoded in the localized orthogonal category We may pursue the reformulation a bit further by considering the left adjoint L ! to the pullback L * in more detail. Instead of working on all of Loc, we consider only the subcategory that is relevant for determining the relative Cauchy evolution induced by a pair (M, h). Because C does not contain any causally disjoint pairs of morphisms, i.e. the restriction of ⊥ Loc to C is empty, the category AQFT(C) ≃ Fun(C, Alg K ) of AQFTs on C is simply the category of all functors from C to the category of associative and unital K-algebras Alg K . Furthermore, because all morphisms in C are Cauchy morphisms, the analog of the orthogonal localization Loc[W −1 ] in the present context is given by the localization C[All −1 ] at all C-morphisms, which also has an empty orthogonality relation.
The localization C[All −1 ] can be described very explicitly. Let us denote by BZ the category with a single object * and morphisms given by the group Z of integers. It is useful to note that any functor B : BZ → D to any category D can be described equivalently as an object B( * ) ∈ D, which we shall often denote simply as B = B( * ), equipped with a Z-action Z ∋ k → B(k) ∈ Aut(B( * )). Now consider the functor (2.8) Lemma 2.2. The functor (2.8) is a localization of the category C given in (2.7) at all morphisms.
Proof. We have to confirm that the functor L : C → BZ satisfies the properties of a localization functor, see e.g. [KS06, Section 7.1]. Because BZ is a groupoid, it is clear that L sends every C-morphism to an isomorphism. Let now F : C → D be any functor to a category D that sends every C-morphism to an isomorphism in D. We have to construct a functor F : BZ → D and a natural isomorphism F L ∼ = F . Let us define F : BZ → D by F ( * ) := F (M ) and, for n ∈ Z, . Then a natural isomorphism η : F → F L can be defined by the components It remains to prove that the pullback functor In the present context, the adjunction from Proposition 2.1 reduces to the adjunction where the right adjoint L * is the pullback functor along L : C → BZ and the left adjoint L ! is the left Kan extension along L : C → BZ. This adjunction restricts to an adjoint equivalence between Fun(BZ, Alg K ) and the full subcategory Fun(C, Alg K ) All ⊆ Fun(C, Alg K ) of all functors that assign to every C-morphism an isomorphism. The left Kan extension L ! (A) : BZ → Alg K of any functor A : C → Alg K can be computed as the colimit where L/ * denotes the comma category of L : C → BZ over the single object * ∈ BZ and π : L/ * → C is the forgetful functor. More explicitly, the objects of the category L/ * are pairs (n, N ) ∈ Z × C consisting of an object N ∈ C and a morphism n : L(N ) = * → * in BZ, and the morphisms (n, N ) → (n ′ , N ′ ) are C-morphisms f : N → N ′ such that n = L(f ) + n ′ . Furthermore, the forgetful functor π : L/ * → C , (n, N ) → N forgets the integers. Observe that the functor π : L/ * → C may be visualized as a kind of "universal cover" over the category C given in (2.7), which takes the form of a spiral of zig-zags lying over C. Straightening this spiral to a line, L/ * may be displayed as The Z-action on L ! (A) (2.12) is defined through the universal property of colimits by for all k ∈ Z and (n, N ) ∈ L/ * , where the ι's denote the canonical morphisms into the colimit. Note that this may be visualized by moving k ∈ Z levels up in the spiral (2.13). This completes the description of L ! on objects. If ζ : A → A ′ is a morphism in Fun(C, Alg K ) (i.e., a natural transformation) then L ! (ζ) : L ! (A) → L ! (A ′ ) is a natural transformation with the single component L ! (ζ) * determined by functoriality of the colimit (2.12). Concretely, are the canonical morphisms for A ′ . Given now any A ∈ Fun(C, Alg K ) All that satisfies the time-slice axiom, we know that (2.12) provides an equivalent description of this theory because restricting (2.11) to Fun(C, Alg K ) All is an adjoint equivalence. From this perspective, the relative Cauchy evolution is given (applying the analogs of (2.6) and (2.5) to B = L ! (A) and using the definition (2.8)) as the action of the generator 1 ∈ Z, i.e.
(2.15) which may be visualized by moving 1 level up in the spiral (2.13). The concept of relative Cauchy evolution for homotopy AQFTs that we develop in this paper is based on a higher categorical generalization of this particular perspective. To conclude, let us briefly note that (2.15) can also be related more directly to the ordinary relative Cauchy evolution in (2.4). Take any object of L/ * , for instance (0, M ) ∈ L/ * at level 0. Then the canonical morphism ι (0,M ) : A(M ) → L ! (A) is an isomorphism which intertwines the two descriptions of the relative Cauchy evolution given in (2.4) and (2.15), i.e. the diagram commutes.

Relative Cauchy evolution for homotopy AQFTs
The aim of this section is to propose a general strategy for how the relative Cauchy evolution may be defined for homotopy AQFTs in the sense of [BSW19]. The latter are higher categorical refinements of traditional AQFTs that are designed to capture the higher categorical structures of gauge theories. Models constructed via the BRST/BV formalism [FR12,FR13] define examples of homotopy AQFTs on Loc. A very special class of such examples, the linear quantum gauge theories, admit a particularly elegant and rigorous construction as homotopy AQFTs using suitable techniques from homotopical algebra and derived geometry, see [BBS19] and also [BS19b]. Let us also refer the reader to [BS19a] for a less technical introduction to this subject.
We start by considering the relative Cauchy evolution induced by a single pair (M, h) which, as we have explained in the previous section, is controlled by the subcategory C ⊆ Loc displayed in (2.7). Because this particular C has an empty orthogonality relation, the category of homotopy AQFTs on C = (C, ∅) admits a simple description in terms of functors from C to the category dgAlg K of associative and unital dg-algebras over K.
Under this equivalence, the model category structure on AQFT ∞ (C) from [BSW19] gets identified with the projective model structure on the functor category. More explicitly, a morphism ζ : A → A ′ in Fun(C, dgAlg K ), i.e. a natural transformation, is a weak equivalence (respectively, a fibration) iff each component ζ N : is a quasi-isomorphism between the underlying chain complexes (respectively, a degree-wise surjection). The cofibrations are determined by the lifting properties in model categories, see e.g. [Hov99]; they are the morphisms with the left lifting property against all weak equivalences that are also fibrations.
As evidenced by the concrete examples constructed in [BBS19] and also in [FR12,FR13], homotopy AQFTs satisfy a priori only a higher categorical relaxation of the time-slice axiom, which was called the homotopy time-slice axiom. In the present context (3.1), this means that the functor A : C → dgAlg K assigns to every (necessarily Cauchy) morphism f : N → N ′ in the category C given in (2.7) a weak equivalence A(f ) : A(N ) → A(N ′ ) in the model category dgAlg K , i.e. a dg-algebra morphism whose underlying chain map is a quasi-isomorphism. We denote by Fun(C, dgAlg K ) hoAll ⊆ Fun(C, dgAlg K ) the full subcategory of such functors. (One could also introduce as in [Car21] a Bousfield localized model structure on Fun(C, dgAlg K ) in order to capture the homotopy time-slice axiom. This provides in general a more refined structure, which however is not necessary for the purpose of the present paper.) Given any homotopy AQFT A ∈ Fun(C, dgAlg K ) hoAll satisfying the homotopy time-slice axiom, the traditional construction of the RCE automorphism in (2.4) is obstructed by the fact that weak equivalences A(f ) : A(N ) → A(N ′ ) between dg-algebras do not in general admit strict inverses. While they do admit quasi-inverses in the form of A ∞ -quasi-isomorphisms [LV12], these have the disadvantage of being difficult to work with.
Our strategy is therefore to adapt the third of the equivalent perspectives on the ordinary relative Cauchy evolution from Section 2 to the setting of homotopy AQFTs. The analog of the adjunction (2.11) for homotopy AQFTs on C is given by the Quillen adjunction where L * is the pullback functor and L ! the left Kan extension along the localization functor L : C → BZ. Suppose for the moment that we could prove that the restriction of the derived unit to Fun(C, dgAlg K ) hoAll ⊆ Fun(C, dgAlg K ) is a natural weak equivalence. Then each homotopy AQFT A ∈ Fun(C, dgAlg K ) hoAll satisfying the homotopy time-slice axiom is weakly equivalent to the object L * LL ! (A) ∈ Fun(C, dgAlg K ), where LL ! denotes the left derived functor. Observe that the theory L * LL ! (A) satisfies the strict time-slice axiom because it lies in the image of the pullback functor L * , and hence its relative Cauchy evolution is simply given in complete analogy to (2.15) by the action LL ! (A)(1) : LL ! (A) → LL ! (A) of the generator 1 ∈ Z. In particular, this is a strict Z-action in terms of strict dg-algebra automorphisms, in contrast to the homotopy coherent Z-action in terms of A ∞ -quasi-automorphisms that would arise by generalizing directly the traditional construction (2.4) to homotopy AQFTs. In other words, proving that the derived unit of the Quillen adjunction (3.2) restricts on Fun(C, dgAlg K ) hoAll ⊆ Fun(C, dgAlg K ) to a natural weak equivalence would provide a rectification theorem that allows us to strictify the homotopy time-slice axiom.
Proving such a result, and thereby obtaining the simple description of the relative Cauchy evolution for homotopy AQFTs outlined above, is a highly technical task that we currently do not know how to complete in full generality. In this work we present a concrete solution for a simplified version of the problem that allows us to discuss the relative Cauchy evolution in the context of linear homotopy AQFTs [BBS19]. The latter are functors A : C → dgAlg K that are given by a composition of the canonical commutation relation (CCR) quantization functor CCR (in the context of chain complexes [BBS19]) and a functor (L, τ ) that assigns chain complexes of linear observables together with their Poisson structure. More precisely, the category PoCh K of Poisson chain complexes is defined as follows: Objects are pairs (V, τ ) consisting of a chain complex V ∈ Ch K and a chain map τ : V ∧ V → K from the antisymmetrized tensor product, while morphisms Our strategy is to prove an analog of the desired rectification theorem for linear observables, which leads to a very explicit and strict model for the relative Cauchy evolution for L. This will be achieved in Section 4. Then, in Section 5, we will endow this strictified model with a suitable Poisson structure and lift our construction along the CCR-functor to obtain a relative Cauchy evolution for the linear homotopy AQFT A = CCR(L, τ ). We emphasize that this relative Cauchy evolution can be computed explicitly as we will do in Section 6 for linear Yang-Mills theories.
To conclude this section, we would like to mention briefly that the main idea behind our approach admits a potential generalization to describe the whole coherent family {RCE (M,h) } of RCE automorphisms in the context of homotopy AQFT, in contrast to only a single RCE automorphism RCE (M,h) . For this we consider the Quillen adjunction BS19b], which relates homotopy AQFTs on Loc to homotopy AQFTs on the orthogonal localization Loc[W −1 ] at all Cauchy morphisms. If one could prove that the derived unit of (3.4) restricts on AQFT ∞ (Loc) hoW ⊆ AQFT ∞ (Loc) to a natural weak equivalence, then L * LL ! (A) would provide a strictified model for the whole coherent family {RCE (M,h) } of RCE automorphisms for a theory A ∈ AQFT ∞ (Loc) hoW .

Rectification theorem for linear observables
In this section we shall prove a rectification theorem for linear observables. The latter are modeled by functors L : C → Ch K from the category C (2.7) to the model category of (possibly unbounded) chain complexes of vector spaces over K. In analogy to (3.2), the localization functor (2.8) defines a Quillen adjunction where both sides are endowed with the projective model structure. Again, L * is simply the pullback functor and L ! the left Kan extension along the localization functor L : C → BZ. The analog of the homotopy time-slice axiom for functors L : C → Ch K assigning chain complexes of linear observables is given by the property that L assigns to every morphism in C a weak equivalence (i.e. a quasi-isomorphism) in Ch K . We denote by Fun(C, Ch K ) hoAll ⊆ Fun(C, Ch K ) the full subcategory of such functors. The rectification problem for linear observables on C is then given by the question of whether the derived unit of (4.1) restricts on Fun(C, Ch K ) hoAll ⊆ Fun(C, Ch K ) to a natural weak equivalence. In order to prove our rectification theorem, we require explicit models for the derived functors associated with the Quillen adjunction (4.1) and also for their derived unit and counit. Recall that the left and right derived functors, in the context of a Quillen adjunction between model categories, are compositions LL ! = L ! Q, RL * = L * R where Q and R are cofibrant and fibrant replacement functors on Fun(C, Ch K ) and Fun(BZ, Ch K ) respectively. Because each object in Fun(BZ, Ch K ) is fibrant in the projective model structure, we may take R = id, so the right derived functor is the ordinary right adjoint, RL * := L * . Concerning the left derived functor we shall use the bar resolution techniques developed by Fresse in [Fre09, Theorem 17.2.7 and Section 13.3]. Following the general construction outlined in Appendix B, we obtain that the action of the left derived functor on an object X ∈ Fun(C, Ch K ) is given by where we abbreviate by X := B ∆ (BZ, C, X) ∈ Fun(BZ, bCh K ) the bicomplex-valued functor described in (B.3) and Tot ⊕ denotes the -totalization of bicomplexes (see also Appendix A).
In the present case, the functor X admits the following very explicit description as a bicomplex with a Z-action: The underlying bicomplex is concentrated only in vertical degrees 0 and 1 because, due to the specific form of the category C in (2.7), there exist no composable m-tuples (f 1 , . . . , f m ) ∈ Mor m (C) of C-morphisms with each f i = id in the case of m > 1. It is concretely given by direct sums of chain complexes where sf, tf ∈ C denote the source/target of the morphism (f : sf → tf ) ∈ Mor(C). (Here, we have simplified the triple direct sum appearing in (B.3b) in an obvious way.) Note that the direct sum in X 0,• is over the objects of the spiral (2.13) and that the direct sum in X 1,• is over its nonidentity morphisms. We shall use the notation (n, N, x) ∈ X 0,• to denote the element x ∈ X(N ) in the summand indexed by (n, N ), and similarly (n, f, x) ∈ X 1,• . The vertical differential δ is given by for all (n, f, x) ∈ X 1,• , where |x| denotes the degree of x ∈ X(sf ), and the Z-action on the bicomplex X is given by addition, i.e. X(k) : We still have to provide explicit models for the derived unit and counit of the Quillen adjunction (4.1). The component at Y ∈ Fun(BZ, Ch K ) of the derived counit is given by (see which clearly is Z-equivariant, i.e. a natural transformation between functors from BZ to Ch K . (Note that every complex L * Y (N ) is a copy of Y ( * ).) The component at X ∈ Fun(C, Ch K ) of the derived unit, η X : Q(X) → L * LL ! (X), is a morphism whose source is a resolution of X that is determined by the bar construction for id : C → C. Following again the general construction outlined in Appendix B, the object X ∆ := B ∆ (C, C, X) ∈ Fun(C, bCh K ) is given explicitly by, for all N ∈ C, with (simplifying the appropriate double and triple direct sums from (B.3)) where Mor 2 (C) denotes the set of composable pairs of morphisms, and vertical differential The functor structure on X ∆ is given by post-composition, i.e. for every morphism h : The component at X ∈ Fun(C, Ch K ) of the derived unit is then given by the components for all N ∈ C, which determine a natural transformation between functors from C to Ch K .
Remark 4.1. We would like to emphasize that Q(X) = Tot ⊕ (X ∆ ) is indeed naturally weakly equivalent to X ∈ Fun(C, Ch K ) via the natural weak equivalence q X : Q(X) → X determined by the components which is of course a quasi-isomorphism too. △ With these preparations, we can now prove the main result of this section.
Theorem 4.2. The derived counit of the Quillen adjunction (4.1) is a natural weak equivalence. The restriction of the derived unit to the full subcategory Fun(C, Ch K ) hoAll ⊆ Fun(C, Ch K ) of functors that send every C-morphism to a quasi-isomorphism is a natural weak equivalence too.
Proof. We first prove that, for every Y ∈ Fun(BZ, Ch K ), the component ǫ Y : LL ! (L * Y ) → Y of the derived counit (4.5) is a weak equivalence. Our strategy is to construct an explicit quasiinverse of its underlying chain map. We define the chain map where M denotes the left object in the category C in (2.7) (i.e., the unperturbed spacetime), which satisfies ǫ Y κ = id. The other composition κ ǫ Y is homotopic to the identity, i.e.
and write for the i-th morphism. We define the chain homotopy ρ by where the second equality holds because Y (k) is a chain map, and the last step follows from the observation that the terms associated with the inner nodes in (4.13) cancel out, leaving only the two boundary terms associated with the ends of (4.13). By a similar calculation, one checks that ∂ρ(n, f, y) = ρδ(n, f, y) = −(n, f, y) , (4.16) for all elements (n, f, y) ∈ LL ! (L * Y ) of vertical degree 1. This completes the proof that all components ǫ Y of the derived counit are weak equivalences.
Let us now prove that, for all X ∈ Fun(C, Ch K ) hoAll , the component η X : Q(X) → L * LL ! (X) of the derived unit (4.8) is a weak equivalence in Fun(C, Ch K ). This amounts to proving that the underlying component chain maps η X,N : Q(X)(N ) → LL ! (X) are quasi-isomorphisms, for all objects N ∈ C. Because the category C in (2.7) is connected and Q(X)(f ) is a quasi-isomorphism for each C-morphism f , naturality of η X and the 2-out-of-3 property of weak equivalences implies that it suffices to prove that one of the components of η X is a quasi-isomorphism. We consider the left object M in the category C in (2.7) and shall prove the equivalent condition that the chain map obtained by pre-composing with the quasi-isomorphism (4.10) is a quasi-isomorphism. Because φ is injective, we obtain a short exact sequence in Ch K . Our strategy is to prove, via a spectral sequence argument, that the homology of coker(φ) is trivial, which implies via the long exact homology sequence associated to (4.18) that φ is a quasiisomorphism. The cokernel of (4.17) admits a direct sum decomposition coker(φ) = F L ⊕ F R , where F L ∈ Ch K is the chain complex associated with the objects and morphisms in (2.13) that are to the left of (0, M ) and F R ∈ Ch K is associated with the objects and morphisms in (2.13) that are to the right of (0, M ). Explicitly, using the following convenient notation for the objects and morphisms to the right of (0, M ), we can write (4.20a) The vertical differential δ R on (k, x) ∈ ∞ k=0 X(sf R k ) is given by where the difference to (4.4) for k = 0 is due to the fact that the object associated with (0, M ) is divided out in the cokernel. The chain complex F L is defined analogously by considering objects and morphisms that are to the left of (0, M ). To prove that the homology of F R is trivial, we consider the (bounded below and exhaustive) filtration 0 ⊆ F R 0 ⊆ F R 1 ⊆ · · · ⊆ F R p ⊆ · · · ⊆ F R given by restricting the direct sums in (4.20) to k ≤ p. Observe that the quotients are mapping cone complexes associated with quasi-isomorphisms. (It is in this step where we use that X ∈ Fun(C, Ch K ) hoAll satisfies the homotopy time-slice axiom.) Using [Wei94, Corollary 1.5.4], we obtain that H • F R p /F R p−1 = 0, for all p, and the convergence theorem for spectral sequences in [Wei94, Theorem 5.5.1] then implies that H • (F R ) = 0. Repeating the same argument for F L we obtain H • (F L ) = 0, from which we conclude that H • (coker(φ)) = 0. This completes the proof.
It is worthwhile to recall and explain in more detail the precise sense in which the results of this section provide a strictification of the homotopy time-slice axiom for linear observables and how they are useful for defining a concept of relative Cauchy evolution. Suppose that L : C → Ch K is a functor that assigns the chain complexes of linear observables of a linear homotopy AQFT. (Recall that the Poisson structure τ : L ∧ L → K is neglected in the present section and will be considered later in Section 5.) By definition, such L satisfies the homotopy time-slice axiom if and only if it defines an object L ∈ Fun(C, Ch K ) hoAll . Combining Remark 4.1 and Theorem 4.2, we obtain a zig-zag of weak equivalences in the model category Fun(C, Ch K ). With this zig-zag we can replace our original L, which satisfies the homotopy time-slice axiom, by the equivalent object L * LL ! (L), which satisfies the strict time-slice axiom because it arises via the pullback functor L * . From this equivalent perspective, the relative Cauchy evolution takes an extremely simple form because it is implemented by the Z-action of the generator 1 ∈ Z. Explicitly, we obtain the RCE automorphism for linear observables 1, N, x) ,  where ∂ denotes the differential on mapping complexes. (Here, we recall that if κ ∈ hom(V, W ) n , representing a collection of maps κ m : V m → W m+n for all m ∈ Z, then ∂κ ∈ hom(V, W ) n−1 is given by (∂κ) m = d W • κ m − (−1) n κ m−1 • d V .) Using previous results on homotopical properties of the CCR-functor from [BS19b, BBS19], it then follows that there exists a zig-zag of weak equivalences between CCR(L, τ ) and CCR(L * LL ! (L), L * τ L ), which provides an explicit strictification construction of the homotopy time-slice axiom of linear homotopy AQFTs and hence allows us to study their relative Cauchy evolution. In order to introduce a Poisson structure τ L on LL ! (L), we make use of the pictorial interpretation from (2.13). Let us also choose, for each quasi-isomorphism L(f ) : L(N ) → L(N ′ ), a quasi-inverse chain map L(f ) −1 : L(N ′ ) → L(N ), two chain homotopies λ f ∈ hom(L(N ′ ), L(N ′ )) 1 and γ f ∈ hom(L(N ), L(N )) 1 , and a chain 2-homotopy ξ f ∈ hom(L(N ), L(N ′ )) 2 , such that For the case of f = id N being an identity morphism, we choose L(id N ) −1 = id L(N ) the identity, together with λ id N = 0, γ id N = 0 and ξ id N = 0.
Remark 5.1. The homotopy coherence data in (5.2) always exists by the following argument: The category Ch K may be enriched to a strict 2-category whose 2-morphisms are homotopy classes of chain homotopies, i.e. equivalence classes [θ] of elements θ ∈ hom(V, W ) 1 modulo elements of the form ∂ℓ for all ℓ ∈ hom(V, W ) 2 . Because K has characteristic 0, quasi-isomorphisms of chain complexes are precisely the equivalences in this 2-category. Using the standard result in 2category theory that each equivalence in a 2-category can be improved to an adjoint equivalence, see e.g. [Lac10, Exercise 2.2], we obtain for each quasi-isomorphism L(f ) : L(N ) → L(N ′ ) the data in (5.2), where L(f ) −1 plays the role of the right adjoint, γ f the role of the adjunction unit and −λ f the role of the adjunction counit. The last identity (5.2c) states one of the two triangle identities for this adjunction, while the other one reads as for some chain 2-homotopy ξ f ∈ hom(L(N ′ ), L(N )) 2 . We did not include this additional coherence condition (5.3) in the main text because it is not needed for our construction. We already would like to emphasize at this point that for concrete examples, such as the linear Yang-Mills model [BBS19], the homotopy coherence data in (5.2) may be constructed from Green operators and a partition of unity. See Section 6 for the details. In general, the 2-categorical argument automates the explicit but tedious linear algebra that constructs quasi-inverses and homotopies, based on choices of linear projections onto ker d i within X i and onto H i (X) within ker d i . △ Now consider the diagram formed by applying L to the spiral in (2.13). Given any two (n, N ), (n ′ , N ′ ) ∈ Z × C, we note that there exists a unique chain of zig-zags of minimal length between them. Using the chosen quasi-inverses L(f ) −1 to reverse arrows as needed, we obtain zig-zagging chain maps for all k ∈ Z and (n, N ), (n ′ , N ′ ) ∈ Z × C.
In Appendix C, we construct, for every (n, f ) ∈ Z × Mor(C) and (n ′ , N ′ ) ∈ Z × C, from the chain homotopy λ f a zig-zagging homotopy (for left composition) Similarly, we construct, for every (n, N ) ∈ Z × C and (n ′ , f ′ ) ∈ Z × Mor(C), from the chain homotopy γ f a zig-zagging homotopy (for right composition) Γ (n,N ) Finally, using also the chain 2-homotopies ξ f , the construction in Appendix C defines, for every (n, f ), (n ′ , f ′ ) ∈ Z × Mor(C), a zig-zagging 2-homotopy The role of this homotopy coherence data is to relate the shortest zig-zags in (2.13) to longer zig-zags that are obtained by additional compositions with morphisms and their quasi-inverses. Note that this is just part of a higher tower of homotopy coherence data for relating arbitrary zig-zags to the shortest ones, which however will not be required in our work.
With these preparations, we can now define a Poisson structure τ L : LL ! (L) ∧ LL ! (L) → K on LL ! (L) = Tot ⊕ ( L). Recall that τ L should be a chain map, therefore obeying τ L • d Tot ⊕ ( L∧ L) = d K • τ L = 0. Using a direct sum decomposition into the vertical degrees of L, such a Poisson structure is given by a family of n-chains τ L,n ∈ hom ( L ∧ L) n,• , K n n∈Z (5.10a) that are invariant under the Z-action on L and (as d Tot ⊕ ( L∧ L) = δ L∧ L +d L∧ L ) satisfy the conditions for all n ∈ Z. Because L ∈ bCh K is concentrated in vertical degrees 0 and 1 (see (4.4)) and hence L ∧ L is concentrated in degrees 0, 1 and 2, it follows that τ L,n = 0 , for all n ∈ {0, 1, 2} . (5.11a) We define the degree 0 component by for all (n, N, x), (n ′ , N ′ , x ′ ) ∈ L 0,• , where τ N and τ N ′ denote the components of the given Poisson structure τ : L ⊗ L → K. The degree 1 component is defined by for all (n, f, x) ∈ L 1,• and (n ′ , N ′ , x ′ ) ∈ L 0,• , and for all (n, N, x) ∈ L 0,• and (n ′ , f ′ , x ′ ) ∈ L 1,• . Finally, the degree 2 component is defined by for all (n, f, x), (n ′ , f ′ , x ′ ) ∈ L 1,• . Proof. Invariance under the Z-action follows directly from (5.6) and the explicit formulas for the Λ, Γ and Ξ given in (C.4), (C.5) and (C.6). The tower of conditions (5.10) reduces in the present case to The first condition is immediate because τ L,0 given in (5.11) is clearly a chain map. By slightly lengthy but straightforward calculations, the second condition follows from (5.7) and (5.8), and the third condition follows from (5.9). (Note: Because the τ L,n are obtained by antisymmetrization with respect to the total degrees and the conditions (5.12) are invariant under the symmetric braiding in bCh K , it is sufficient to consider in these calculations only the first of the two terms in (5.11).) We shall now construct the components ρ N ∈ hom(Q(L)(N ) ∧ Q(L)(N ), K) 1 of the natural chain homotopy ρ from (5.1), for all N ∈ C. Using that Q(L)(N ) = Tot ⊕ (L ∆ (N )) is the totalization of the bicomplex (4.7), we can use again a direct sum decomposition into the vertical degrees of L ∆ (N ) to define ρ N by a family of n + 1-chains where τ N,n denotes the decomposition of the original Poisson structure τ N : L(N ) ∧ L(N ) → K into vertical degrees, i.e. τ N,0 = τ N and τ N,n = 0 for all n = 0. Here, we have used the formula N )) . We define ρ N,n = 0 , for all n ∈ {0, 1} . (5.14a) The degree 0 component is defined by for all (g, x), (g ′ , x ′ ) ∈ L ∆ (N ) 0,• , and the degree 1 component is defined by (Recall that, due to the fact that the nerve of C is degenerate in degrees ≥ 2, we necessarily have that (g, f, x) = (id N , f, x) and Proposition 5.3. The components (5.14) define a natural chain homotopy ρ ∈ hom(Q(L) ∧ Q(L), K) 1 satisfying (5.1).
Proof. This is a straightforward check using the properties (5.7), (5.8) and (5.9) of the zig-zagging homotopies and naturality of the original Poisson structure τ : L ∧ L → K.

Let us recall from [BBS19, Section 5] the functor
that assigns to a Poisson chain complex (V, τ ) ∈ PoCh K its associated differential graded CCRalgebra CCR(V, τ ) ∈ dgAlg K . Composing this functor with our given functor (L, τ ) : C → PoCh K that models the Poisson chain complexes of linear observables of a linear homotopy AQFT, we obtain a homotopy AQFT on the category C given in (2.7). Furthermore, composing CCR with the pullback of the functor (LL ! (L), τ L ) : BZ → PoCh K along the localization functor L : C → BZ defines another homotopy AQFT on C which, due to the fact that it is obtained as a pullback along the localization functor, satisfies the strict time-slice axiom. The following rectification theorem is the main result of this paper.
Theorem 5.4. Let (L, τ ) : C → PoCh K be any functor on the category C in (2.7) that satisfies the homotopy time-slice axiom, i.e. L ∈ Fun(C, Ch K ) hoAll . Then its associated homotopy AQFT (5.16), which satisfies the homotopy time-slice axiom, is equivalent via a zig-zag of weak equivalences in the model category Fun(C, dgAlg K ) to the homotopy AQFT (5.17) that satisfies the strict time-slice axiom.
We conclude by emphasizing that the relative Cauchy evolution for the strictified homotopy AQFT (5.17) is given by applying the CCR-functor on the RCE automorphism for linear observables from (4.23), i.e. In particular, the relative Cauchy evolution of the strictified model is given by a strict automorphism of dg-algebras, in contrast to an A ∞ -quasi-automorphism.
Remark 5.5. Throughout the bulk of this paper we have neglected * -involutions on the homotopy AQFTs in order to streamline our presentation. However, we would like to emphasize that our main result in Theorem 5.4 also holds true for homotopy AQFTs with * -involutions. The proof is obtained by choosing up to Proposition 5.3 the ground field K = R and recalling that the homotopical properties of the CCR-functor entering the proof of Theorem 5.4 were shown in [BBS19, Proposition 5.3] also for the complexified CCR-functor with * -involutions over C. △ 6 Example: Linear quantum Yang-Mills theory The aim of this section is to illustrate our construction of the relative Cauchy evolution for linear homotopy AQFTs through a simple example, the linear Yang-Mills model presented in [BBS19]. This example is based on the solution complexes for each spacetime M ∈ Loc, where the round brackets indicate homological degrees, d M is the de Rham differential and δ M the codifferential. To be explicit, we work in arbitrary spacetime dimension and our convention for the metric signature is (+ − − · · · −). The Hodge operator * M is defined so that where the subscript c denotes differential forms of compact support.  4) is constructed by chain complex analogs of retarded/advanced Green operators, which were called retarded/advanced trivializations in [BBS19] and were shown to be unique up to contractible choices. These structures provide chain homotopies trivializing the inclusion chain map from the chain complex of linear observables to the shifted solution complex. (In (6.6) we have chosen sign conventions for the vertical inclusion maps that are opposite to the ones in [BBS19]. Our present choice is convenient to fix a minor sign error in Eqn. (4.26) of the latter paper.) A concrete model for such retarded/advanced Green homotopies can be given in terms of the retarded/advanced Green operators G ± M of the d'Alembert operator M := δ M d M + d M δ M on differential forms, 4 reading explicitly as between the retarded and advanced Green homotopy defines a chain map to the unshifted solution complex, which plays a similar role to the causal propagator in ordinary AQFT. Explicitly, this chain map reads as where G M := G + M −G − M denotes the causal propagator for the d'Alembert operator on differential forms. The Poisson structure is then defined by the chain map (6.10a) whose non-zero components explicitly read as for all χ ∈ L(M ) −1 , ϕ 1 , ϕ 2 ∈ L(M ) 0 and α ∈ L(M ) 1 . (As already mentioned above, there is a minor sign error in the corresponding equation (4.26) in [BBS19]. Our opposite sign convention for the vertical maps in (6.6) implies that our Poisson structure (6.10) has a positive sign for linear observables in degree 0, in contrast to the negative sign in the corrected version of Eqn.
(4.26a) in [BBS19].) Naturality of (6.9) and (6.5) implies that the assignment M → (L(M ), τ M ) can be promoted to a functor (L, τ ) : Loc → PoCh R . To a Loc-morphism f : M → N , this functor assigns the chain map L(f ) := f * : (L(M ), τ M ) → (L(N ), τ N ) of Poisson chain complexes given by pushforward (i.e. extension by zero) of compactly supported differential forms. By [BBS19, Theorem 6.19], the composition of this functor with the CCR-functor CCR : PoCh R → dg * Alg C (in the context of chain complexes and with * -involutions) defines a homotopy AQFT A := CCR(L, τ ) ∈ * AQFT ∞ (Loc) hoW with * -involution that satisfies the homotopy time-slice axiom. With an abuse of notation, we denote by the same symbol A : C → dg * Alg C the restriction of this homotopy AQFT to the category C in (2.7) that is relevant for the discussion of the relative Cauchy evolution induced by the pair (M, h) consisting of a spacetime M and a compactly supported metric perturbation h.
Our main Theorem 5.4, see also Remark 5.5, provides us with a concept of relative Cauchy evolution for this homotopy AQFT. Let us recall that our construction is slightly abstract as it involves a replacement of the theory A by a weakly equivalent theory A st : C → dg * Alg C (see (5.17)) that satisfies the strict time-slice axiom and hence admits an RCE automorphism in the ordinary sense (5.20). In order to physically interpret the resulting concept of relative Cauchy evolution, and to compute its associated stress-energy tensor, we make use of a concept similar to the quantum fields in the sense of [BFV03,FV15], where they are understood as natural transformations between the functors assigning test function spaces and field algebras to spacetimes. We shall use the chain complex L(M ) of linear observables on the object M in (2.7) for labeling these quantum fields and consider the chain map to the chain complex of linear observables for the strictified theory given in (4.3) and (4.4). In words, we embed the linear observables on M at the level n = 0 in the spiral (2.13) controlling our strictification construction. (Due to Z-invariance of LL ! (L) and its Poisson structure τ L , it does not matter at which level n ∈ Z we embed L(M ).) Using further the chain map LL ! (L) → CCR(LL ! (L), τ L ) that assigns to linear observables their corresponding generators in the CCRalgebra, we can compose with (6.11) to obtain a chain map from L(M ) to CCR(LL ! (L), τ L ) that plays a similar role as the quantum fields in [BFV03,FV15]. We can now introduce a concept of relative Cauchy evolution for this particular quantum field, which can be interpreted as a combination of the gauge field, ghost and antifields into a single multicomponent field.
Proposition 6.1. The diagram in Ch R is homotopy commutative, where (1,M ) = ∂θ = (δ L + d L )θ + θ d L(M ) , where we recall that LL ! (L) = Tot ⊕ ( L) is obtained as the -totalization of a bicomplex (4.4) whose horizontal and vertical differentials are denoted here by d L and δ L . Evaluating on any x ∈ L(M ) and recalling the RCE automorphism on linear observables (4.23), the relevant homotopy relation is given by (6.14) We construct the chain homotopy θ by transporting the element (1, M, x) ∈ LL ! (L) step by step from right to left along the relevant part of the zig-zags in (2.13).
To describe the chain homotopies for the individual steps, let f : (n + L(f ), sf ) → (n, tf ) be any of the morphisms in (6.15). If f points from right to left in (6.15), we define the homotopy for all x ∈ L(sf ). (Observe that this homotopy moves the element (n + L(f ), sf, x) from right to left in (6.15).) If f points from left to right in (6.15), we define the homotopy θ− → f ∈ hom(L(tf ), LL ! (L)) 1 by where L(f ) −1 is a quasi-inverse of L(f ) and λ f its associated homotopy coherence data from (5.2), which satisfies for all x ∈ L(tf ). (Observe that this homotopy moves the element (n, tf, x) from right to left in (6.15).) Composing these basic homotopies defines the required chain homotopy in (6.14).
Remark 6.2. Because ι : L(M ) → LL ! (L) is a quasi-isomorphism, our model given in Proposition 6.1 for the relative Cauchy evolution for this linear quantum field is unique up to homotopy. Note further that this model agrees with the "naive approach" consisting of quasi-inverting the quasi-isomorphisms associated with the Cauchy morphisms in (2.7) since the relevant zig-zagging chain map (6.13) is of the same form as the usual relative Cauchy evolution formula in (2.4).
It is important to stress that this "naive approach" works only at the level of linear quantum fields because the chain map (6.13) does in general not preserve the Poisson structure Hence, the role of our rectification Theorem 5.4 is to establish existence of a well-behaved concept of relative Cauchy evolution for linear homotopy AQFTs, while the role of Proposition 6.1 is to provide explicit computational tools to study the relative Cauchy evolution at the level of linear quantum fields. △ In order to obtain an explicit formula for the relative Cauchy evolution for linear quantum fields determined by Proposition 6.1, we have to construct explicitly the necessary quasi-inverses in (6.13) and their corresponding homotopy coherence data (5.2). Without much extra effort, we will perform this construction for a general Cauchy morphism f : N → N ′ in Loc. Let us choose any two Cauchy surfaces Σ ± ⊆ N in the source spacetime N such that Σ + lies in the future of Σ − , i.e. Σ + ⊆ I + N (Σ − ). We denote by Σ ′ ± := f (Σ ± ) ⊆ N ′ their images under f , which are Cauchy surfaces in N ′ because f is by hypothesis a Cauchy morphism. We then pick any partition of unity ρ ′ ± subordinate to the open cover its pullback along f , which defines a partition of unity subordinate to the open cover {I + N (Σ − ), I − N (Σ + )} of N . From this data and the retarded/advanced Green homotopies in (6.7), we define the following two chain homotopies In complete analogy, the chain homotopy γ f ∈ hom(L(N ), Sol(N )[1]) 1 factors uniquely through , defining a unique chain homotopy γ f ∈ hom(L(N ), L(N )) 1 such that Proof. The two proofs are identical, hence we shall write out only the one for γ f . First, let us note that uniqueness of γ f follows from the fact that the chain map j N in (6.6) is degree-wise injective. Concerning existence, we have to show that the image of every element ω ∈ L(N ) is a compactly supported differential form. Recalling (6.7c), the support of G ± N (ω) is strictly past/future compact, i.e. supp(G ± N (ω)) ⊆ J ± N (K) for a compact subset K ⊆ N , because G ± N is a composition of a retarded/advanced Green operator and a differential operator. Furthermore, the support of ρ ± is by construction past/future compact. It then follows from [Bär15, Lemma 1.9] that both terms in (6.21) are compactly supported differential forms, hence so is γ f (ω).

Consider now the chain map
. It is easy to see that the image j N ′ (ω)+∂ N ′ λ f (ω) ∈ Sol(N ′ )[1] of every element ω ∈ L(N ′ ) is a differential form supported in the closed time-slab J − where in the first equality we have used that ρ ′ + | U + = 1 and ρ ′ − | U + = 0, and in the second equality we have used (6.7b). A similar argument shows that the restriction to U − := I − N ′ (Σ ′ − ) ⊆ N ′ is zero too. In terms of the untilded chain homotopies from Lemma 6.3, this determines a chain map We will now prove that (6.26) is indeed a quasi-inverse of L(f ) : L(N ) → L(N ′ ) and that the chain homotopies from Lemma 6.3 provide the necessary homotopy coherence data in (5.2) with trivial 2-homotopy ξ f = 0.
Proposition 6.5. For every Cauchy morphism f : N → N ′ in Loc, the following three identities hold true where the chain map L(f ) −1 is defined in (6.26) and the chain homotopies λ f and γ f are defined in Lemma 6.3.
Proof. The first identity is a simple check where in the second step we have used that f * f * = id on L(f (N )). Because the chain map j N is degree-wise injective, the second identity is equivalent to j N L(f ) −1 L(f ) − j N = ∂ N γ f . Using (6.20) and the fact that j N is a chain map, one checks the latter identity as follows where in the second step we have also used that f * f * = id on L(N ) and naturality of j N and ∂ N . The last step follows from (6.18) and naturality of the Green homotopies. Explicitly, where we recall that, by definition, f * (ρ ′ ± ) = ρ ± . To prove the third identity, we use the explicit expressions in (6.22) and (6.23) for the untilded chain homotopies. Because both terms in L(f )γ f − λ f L(f ) take values in differential forms supported in the image f (N ) ⊆ N ′ of f , we may prove the third identity by post-composing with f * . The relevant calculation is then given by In the second step we have used f * (ρ ′ ± ) = ρ ± and naturality of Q N and G ± N . The last step follows from the fact that f * f * = id on L(N ).
With these preparations, we can now write down an explicit formula for the zig-zagging chain map (6.13) that models the relative Cauchy evolution at the level of linear quantum fields. Because each of the morphisms f in the category C in (2.7) is a subset inclusion, we can suppress all occurrences of pullbacks f * and pushforwards f * of differential forms as these are simply restrictions and extensions by zero. The explicit formula is then given by Because homotopy commutativity of the diagram in Proposition 6.1 determines the chain map Z Proposition 6.6. The chain map (6.33) can be simplified as where G M h is the causal propagator for the d'Alembert operator on differential forms and Q M h is the differential operator defined in (6.22b) on the perturbed spacetime M h ∈ Loc.
Proof. This is a straightforward argument considering the supports of the differential forms involved. By (6.25), we have that id + ∂ M λ i + : L(M ) → L(M + ) takes values in the sub-chain complex of linear observables supported in M + ⊆ M . We then compute using (6.22) Remark 6.7. Using (6.4) and (6.22b), we can spell out explicitly the components of the chain map (6.34). For a linear ghost field observable χ ∈ L(M ) −1 = Ω 0 c (M ), we obtain for all k + 1-forms ω = ω a 0 a 1 ···a k dx a 0 ∧ dx a 1 ∧ · · · ∧ dx a k ∈ Ω k+1 (M ). In contrast to the situation in ordinary AQFT, the chain map t (  for all ω 1 , ω 2 ∈ L(M + ). By another straightforward calculation using integration by parts, we find the explicit formula T ab M (ω 1 , ω 2 ) = 1 4 g ab (F ϕ 1 ) cd (F ϕ 2 ) cd − (F ϕ 1 ) ac (F ϕ 2 ) b c , (6.46) where ϕ i := pr 0 (ω i ) ∈ L(M + ) 0 denotes the degree 0 component of ω i and F ϕ i := d M G M ϕ i ∈ Ω 2 (M ) denotes its associated field strength 2-form. Observe that our polarized stress-energy tensor does not receive contributions from the ghosts and the antifields, and that in degree 0 it agrees with the usual Maxwell stress-energy tensor (in our (+ − · · · −) signature) on setting ω 2 = ω 1 . This result shows how these physically expected results can be provided with a firstprinciples justification on quite abstract grounds.

A Conventions for bicomplexes
A bicomplex V is a bigraded family of K-vector spaces {V p,q } p,q∈Z equipped with a vertical differential δ : V p,q → V p−1,q and a horizontal differential d : V p,q → V p,q−1 satisfying δ 2 = 0 , d 2 = 0 , δ d + d δ = 0 . (A.1) Observe that the vertical and horizontal differentials are required to anti-commute, which is more convenient for discussing the symmetric monoidal structure and totalization of bicomplexes than the alternative convention of commuting differentials. We refer to e.g. [MR19] for more details and the relevant argument that both conventions are equivalent. A morphism of bicomplexes f : V → W is a family of linear maps {f p,q : V p,q → W p,q } p,q∈Z that commutes with both the vertical and the horizontal differential. We denote by bCh K the category of bicomplexes of K-vector spaces. The category bCh K is symmetric monoidal with respect to the tensor product defined by together with the differentials where |v| tot = i + j denotes the total degree of v ∈ V i,j . The monoidal unit is K concentrated in degree (0, 0) and the symmetric braiding is given by the Koszul sign rule with respect to the total degrees, i.e.
The -totalization of bicomplexes is given by the functor It is easy to check that Tot ⊕ is a strong symmetric monoidal functor with respect to the obvious structure maps.

B Bar construction
The bar construction is a powerful and efficient tool to obtain derived functors. In this appendix we shall spell out some computational details that will help the reader to understand better our explicit formulas in Section 4. We refer to [Fre09] and also [Rie14] for a more detailed presentation of the bar construction.
Let F : C → D be a functor between any two categories C and D. (The case of interest in the bulk of the paper is the localization functor L : C → BZ from (2.8).) The associated bar construction is then a functor B ∆ (D, C, −) : Fun(C, Ch K ) −→ Fun(D, sCh K ) (B.1) that assigns to each chain complex-valued functor X : C → Ch K on C a functor B ∆ (D, C, X) : D → sCh K on D with values in simplicial chain complexes. In order to avoid simplicial technology, which may be unfamiliar to some readers, we prefer to present this construction in the more familiar language of bicomplexes, see Appendix A for our conventions. (Technically, this uses the Dold-Kan correspondence between simplicial chain complexes and bicomplexes.) From this perspective the bar construction is a functor We shall also need concrete models for the derived counit and unit associated with the derived left Kan extension. Denoting by F * : Fun(D, Ch K ) → Fun(C, Ch K ) the pullback functor, which we recall does not have to be derived because each object in Fun(D, Ch K ) is fibrant in the projective model structure, the component at Y ∈ Fun(D, Ch K ) of the derived counit is the natural transformation