Differential calculus of Hochschild pairs for infinity-categories

In this paper, we provide a conceptual new construction of the algebraic structure on the pair of the Hochschild cohomology spectrum (cochain complex) and Hochschild homology spectrum, which is analogous to the structure of calculus on a manifold. This algebraic structure is encoded by a two-colored operad introduced by Kontsevich and Soibelman. We prove that for a stable idempotent-complete infinity-category, the pair of its Hochschild cohomology and homology spectra naturally admits the structure of algebra over the operad. We also prove a generalization to the equivariant context.

Theorem 8.2 and Proposition 8. 8. In other words, the underlying morphism given by actions is an expected one. By considering Cartan's homotopy formula built in KS, an analogue L : HH • (C) [1] ⊗ HH • (C) → HH • (C) of the Lie derivative map l is also an expected morphism, cf. Remark 8. 10.
In Section 2, we briefly describe the idea and approach of our construction, which is based on a simple observation. To achieve this, we prove the following (see Corollary 4.21 for details): See Section 4 for the definition of the fiber product on the right-hand side.
This result means that Alg E2 (M), Alg As (Fun(BS 1 , M)) and LMod(Fun(BS 1 , M)) form building blocks for KS-algebras. This allows us to describe the structure of a KS-algebra as a collection of more elementary algebraic data involving associative algebras, left modules, circle actions, and E 2 -algebras. As for E 2 -algebras, thanks to Dunn additivity theorem for ∞-operads proved by Lurie [26], a canonical equivalence Alg E2 (M) ≃ Alg As (Alg As (M)) exists. While we make use of Theorem 1.2 in the construction process, it would be generally useful in the theory of KS-algebras since the notion of KS-algebras is complicated as is. For example, when M is the derived ∞-category D(k) of vector spaces over a field k of characteristic zero, i.e., in the differential graded context, there is a quite elementary interpretation. One may take Fun(BS 1 , D(k)) to be the ∞-category obtained from the category of mixed complexes in the sense of Kassel (see e.g. [23]) by localizing quasi-isomorphisms. Therefore, an object of Alg As (Fun(BS 1 , D(k))) may be regarded as an associative algebra in the monoidal (∞-)category of mixed complexes. Objects of LMod(Fun(BS 1 , D(k))) can be described in a similar way. Moreover, dg E 2 -operad is formal in characteristic zero.
As we will describe in the following section, our method consists of only natural procedures. In particular, by contrast with previous work, it does not involve/use complicated resolutions of operads or genuine chain complexes. Thus, we hope that our proposed approach can be applicable to other settings and generalizations such as (∞, n)-categories. Indeed, we prove an equivariant generalization of Theorem 1.1 (see Theorem 9.1 for details): Theorem 1.3. Let G be a group object in the ∞-category S of spaces, that is, a group-like E 1 -space. Let C be a small R-linear stable indempotent-complete ∞-category. Suppose that G acts on C (namely, it gives a left action). Then (HH • (C), HH • (C)) is promoted to a KS-algebra in Fun(BG, Mod R ). Namely, (HH • (C), HH • (C)) is a KS-algebra in Mod R , which comes equipped with a left action of G.
We would like to view our results from the perspective of noncommutative algebraic geometry. As mentioned above, the notion of KS-algebra structures is a counterpart to the calculus on manifolds. Thus, KS-algebras are central objects in "noncommutative calculus". We refer the reader to [12] and references therein for this point of view.
Recall the algebro-geometric interpretations of the Hochschild cohomology HH • (C) and Hochschild homology HH • (C) for stable ∞-categories C or dg categories (somewhat more precisely, we assume that they are "linear" over a field of characteristic zero). The E 2 -algebra HH • (C) governs the deformations theory of the stable ∞-category C in the derived geometric formulation. The Hochschild homology HH • (C) (more precisely, the Hochschild chain complex) inherits an S 1 -action that corresponds to the Connes operator. Then HH • (C) with S 1 -action gives rise to an analogue of the Hodge filtration: the pair of the negative cyclic homology and the periodic cyclic homology can be thought of as such a structure. (These algebraic structures are contained in the KS-algebra (HH • (C), HH • (C)).) As revealed in [19] in the case of associative (dg) algebras A, the action of HH • (A) on HH • (A) encoded by the KS-algebra structure at the operadic level is a key algebraic datum that describes variations of the (analogue of) Hodge filtration along noncommutative (curved) deformations. Namely, the period map for noncommutative deformations (of an associative algebra) is controlled by the KS-algebra of the Hochschild cohomology and Hochschild homology. Therefore, the KS-algebra (HH • (C), HH • (C)) will provide a crucial algebraic input for the theory of period maps for deformations of the stable ∞-category C. The motivations of the equivariant generalization Theorem 1.3 partly come from mirror symmetry. For example, stable ∞-categories endowed with S 1 -actions, that are interesting from the viewpoint of S 1 -equivariant deformation theory, naturally appear from Landau-Ginzburg models in the context of matrix factorizations. As a second example, if X is a sufficiently nice algebraic stack (more generally, a derived stack), one can consider the derived free loop space LX = Map(S 1 , X) of X (see e.g., [4]). The stable ∞-category Perf(LX) of perfect complexes on LX comes equipped with the natural S 1 -action.
We define Hochschild cohomology spectrum HH • (C) = HH • (D) to be E(End(D) ⊗ ) ∈ Alg E2 (Sp). The underlying associative algebra HH • (C) is the endomorphism algebra of the identity functor D → D in Fun L (D, D).
Consider the counit map of the adjunction: In other words, D is a left RMod ⊗ HH • (D) -module object in Pr L St . Let RPerf ⊗ HH • (C) ⊂ RMod ⊗ HH • (D) be the monoidal full subcategory that consists of compact objects. By the restrictions, it gives rise to a left RPerf ⊗ HH • (C) -module object C: in the ∞-category St of small stable idempotent-complete ∞-categories in which morphisms are exact functors (St also admits a suitable symmetric monoidal structure). Informally, we think of it as a categorical associative action of HH • (C) on C. This is induced by the adjunction so that it has an evident universal property. Construct a functor St → Sp which carries C to the Hochschild homology spectrum HH • (C). In the classical differential graded context, Hochschild chain complex comes equipped with the Connes operator. In our general setting, it is natural to encode such structures by means of circle actions: Hochschild homology spectrum HH • (C) is promoted to a spectrum with an S 1 -action, that is, an object of Fun(BS 1 , Sp). Thus we configure the assignment C → HH • (C) as a symmetric monoidal functor where Fun(BS 1 , Sp) inherits a pointwise symmetric monoidal structure from the structure on Sp.
2.2. This paper is organized as follows: Section 3 collects conventions and some of the notation that we will use. In Section 4, we discuss algebras over the Kontsevich-Soibelman operad. The main result of Section 3 is Corollary 4.21 (=Theorem 1.2). Along the way, we introduce several topological colored operads (∞-operads). In Section 5, we give a brief review of Hochschild cohomology spectra that we will use. In Section 6, we give a construction of the assignment C → HH • (C) which satisfies the requirements for our goal (partly because we are not able to find a suitable construction in the literature). The results of this section will be quite useful for various purposes other than the subject of this paper. In Section 7, we prove Theorem 7.14 (=Theorem 1.1), namely, we construction a KS-algebra (HH • (C), HH • (C)). In Section 8, we study the action morphisms determined by the structure of the KS-algebra on (HH • (C), HH • (C)). In Section 9, we give a generalization to an equivariant setting (cf. Theorem 1.3): C is endowed with the action of a group (a group object in the ∞-category of spaces).

Notation and convention
Throughout this paper, we use the theory of quasi-categories. A quasi-category is a simplicial set which satisfies the weak Kan condition of Boardman-Vogt. The theory of quasi-categories from the viewpoint of models of (∞, 1)-categories were extensively developed by Joyal and Lurie [20], [25], [26]. Following [25], we shall refer to quasi-categories as ∞-categories. Our main references are [25] and [26]. For the brief introduction to ∞-categories, we refer to [25,Chapter 1], [14]. Given an ordinary category C, by passing to the nerve N(C), we think of C as the ∞-category N(C). We usually abuse notation by writing C for N(C) even when C should be thought of as a simplicial set or an ∞-category.
We use the theory of ∞-operads which is thoroughly developed in [26]. The notion of ∞-operads gives one of the models of topological colored operads (multicategories). Thanks to Hinich [16], there is a comparison between algebras over differential graded operads and algebras over ∞-operads in values in chain complexes. In particular, in characteristic zero, [16] establishes an equivalence between two notions of algebras, see loc. cit.
Here is a list of some of the conventions and notation that we will use: • Z: the ring of integers, R denotes the set of real numbers which we regard as either a topological space or a ring.

5])
• S: ∞-category of small spaces. We denote by S the ∞-category of large spaces (cf. [25, 1.2.16]). • C ≃ : the largest Kan subcomplex of an ∞-category C • C op : the opposite ∞-category of an ∞-category. We also use the superscript "op" to indicate the opposite category for ordinary categories and enriched categories. • Cat ∞ : the ∞-category of small ∞-categories • Sp: the stable ∞-category of spectra. • Map C (C, C ′ ): the mapping space from an object C ∈ C to C ′ ∈ C where C is an ∞-category. We usually view it as an object in S (cf. [25, 1.2.2]). • Fin * : the category of pointed finite sets 0 , 1 , . . . n , ... where n = { * , 1, . . . n} with the base point * . We write Γ for N(Fin * ). n • = n \ * . Notice that the (nerve of) Segal's gamma category is the opposite category of our Γ. • P act : If P is an ∞-operad, we write P act for the subcategory of P spanned by active morphisms.
Informally, an As-algebra (an algebra over As ⊗ ) is an unital associative algebra. For a symmetric monoidal ∞-category C ⊗ , we write Alg As (C) for the ∞-category of As-algebra objects. We refer to an object of Alg As (C) as an associative algebra object in C ⊗ . We refer to a monoidal ∞-category over As ⊗ as an associative monoidal ∞-category. • LM ⊗ : the ∞-operad defined in [26, 4.2.1.7]. An algebra over LM ⊗ is a pair (A, M ) such that an unital associative algebra A and a left A-module M . For a symmetric monoidal ∞-category C ⊗ → Γ, we write LMod(C ⊗ ) or LMod(C) for Alg LM ⊗ (C ⊗ ). • E ⊗ n : the ∞-operad of little n-cubes. For a symmetric monoidal ∞-category C ⊗ , we write Alg En (C) for the ∞-category of E n -algebra objects.
4. operad 4.1. We will define several simplicial colored operads which are relevant to us. By a simplicial colored operad, we mean a colored operad in the symmetric monoidal category of simplicial sets. A simplicial colored operad is also referred to as a symmetric multicategory enriched over the category of simplicial sets.
Definition 4.2. Let Cyl be a simplicial colored operad defined as follows: (i) The set of colors of Cyl has a single element, which we will denote by C.
(ii) Let I = r • be a finite set and let {C} I be a set of colors indexed by I. By abuse of notation, we write C ⊔r for {C} I where r is the number of elements of I. We remark that C ⊔r does not mean the coproduct. We define Mult Cyl ({C} I , C) = Mult Cyl (C ⊔r , C) to be the singular simplicial complex of the space Emb rec (((0, 1) × S 1 ) ⊔r , (0, 1) × S 1 ) of embeddings ((0, 1)×S 1 ) ⊔r → (0, 1)×S 1 such that the restriction to each component (0, 1)×S 1 → (0, 1)×S 1 is rectilinear. Here ((0, 1)×S 1 ) ⊔r is the disjoint union of (0, 1)×S 1 , whose set of connected components is identified with I. The space Emb rec (((0, 1) × S 1 ) ⊔r , (0, 1) × S 1 ) is endowed with the standard topology, that is, the subspace of the mapping space with compact-open topology. (iii) The composition law in Cyl is given by the composition of rectilinear embeddings, and a unit map is the identity map. The color C together with Mult Cyl ({C} I , C) constitutes a fibrant simplicial colored operad. By a fibrant simplicial colored operad we mean that every simplicial set Mult Cyl ({C} I , C) is a Kan complex. Note that the singular simplicial complex of a topological space is a Kan complex. Definition 4.3. Let Cyl be a simplicial colored operad defined as follows: (i) The set of colors of Cyl has two elements denoted by C and C M .
(iii) We set Mult Cyl (C ⊔m , C) = Mult Cyl (C ⊔m , C). If n = 0, Mult Cyl (C ⊔m ⊔ C ⊔n M , C) is the empty set. (iv) The composition law is given by the composition of rectilinear embeddings, and a unit map is the identity map. The colors C, C M together with simplicial sets of maps constitute a fibrant simplicial colored operad.
Definition 4.4. Let DCyl be a simplicial colored operad defined as follows: (i) The set of colors of DCyl has two elements, which we denoted by D and C.
(ii) Let I = r • be a finite set and let {D, C} I be a set of colors indexed by I, that is, a map p : I → {D, C}. By abuse of notation we write D ⊔l ⊔C ⊔m for {D, C} I when p −1 (D) (resp. p −1 (C)) has l elements (resp. m elements). We define Mult DCyl (D ⊔l , D) to be the singular simplicial complex of the space Emb rec (((0, 1) 2 ) ⊔l , (0, 1) 2 ) of embeddings from the disjoint union (0, 1) 2 × p −1 (D) to (0, 1) 2 such that the restriction to each component is rectilinear, where the space comes equipped with the subspace topology of the mapping space with compact-open topology.
If m ≥ 1, Mult Cyl (D ⊔l ⊔ C ⊔m , D) is the empty set. (iii) We define Mult DCyl (D ⊔l ⊔ C ⊔m , C) to be the singular complex of the space of embeddings such that the rescriction to a component (0, 1) 2 is rectilinear, and the restriction to a component (0, 1) × S 1 is rectilinear. (iv) The composition law and the unit are defined in an obvious way. The colors D, C together with simplicial sets of maps constitute a fibrant simplicial colored operad.
Definition 4.5. Let DCyl be a simplicial colored operad defined as follows.
(i) The set of colors of DCyl has three elements, which we denote by D, C, and C M .
(ii) Let I = r • be a finite set and let {D, C, C M } I be a set of colors indexed by I, that is, a map p : I → {D, C, C M }. By abuse of notation, we write D ⊔l ⊔ C ⊔m ⊔ C ⊔n M for {D, C, C M } I when p −1 (D) (resp. p −1 (C), p −1 (C M )) has l elements (resp. m elements, n elements). We set (v) The composition law and the unit map are defined in an obvious way. The colors D, C, C M together with simplicial sets of maps constitute a fibrant simplicial colored operad.
Remark 4.6. There is a commutative diagram of inclusions of simplicial colored operads: Each inclusion determines a simplicial colored full suboperad.
We obtain an ∞-operad from a fibrant simplicial colored operad. We recall the construction from [26, Definition 4.7. Let P be a simplicial colored operad. Let P col be the set of colors of P . We let P ∆ be a simplcial category defined as follows: (i) The objects of P ∆ are maps a : n • → P col , that is, pairs ( n , (C 1 , . . . , C n )) where n ∈ Fin * and (C 1 , . . . , C n ) is a finite sequence (a(1), . . . , a(n)) of colors. (ii) Let C = ( n , (C 1 , . . . , C n )) and C ′ = ( m , (C ′ 1 , . . . , C ′ m )) be two objects. The hom simplicial set Map P∆ (C, C ′ ) is given by (iii) Composition is determined by the composition laws on Fin * and on P in an obvious way. There is a canonical simplicial functor P ∆ → Fin * which sends ( n , (C 1 , . . . , C n )) to n . If P is fibrant, the map of simplicial nerves P := N(P ∆ ) → N(Fin * ) = Γ constitutes an ∞-operad (cf. [26, 2.1.1.27]). We shall refer to N(P ∆ ) → N(Fin * ) = Γ (or N(P ∆ )) as the operadic nerve of P . We shall denote by P n the fiber P × Γ { n } over n . We usually identify colors with objects in P 1 .
• Let Cyl be the operadic nerve of Cyl.
• Let Cyl be the operadic nerve of Cyl.
• Let DCyl be the operadic nerve of DCyl.
• Let DCyl be the operadic nerve of DCyl.
We now recall Kontsevich-Soibelman operad [22]. We refer to KS as Kontsevich-Soibelman operad. Let KS be the operadic nerve of KS (the notation is slightly diffrent from Introduction). We abuse terminology by referring to it as Kontsevich-Soibelman operad. In a nutshell, KS ⊂ DCyl is the maximal simplicial subcomplex spanned by vertices correponding to those tuples which do not contain the color C. It is not difficult to check that KS is equivalent to that of [22, 11.2] or [15]. In [11], a version of KS is called the cylinder operad. . Let E ⊗ 2 be the operadic nerve of D , which we shall refer to as the ∞-operad of little 2-cubes.
Remark 4.11. We have the diagram in Remark 4.6 and inlusions KS ⊂ DCyl, D ⊂ DCyl. These inclusions determine the following diagram of ∞-operads: Let E 1 be the simplicial operad of little 1-cubes. The definition is similar to the case of little 2-cubes (see e.g. [26, 5.1.0.3]). Namely, E 1 has a single color D 1 , and for a finite sequence (D 1 , . . . , D 1 ), the simplicial set Mult E1 (D ⊔n 1 , D 1 ) is defined to be the singular simplicial complex of the space Emb rec ((0, 1) ⊔n , (0, 1)) of rectilinear embeddings. The composition law and the unit are defined in the obvious way. Let E ⊗ 1 denote the ∞-operad of little 1-cubes, that is, the operadic nerve of E 1 .
Definition 4.12. Let E 1 be a simplicial colored operad defined as follows.
(i) The set of colors of E 1 has two elements which we denote by D 1 and M .
(ii) Let I = r • be a finite set and let {D 1 , M } I be a set of colors indexed by I, which is a map p : I → {D 1 , M }. We write D ⊔m 1 ⊔ M ⊔n for {D 1 , M } J when p −1 (D 1 ) (resp. p −1 (M )) has m elements (resp. n elements). Let Emb rec ((0, 1) ⊔m ⊔ (0, 1), (0, 1)) be the topological space of embeddings (0, 1) ⊔m ⊔ (0, 1) → (0, 1) such that the restriction to each component is rectilinear. We define Mult E1 (D ⊔m 1 ⊔ M, M ) to be the singular simplicial complex of the subspace The subspace consists of those rectilinear embeddings such that the restriction to (0, 1)×p −1 (M ) ≃ (0, 1) is shrinking. If n = 1, Mult E1 (D ⊔m (iv) The composition law is given by the composition of embeddings, and a unit map is the identity map.
Let E ⊗ 1 be the operadic nerve of E 1 .

4.2.
Following [26], we recall the notion of algebras over an ∞-operad. Let O → Γ be an ∞-operads. Let M ⊗ → Γ be a symmetric monoidal ∞-category whose underlying ∞-category is Let E ⊗ 1 → Γ be the ∞-operad of little 1-cubes with the natural projection. Let (BS 1 ) ∆ be the simplicial category having a single object * and Hom simplicial set Hom (BS 1 )∆ ( * , * ). The simplicial set Hom (BS 1 )∆ ( * , * ) is the simplicial complex of S 1 = R/Z, and the composition is induced by the ordinary multiplication S 1 × S 1 → S 1 . We denote by BS 1 the simplicial nerve of (BS 1 ) ∆ . It can also be regarded as the classifying space of S 1 in S. Let p : Let M ⊗ → Γ be a symmetric monoidal ∞-category. Though the above definition of algebra objects is not applicable to E ⊗ 1 × BS 1 → Γ, we define Alg E ⊗ 1 ×BS 1 (M ⊗ ) as follows (cf. [26, 2.3.3.20]). Let ρ i : n → 1 be the unique inert morphism which sends i ∈ n to 1 ∈ 1 . Then Alg E ⊗ 1 ×BS 1 (M ⊗ ) is the full subcategory of Fun Γ (E ⊗ 1 × BS 1 , M ⊗ ) spanned by those maps F : E ⊗ 1 × BS 1 → M ⊗ satisfying the condition: If C is an object of E ⊗ 1 × BS 1 lying over n , and for 1 ≤ i ≤ n α i : C → C i is a locally p-coCartesian morphism covering ρ i : Proof. We prove that there is an isomorphism of simplicial sets M)). Observe that the symmetric monoidal ∞-category Fun(BS 1 , M) ⊗ is defined by the following universal property: for a simplicial set K, there is a natural bijection of Hom Set∆ (K, Fun(BS 1 , M) ⊗ ) with the set of pairs (α, β) which makes the diagram commute The assignment (α, β) → α induces Fun(BS 1 , M) ⊗ → Γ. Therefore, for a simplicial set L, a map L → Alg E1 (Fun(BS 1 , M) ⊗ ) amounts to a map f : BS 1 × L × E ⊗ 1 → M ⊗ over Γ such that for any vertex (a, l) in BS 1 × L and for any inert morphism i in E ⊗ 1 , the image f ((a, l, i)) is an inert morphism in M ⊗ (note also that by construction BS 1 has a single vertex). Next, we consider the universal property of Alg E ⊗ 1 ×BS 1 (M ⊗ ). By the observation before this Lemma, for a simplicial set L, a map commutes and for any vertex (l, a) in L × BS 1 and for any inert morphism i in E ⊗ 1 , the image g((l, i, a)) is an inert morphism in M ⊗ . Comparing universal properties of Alg E1 (Fun(BS 1 , M)) and The final assertion also follows from an argument similar to this proof. ✷ Construction 4.15. We will define a functor E ⊗ 1 × BS 1 → Cyl over Γ. To this end, we consider the following simplicial categories (E 1 × BS 1 ) ∆ and Cyl ∆ . Let (E 1 × BS 1 ) ∆ be a simplicial category defined as follows.
Observe that Mult Cyl (C ⊔n ⊔ C M , C M ) is the singular complex of the space which is homeomorphic to . is an equivalence of ∞-categories.
Proof. It follows from Lemma 4.14, Proposition 4. 16 is an equivalence. We first prove (ii). Let z 1 : (E ⊗ 1 × BS 1 ) 1 → Cyl 1 be the map of fibers over 1 . Both fibers consist of a unique object (here we denote it by * whose mapping space Map( * , * ) is (homotopy) equivalent to S 1 ). Taking into account our construction of Z : (E 1 × BS 1 ) ∆ → Cyl ∆ , we see that z 1 is a homotopy equivalence BS 1 → BS 1 . This proves (ii). Next we will prove (i). Let p : E ⊗ 1 × BS 1 → Γ be the projection. Let Tup n be the subcategory of Γ / n whose objects are active morphisms m → n and whose morphisms are equivalences. According to a criterion [26, 2.3.3.14], to prove (i), it is enough to prove that for any X ∈ E ⊗ 1 × BS 1 with n = p(X), z induces a weak homotopy equivalence ). Note that both domain and target are ∞-categories. Unwinding mapping spaces, we see that both ∞categories are Kan complexes. Consequently, it will suffice to show that u is a categorical equivalence. Clearly, u is essentially surjective. We prove that u is fully faithful. The general case is essentially the same as the case n = 1 except for a more complicated notation, so that we treat the case of n = 1. We think of D 1 as the unique object of (E ⊗ 1 × BS 1 ) 1 . Also, we write D m 1 for the unique object of the fiber (E ⊗ 1 × BS 1 ) m over m (namely, D 1 = D 1 1 ). Let f : D m 1 → D 1 be a map in E ⊗ 1 × BS 1 lying over an active morphism α : m → 1 of Γ. We regard f as the product g × h : (0, 1) ⊔m × S 1 → (0, 1) × S 1 of a rectilinear map φ : (0, 1) ⊔m → (0, 1) and a rectilinear map h : Here Σ m denotes the symmetric group (which comes from permutations of components). Thus, the mapping space is contractible because (S 1 × Σ m ) → (S 1 × Σ m ) is an equivalence. Next, we regard the color C in Cyl as an object in Cyl that lies over 1 . We denote by C m the unique object of Cyl that lies over m . Let z(f ), z(f ′ ) : C m → C be the images of f and f ′ respectively. Then we have equivalences in S Cyl (C m , C m ) is the full subcategory of Map Cyl (C m , C m ) spanned by equivalences. It follows from the canonical equivalence ( is an equivalence. We conclude that u is a categorical equivalence. . Moreover, by Corollary 4.18, the ∞-category on the left-hand side is equivalent to M)). In particular, we have an equivalence of ∞-categories This equivalence commutes with projections to Alg E2 (M) in the natural way.
The proof of Proposition 4.19 requires Lurie-Barr-Beck theorem [26, 4.7.4.16]. Let us consider the comutative diagram Proof of Proposition 4.19. We will prove the conditions (i), is an equivalence if and only if the evaluations at D, C, C M are equivalences. Similarly, a morphism f in Alg DCyl (M) is an equivalence if and only if the evaluations at objects D, C are equivalences. It follows that G is conservative. Similarly, we see that G ′ is conservative. Hence (iv) is proved. The conditions (ii) and (iii) follow from the existence and the compatibility of sifted colimits [26, 3.2.3.1] and the conservativity in (iv). (If M ⊗ is a presentably symmetric monoidal ∞-category, the condition (i) follows from adjoint functor theorem since G and G ′ preserve small limit and filtered colimit. ) We prove the condition (v). For this purpose, we first consider the left adjoint F ′ of G ′ . The values under F ′ can be described in terms of operadic colimits if we assume the existence of a left adjoint F ′ . Let j : Cyl → Cyl be the canonical inclusion. Let j ′ : Triv ⊗ → Cyl be a morphism from the trivial ∞-operad to Cyl that is determined by C M . Let Cyl ⊞ Triv ⊗ denote the coproduct of Cyl and Triv ⊗ . Namely, it is a coproduct of Cyl and Triv ⊗ in the ∞-category of ∞-operads, but we use its explicit construction in [26, 2.2.3.3]. By the universal property, the morphism j and j ′ induces k : Cyl ⊞ Triv ⊗ → Cyl. By [26, 3.1.3.5], we have an adjoint pair Here we use the canonical categorical equivalence Alg Triv (M) ∼ → M. There are categorical fibrations Thus (k ! , k * ) is an adjunction relative to Alg Cyl (M). See [26, 7.3.2.2] for the notion of relative adjunctions. The base change of (k ! , k * ) along Alg DCyl (M) → Alg Cyl (M) gives rise to an adjunction Let Cyl ⊞ Triv ⊗ → Cyl be a morphism determined by Cyl → Cyl and the morphism Triv ⊗ → Cyl classified by the object C M in the fiber Cyl 1 (by using the same symbol C M we abuse notation). Let Cyl act be the subcategory spanned by those morphisms whose images in Γ are active be an object of Cyl ⊞ Triv ⊗ lying over n such that ( n , S, T ) is an object of Sub, C n−1 is the unique object of Cyl n−1 (lying over n − 1 ), and ⋆ ∈ Triv = Triv ⊗ 1 . This presentation is based on the explicit construction of the coproducts in [26, 2.2.3.3]. For our purpose below, we may assume that T ⊂ n is of the form T = { * , i} so that by default T in the above object is of the form T = { * , i}. The mapping space from ( n , S, T, C n−1 , ⋆) to ( m , S ′ , T ′ , C m−1 , ⋆) is given by which we regard as an object in S, and * indicates the contractible space which we regard as the mapping space from ⋆ to ⋆. Using this description we consider mapping spaces in (Cyl ⊞ Triv ⊗ ) act /CM . We abuse notation by writing ( n , S, T, C n−1 , ⋆, (C n−1 , C M ) → C M ) for an object of K, where ( n , S, T, C n−1 , ⋆) ∈ Cyl ⊞ Triv ⊗ and (C, . . . , C, C M ) = (C n−1 , C M ) → C M is a morphism in Cyl act lying over the active morphism n → 1 where (C n−1 , C M ) is a sequence of n − 1 C's and a single C M which we regard as an object in Cyl n . Now it is easy to compute the mapping space from H = ( n , S, T, C n−1 , ⋆, (namely, the restricition to the "right component" (0, 1) × S 1 is shrinking). Consider the restriction ((0, 1) × S 1 ) ⊔n−1 → (0, 1) × S 1 and its projection f : (0, 1) ⊔n−1 → (0, 1) obtained by forgetting the S 1 -factor, which is a rectilinear embedding. If we denote by D n 1 the unique object in the fiber (E ⊗ 1 ) n over n ∈ Γ, we can regard f as a map D n−1 Taking account of definitions of ∞-operads Cyl ⊞ Triv ⊗ and Cyl, we see that is an equivalence in S. Let L be the full subcategory of K spanned by the single object (a morphism j is uniquely determined up to homotopy). We now claim that L ⊂ K is cofinal. It will suffice to prove that for each V ∈ K, the ∞-category L × K K V/ is weakly contractible, see [26, 4.1 By the above discussion about mapping spaces, a morphism Indeed, colim p ′′ | K is equivalent to B ⊗ M ⊗ S 1 (this computation is not necessary to the proof so that the reader may skip this paragraph, but it may be helpful to get feeling for the operadic left Kan extension F ′ ). By construction, the composite L ֒→ K Another way to compute it is as follows. By Corollary 4.18, we have Alg Cyl (M) ≃ Alg As (Fun(BS 1 , M)) and Alg Cyl (M) ≃ LMod(Fun(BS 1 , M)). These equivalences commute with forgetful functors arising from the inclusions As → LM and Cyl → Cyl. The adjunction (k ! , k * ) can be identified with the composite of adjunctions Next we will consider F (A, B, M ). Let r : DCyl ⊞ Triv ⊗ → DCyl be a morphism of ∞-operads induced by DCyl ֒→ DCyl and Triv ⊗ → DCyl determined by C M ∈ DCyl 1 correpondings to the color C M (we slightly abuse notation again). By We infomally denote by ( n , S, T, D d , C c , ⋆) an object of DCyl ⊞ Triv ⊗ , where ⋆ ∈ Triv, and (D d , C c ) indicates the sequence of colors which consists of d D's and c C's which we regard as an object in DCyl n−1 (d + c = n − 1). By abuse of notation, we write for an object of P , where f : (D d , C c , C M ) → C M is a morphism in DCyl act that lies over the active morphism n → 1 . We compute the mapping space from R to another object Given a morphism φ : In this way, we obtain the induced morphism As in the case of K, the restriction to C M gives rise to a morphism It gives rise to an equivalence in S: Let Q ⊂ P be the full subcategory spanned by Z which we think of as an object of P in the obvious way. As in the case of K, using the above description of Map P (R, R ′ ) we see that for any V ∈ P , Q × P P V/ is weakly contractible so that Q ⊂ P is cofinal. Let e ′ : P ⊲ → M ⊗ be an operadic q-colimit diagram that extends e. Let e ′′ : P ⊲ → M = M 1 be the diagram obtained by a q-coCartesian natural transformation from e ′ . Then the image of the cone point under e ′ is colim e ′′ | P ≃ colim e ′′ | Q . (We can also deduce that is an equivalence since evaluations at D, C, and the projection to M are equivalences. This proves (v). We also have proved the existence of F and F ′ , that is, (i). The final assertion is clear.  By the comparison of operadic q-colimit diagrams from K ⊲ and L ⊲ , the counit map of this adjunction is an equivalence. Therefore, we obtain a categorical equivalence Alg KS (M) ≃ Alg D DCyl (M) induced by j ! (or j * ). ✷

Hochschild cohomology
In this Section, we recall Hochschild cohomology spectra of stable ∞-categories C. The definition is based on the principle that, under a suitable condition on C, Hochschild cohomology of C is the endomorphism algebra of the identity functor C → C. Moreover, since Fun(C, C) has the monoidal structure given by the composition, Hochschild cohomology is the endomorphism algebra of the unit object of Fun(C, C) so that it comes equipped with the structure of an E 2 -algebra, cf. [3], [21] (see also references cited in loc. cit. for Deligne conjecture concerning Hochschild cochains). We establish some notation. Let R be a commutative ring spectrum. Let Mod ⊗ R be the symmetric monoidal ∞-category of R-module spectra whose underlying category we denote by Mod R . Let Alg As (Mod R ) be the ∞-category of the associative algebra objects in Mod R . Let Pr L be the ∞-category of presentable ∞-categories whose morphisms are those functors that preserve small colimits. This category Pr L admits a symmetric monoidal structure, see [26, 4.8.1.15, 4.8.1.7]. The ∞-category of small spaces S is a unit object in Pr L . For D, D ′ ∈ Pr L , the tensor product D ⊗ D ′ comes equipped with a functor D × D ′ → D ⊗ D ′ which preserves small colimits separately in each variable and satisfies the following universal property: for any F ∈ Pr L , the composition induces a fully faithful functor ) whose essential image is spanned by those functors D × D ′ → F which preserves small colimits separately in each variable, where Fun L (−, −) indicates the full subcategory of Fun(−, −) spanned by those functors which preserves small colimits. The underlying associative monoidal ∞-category Mod ⊗ R can be regarded as an associative algebra object in Pr L since Mod R is presentable and the tensor product functor Mod R × Mod R → Mod R preserves small colimits separately in each variable. We denote by is an equivalence of ∞-categories (for example, apply Lurie-Barr-Beck theorem to this functor endowed with projections to Sp) so that the notation RMod A is consistent with that of [26]. The category RMod A has a natural left module structure In what follows, when we treat the tensor product of objects in Mod R (over R), we write ⊗ for ⊗ R . The assignment A → RMod A gives rise to a functor which sends A to RMod A and carries a morphism f : A → B to the base change functor RMod A → RMod B ; N → N ⊗ A B, that is, a left adjoint of the forgetful functor RMod B → RMod A , see [26, 4.8.3, 4.8.5.10, 4.8.5.11]. We have the induced functor which sends A to the base change functor Mod R = RMod R → RMod A . The functor I is fully faithful and admits a right adjoint E. A morphism f : an essentially unique way (up to a contractible space of choices). Therefore, an object of LMod Mod ⊗ R (Pr L ) Mod R / is regarded as a pair (D, D) such that D belongs to LMod Mod ⊗ R (Pr L ) and D is an object of D. The essential image of I can naturally be identified with Alg As (Mod R ). Namely, it consists of pairs of the form (RMod A , A): I carries A to (RMod A , A). Put another way, the essential image is spanned by pairs (D, D) such that D is a compactly generated stable ∞-category equipped with a single compact generator D. The right adjoint E sends (D, D) to an endormorphism algebra object End(D) ∈ Alg As (Mod R ) [26, 4.8.5.11]. Since the left adjoint I is fully faithful, the unit map id → E • I is a natural equivalence. Namely, the adjunction ( The functor I is extended to a symmetric monoidal functor. To explain this, note that Alg As (Mod R ) comes equipped with a symmetric monoidal structure induced by that of Mod ⊗ R , see [26, 3.2.4] or Construction 7.9. Since Mod ⊗ R is a symmetric monoidal ∞-category such that Mod R has small colimits and the tensor product functor Mod R × Mod R → Mod R preserves small colimits separately in each variable, we define ) inherits a symmetric monoidal structure. In summary, we have the adjunction whose left adjoint is symmetric monoidal and fully faithful, and whose right adjoint is lax symmetric monoidal. It gives rise to an adjunctuion Alg E2 (Mod R ) ≃ Alg As (Alg As (Mod R )) (we can also use additivity theorem to the equivalence on the right-hand side).
We refer to an object of Pr L R := Mod Mod ⊗ R (Pr L ) as an R-linear presentable ∞-category. Note that the underlying ∞-category of an R-linear presentable ∞-cartegory is stable.
be the tensor product functor. There exists a morphism object from D to itself (i.e., an internal hom object) Proof. According to [26, 4.7.1.40, 4.7.1.41], the second assertion follows from the first assertion. We will show the existence of a morphism object Mor R (D, D). Recall that a morphism object for D and D ′ is an R-linear presentable ∞-category C together with a morphism C ⊗ R D → D ′ such that the composition induces an equivalence constitutes a morphism object for D and D ′ . To prove that is an equivalence, we may and will assume that P is a free object since the tensor operation functor ⊗ R preserves small colimits separately in each variable (see the proof of [26, 5.1.2.9]), and P is a (small) colimit of the diagram of free objects: for example, using the adjunction (F, U ) we have a simplicial diagram of free objects whose colimit is P. When P = C ′ ⊗ Mod R , by the adjunction we see that θ is an equivalence. We put D = colim i∈I D i where each D i is a free object. Then for any P ∈ Pr L R there exist natural equivalences Hence there exists a morphism object which we regarded as an object in S. By the above proof, Mor R (D, D ′ ) ≃ is equivalent to the mapping space Map Pr L R (D, D ′ ). We shall write End R (D) for E ∈ Alg As (Pr L R ).
Definition 5.3. Let D be an R-linear presentable ∞-category. Applying E : Alg As (Pr L R ) → Alg E2 (Mod R ), we define the Hochschild cohomology R-module spectrum of D to be . We often abuse notation by identifying HH • R (D) with its image in Mod R . If no confusion can arise, we write HH • (D) for HH • R (D).
Let St be the ∞-category of small stable idempotent-complete ∞-categories whose morphisms are exact functors. Let C be a small stable idempotent-complete ∞-category and let Ind(C) denote the ∞-category of Ind-objects. Then Ind(C) is a compactly generated stable ∞-category. The inclusion C → Ind(C) identifies the essential image with the full subcategory Ind(C) ω spanned by compact obejcts in Ind(C). Given C, C ′ ∈ St, if we write Fun ex (C, C ′ ) for the full subcategory spanned by exact functors, the left Kan extension [25, 5.3.5.10] gives rise to a fully faithful functor Fun ex (C, C ′ ) → Fun L (Ind(C), Ind(C ′ )) whose essential image consists of those functors that carry C to C ′ . We set Pr L St = Mod Sp ⊗ (Pr L ), which can be regarded as the full subcategory of Pr L that consists of stable presentable ∞-categories. The assignment C → Ind(C) identifies St with the subcategory of Pr L St whose objects are compactly generated stable ∞categories, and whose morphisms are those functors that preserve compact objects. The ∞-category St inherits a symmetric monoidal structure from the structure on Pr L St . The stable ∞-category of compact spectra is a unit object in St. Given two objects C and C ′ of St, the tensor product C ⊗ C ′ is naturally equivalent to the full subcategory (Ind(C) ⊗ Ind(C ′ )) ω ⊂ Ind(C) ⊗ Ind(C ′ ) spanned by compact objects. Consider RMod A for A ∈ Alg As (Mod R ). We let RPerf A be the full subcategory of RMod A spanned by compact objects. This subcategory is the smallest stable subcategory which contains A (regarded as a right module) and is closed under retracts. When A belongs to CAlg(Mod R ), we write Perf A for RPerf A . In this case, Perf A is closed under taking tensor product so that it inherits a symmetric monoidal structure from that of Mod ⊗ A . We usually regard the symmetric monoidal ∞-category Perf ⊗ R as an object of CAlg(St), and we write St R for Mod Perf ⊗ R (St). We refer to an object of St R as a small R-linear stable ∞-category.
Definition 5.4. Given C ∈ St R , we define the Hochschild cohomology R-module spectrum HH • R (C) to be HH • R (Ind(C)). If no confusion can arise, we write HH • (C) for HH • R (C).

Hochschild homology
Let R be a commutative ring spectrum. Suppose that we are given a small R-linear stable ∞-category C. In this Section, we assign to C ∈ St R the Hochschild homology R-module spectrum HH • (C) ∈ Mod R . For the main purpose of this paper, we require the following additional structures: • the R-module spectrum HH • (C) has an action of the circle S 1 . Namely, HH • (C) is promoted to an object of Fun(BS 1 , Mod R ), and the assignment C → HH • (C) gives rise to a functor is equipped with a pointwise symmetric monoidal strcuture induced by that of Mod R , then the above functor St R → Fun(BS 1 , Mod R ) is promoetd to a symmetric monoidal functor from St R to Fun(BS 1 , Mod R ). To this end, we will use enriched models of stable idempotent-complete ∞-categories, i.e., spectral categories.
Symmetric spectra. We give a minimal review of the theory of symmetric spectra, introduced and developed in [17]. This theory provides a nice foundation of the homotopy theory of highly structured ring spectra as well as a theoretical basis for spectral categories. We let Sp Σ be the closed symmetric monoidal category of symmetric spectra. We write S for the unit object which we call the sphere spectrum. We use the notation slightly different from [17], [31]: S is S in [17]. We use a symmetric monoidal proper combinatorial model category structure on Sp Σ satisfying the monoid axiom in the sense of [ [31], it is proved that there is another model structure called the stable S-model structure. The difference (relevant to us) between stable model structure and stable S-model structure is that cofibrations in the stable model structure [17, 3.4.4] are contained in the class of cofibrations in the stable S-model structure while both have the same class of weak equivalences. Let CAlg(Sp Σ ) denote the category of commutative algebra objects in Sp Σ . We refer to an object of CAlg(Sp Σ ) as a commutative symmetric ring spectrum. The category CAlg(Sp Σ ) admits a model category structure: we use the model structure on CAlg(Sp Σ ), defined in [31,Theorem 3.2] in which a morphism is a weak equivalence if the underlying morphism in Sp Σ is a stable equivalence. The stable S-model structure on Sp Σ has the following pleasant property: if R is a cofibrant object in CAlg(Sp Σ ), then the underlying object R in Sp Σ is cofibrant with respect to the stable S-model structure, see [31,Section 4].
Let R be a commutative symmetric ring spectrum, which we think of as a model of R ∈ CAlg(Sp). Unless otherwise stated, we assume that R is cofibrant in CAlg(Sp Σ ). We let Sp Σ (R) denote the category of Rmodule objects in Sp Σ , which is endowed with the natural symmetric monoidal structure induced by the structure on Sp Σ . In virtue of [31, Theorem 2.6] (or [28, Theorem 4.1]), there is a combinatorial symmetric monoidal projective model structure on Sp Σ (R) satisfying the monoid axiom, in which a morphism is a weak equivalence (resp. a fibration) if the underlying morphism in Sp Σ is a stable equivalence (resp. a fibration with respect to stable S-model structure). We refer to this model structure as the stable R-model structure.
Definition 6.1. Let R be a commutative symmetric ring spectrum. An R-spectrum category is a category enriched over Sp Σ (R). More explicitly, a (small) R-spectrum category A consists of the data: • A (small) set of objects, satisfying the standard associativity axiom, • S → A(X, X) for each object X that satisfies the standard unit axiom.
Here ∧ R denotes the wedge product over R, which defines the tensor product in Sp Σ (R). A functor of R-spectral categories is an enriched functor, that is, a functor as enriched categories. We refer to them as R-spectral functors. We write Cat R for the category of R-spectral categories whose morphisms are R-spectral functors. We refer to an S-spectral category (resp. an S-spectral functor) as a spectral category (resp. a spectral functor). We write ∧ for ∧ S . Let R be a commutative symmetric ring spectrum. Let us recall the tensor product of R-spectral categories. Suppose that we are given A, B ∈ Cat R . The tensor product A ∧ R B is defined by the following data: • The set of objects of A ∧ R B is the set of pairs (A, B) where A is an object of A, and B is an object of B, This tensor product determines a symmetric monoidal structure on Cat R . A unit object is defined as follows: Let BR be the spectral category which has a single object * together with the morphism ring spectrum BR( * , * ) = R. The composition R ∧ R → R and the unit S → R are determined by the algebra structure on R in an obvious way. Clearly, BR is a unit object in Cat R . Since R is commutative, we can also think of BR as a symmetric monoidal spectral category. Namely, it is a commutative algebra object in the symmetric monoidal category Cat S . Note that an R-spectral category A is regarded as a BR-module in Sp Σ . Namely, there is a canonical equivalence of categories Cat R ∼ → Mod BR (Cat S ) where the target is the category of BR-module objects in Cat S .
For technical reasons, we use the notion of pointwise-cofibrant spectral categories, cf. [6,Section 4]. We say that an R-spectral category A is pointwise-cofibrant if each morphism spectrum A(X, Y ) is cofibrant in Sp Σ (R) with respect to the stable R-model structure. Using the same argument as that in the proof in [6, Proposition 4.1], we have: (i) Every R-spectral category is functorially Morita equivalent to a pointwisecofibrant R-spectral category with the same objects.
(ii) The subcategory of pointwise-cofibrant R-spectral category is closed under the tensor product.
(iii) If A is a pointwise-cofibrant R-spectral category, the tensor operation A ∧ R (−) preserves Morita equivalences and colimits. (iv) If A and B are both pointwise-cofibrant R-spectral categories, then the A ∧ R B computes the derived tensor product.
We denote by Cat pc S the category of small pointwise-cofibrant spectral categories. By Proposition 6.3, Cat pc S admits a symmetric monoidal structure given by tensor products, and the tensor products preserves Morita equivalences in each variable. Similarly, We denote by Cat pc R the category of small pointwisecofibrant R-spectral categories.
Inverting morphisms. We recall the notion of ∞-categories obtained from an ∞-category endowed with a set of morphisms. We refer the readers to [ whose essential image consists of those functors F : C → D which carry edges in S to equivalences in D. We shall refer to C[S −1 ] as the ∞-category obtained from C by inverting S. We note that C[S −1 ] is generally not locally small even when C is so. When C is an ordinary category, an explicit construction of C[S −1 ] is given by the hammock localization [10]. Let C ⊗ be a symmetric monoidal ∞-category. Let S be a set of edges in C such that all equivalences are contained in S. Assume that for any object C ∈ C and any morphism C 1 → C 2 in S, the induced morphisms C ⊗ C 1 → C ⊗ C 2 and C 1 ⊗ C → C 2 ⊗ C belong to S. Then there exists a symmetric monoidal ∞-category C[S −1 ] ⊗ together with a symmetric monoidal functorξ : C ⊗ → C[S −1 ] ⊗ whose underlying functor is equivalent to ξ. There is a universal property: for any symmetric monoidal ∞-category D ⊗ the composition induces a fully faithful functor Example 6.4. Let Sp Σ (R) c be the full subcategory that consists of cofibrant objects. The tensor product ∧ R given by the wedge product over R preserves cofibrant objects. In addition, if C ∈ Sp Σ (R) c and f : C 1 → C 2 is a weak equivalence (i.e., stable equivalence), then C ∧ R f is a weak equivalence.
Let Alg As (Sp Σ (R) c ) be the the category of associative algebra objects in Sp Σ (R) c , which is endowed with the symmetric monoidal structure induced by that of Sp Σ (R) c . Then if Alg As (Sp Σ (R) c )[W −1 ] ⊗ denotes the associated symmetric monoidal ∞-category obtained by inverting weak equivalences, then we have equivalences of symmetric monoidal ∞-categories where the left equivalence follows from the rectification result [26, 4.1.8.4]. In particular, given an associative algebra A ∈ Alg As (Mod R ), there is an associative algebra A ∈ Alg As (Sp Σ (R) c ) together with an equivalence σ : A ≃ A in Alg As (Mod R ). In this case, we say that A (together with σ) represents A. where Mon is the class of monomorphisms of symmetric sequences, and S⊗ i denotes the morphism of symmetric spectrum induced by i, namely, S⊗ (−) is the left adjoint of the forgetful functor from Sp Σ to the category of symmetric sequences, see [31]. The class of cofibrations in Sp Σ (R) with respect to the stable R-model structure is the smallest weakly saturated class of morphisms containing {R ⊗ i = R ∧ (S ⊗ i)} i∈Mon . Note that we assume that R is a cofibrant object in CAlg(Sp Σ ) so that the underlying object R is cofibrant in Sp Σ . It follows that the underlying morphisms R ⊗ i in Sp Σ are cofibrations. Since Sp Σ (R) → Sp Σ preserves colimits, Sp Σ (R) → Sp Σ preserves cofibrations.
where π(A) ⊗ R π(B) indicates the tensor product in Mod PerfR (St). By construction, the morphism Perf R → π(BR) is an equivalence. We prove that π(A) ⊗ R π(B) → π(A ∧ R B) is an equivalence. We here write A pe for the R-spectral full subcategory of Fun R (A op , Sp Σ (R)) cf that consists of cofibrantfibrant objects lying over D pe (A) R ( A pe is DK-equivalent to D Σ pe (A) as spectral categories, see the proof of Lemma 6.11 and Claim 6.11.1 for the notation). For any other R-spectral category P, we define P pe in the same way. By Claim 6.11.1 below, the image of (A ∧ R B) pe in Mod BR (Cat pc S [M −1 ]) is equivalent to π(A ∧ R B) in the natural way. Let {A λ } λ∈Λ be the filtered family (poset) of R-spectral full subcategories of A such that for any A λ , D(A λ ) (or Fun R (A op λ , Sp Σ (R)) cf ) admits a single compact generator (so that ( A λ ) pe is Morita equivalent to BA for some A ∈ Alg As (Sp Σ (R) c ) where BA has one object * with the morphism ring spectrum BA( * , * ) = A). Then we have the filtered family (poset) of R-spectral full subcategories { (A λ ∧ R B) pe } λ∈Λ of (A ∧ R B) pe . The filtered colimit of this family In addition, we note that the tensor product in Mod Perf R (St) preserves small colimits in each variable since St ⊗ is a symmetric monoidal compactly generated ∞-category whose tensor product preserves small colimits in each variable (see [5, 4.25]). Therefore, taking into account Proposition 6.3 (3), we may and will suppose that A = BA for some A ∈ Alg As (Sp Σ (R) c ). Taking the same procedure to B, we may and will suppose that B = BB for some B ∈ Alg As (Sp Σ (R) c ). We write A and B for the images of A and B in Alg As (Mod R ) respectively. In this situation, we have a canonical equiva- Using the equivalence in Proposition 6.7, we obtain equivalences of symmetric monoidal ∞-categories Next, to an R-spectral category we assign Hochschild homology R-module spectrum endowed with circle action. The construction is based on the Hochschild-Mitchell cyclic nerves (cf. [7], [6]).  We let Fun(Λ op , Sp Σ (R)) denote the ordinary functor category from Λ op to Sp Σ (R). The category Fun(Λ op , Sp Σ (R)) inherits a symmetric monoidal structure given by the pointwise tensor product From the definition of the tensor product of R-spectral categories and the construction of HH(A) p , it is straightforward to check that the assignment A → HH(A) • determines a symmetric monoidal functor The image of HH(−) • is contained in Fun(Λ op , Sp Σ (R) c ) since the stable R-model structure satisfies the axiom of symmetric monoidal model categories. Let Sp Σ (R) c [W −1 ] be the symmetric monoidal ∞category obtained from Sp Σ (R) c by inverting stable equivalences. The underlying ∞-category is presentable since Sp Σ (R) is a combinatorial model category. There is a canonical symmetric monoidal functor We recall the following results from [9], [23], [18]: Lemma 6.9.
(i) Let Λ → Λ be the groupoid completion. Namely, it is induced by a unit map of the adjunction Cat ∞ ⇄ S : ι where ι is the fully faithful inclusion. Then Λ is equivalent to BS 1 in S.
(ii) Let C be a presentable ∞-category. Let F : Λ op → C be a cyclic object in C. Let F ′ : BS 1 → C be a functor. Let ∆ 0 → BS 1 be the map determined by the unique object of BS 1 . Consider the commutative diagram where we regard ∆ 0 as the groupoid completion of ∆ op (∆ op is sifted so that the groupoid completion is given by the contractible space ∆ 0 , cf. Proof. Since the right adjoint is a symmetric monoidal functor, the left adjoint is an oplax symmetric monoidal functor. Thus it is enough to show that To this end, note first that L(F ), L(G) : are given by left Kan extensions of F and G respectively along Λ op → BS 1 . By Lemma 6.9 (ii), F ′ : . On the other hand,  Proof. It will suffice to prove that H(F ) is an equivalence in Mod R . By [7, 5.9, 5.11], the image of is an equivalence if F : A → B is a Morita equivalence over R. We explain the notion of a Morita equivalence over R, which we distinguish from the notion of Morita equivalences for the moment. Let Fun R (A op , Sp Σ (R)) be the R-spectral category of R-spectral functors. As in the case of R = S, it admits a combinatorial R-spectral model structure whose weak equivalences (resp. fibrations) are objectwise stable equivalences (resp. fibrations). Let D(A) R denote the homotopy (triangulated) category of the full subcategory Fun R (A op , Sp Σ (R)) cf spanned by cofibrant and fibrant objects. Let D pe (A) R be the smallest thick subcategory that contains the image of the Yoneda embedding A → D(A) R . We define D pe (B) R in a similar way. The functor F induces (LF ! ) R : . We say that F is a Morita equivalence over R if (LF ! ) R is an equivalence. Thus to prove our assertion, it is enough to show the following claim: Proof. We let D Σ pe (A) R be the full subcategory of Fun R (A op , Sp Σ (R)) cf spanned by those objects that belongs to D pe (A) R . The Yoneda emebedding I : . By [5,Proposition 4.11], we deduce that the canonical functor Proof. It follows from the universal property of Cat pc R → Cat pc R [M −1 ] and Lemma 6.11. ✷ Composing with θ, we obtain a sequence of symmetric monoidal functors Definition 6.13. Let C be a small R-linear stable ∞-category. We denote by HH • (C) the image of C in Fun(BS 1 , Mod R ) under the above composite H • θ. We often abuse notation by writing HH • (C) for its image in Mod R . We refer to HH • (C) as Hochschild homology R-module spectrum of C. If A is a pointwise-cofibrant R-spectral category, we refer to the image H(A) in Fun(BS 1 , Mod R ) or Mod R as Hochschild homology R-module spectrum of A.
We record our construction as a proposition: Proposition 6.14. There is a sequence of symmetric monoidal functors which to R-linear stable ∞-categories or pointwise-cofibrant R-spectral categories assigns Hochschild homology R-module spectra. In particular, for any ∞-operad O it gives rise to

Construction
In this Section, we prove Theorem 7.14. Namely, we construct the structure of a KS-algebra on the pair of Hochschild cohomology spectrum and Hochschild homology spectrum. We maintain the notation of Section 6. be the the fibrant simplicial colored operad whose set (Mfld rec 1 ) col of colors consists of (possibly empty) finite disjoint unions of (0, 1) and S 1 . For a finite family {M i } i∈I of colors and N ∈ (Mfld rec 1 ) col , the simplicial hom set Mult Mfld rec 1 ({M i } i∈I , N ) is defined to be the singular complex of the space Emb rec (⊔ i∈I M i , N ) of rectilinear embeddings. The composition is defined in the obvious way. Then from Definition 4.7, we obtain the associated ∞-operad (Mfld rec 1 ) ⊗ → Γ, which is a symmetric monoidal ∞-category by construction. Informally, objects of this symmetric monoidal ∞-category are finite disjoint unions of (0, 1) and S 1 , and the symmetric monoidal structure is given by disjoint union. The empty space is a unit. The mapping spaces are spaces of rectilinear embeddings. Let Mfld rec 1 denote the underlying ∞-category. Let Disk rec 1 ⊂ Mfld rec be the full subcategory spanned by finite disjoint unions of (0, 1). It is closed under taking tensor products so that Disk rec 1 is promoted to a symmetric monoidal ∞-category (Disk red 1 ) ⊗ (it is equivalent to an ordinary symmetric monoidal 1-category). Remark 7.2. There are several variants which are equivalent to Mfld rec 1 . Let Mfld f r 1 be the ∞-category of framed (or oriented) 1-manifolds without boundaries whose mapping spaces are spaces of embeddings of framed manifolds (see e.g. [1, Section 2]). The symmetric monoidal structure is given by disjoint union. It is easy to see that there is an equivalence Mfld rec 1 ∼ → Mfld f r 1 as symmetric monoidal ∞-categories. If we write Disk f r 1 for the full subcategory of Mfld f r 1 spanned by framed 1-disks, it also induces an equivalence Disk rec 1 ∼ → Disk f r 1 of symmetric monoidal ∞-categories. From now on, for ease of notation, we write Mfld 1 , Disk 1 , and Disk ⊗ 1 for Mfld rec 1 , Disk rec 1 , and (Disk rec 1 ) ⊗ respectively.
We set (Disk 1 ) /S 1 := Disk 1 × Mfld 1 (Mfld 1 ) /S 1 . Let S 1 be the full subcategory of Mfld 1 that consists of S 1 . By the equivalence Emb rec (S 1 , S 1 ) ≃ S 1 , it follows that S 1 is equivalent to BS 1 , that is, the ∞-category which has one object * together with the mapping space Map BS 1 ( * , * ) = S 1 endowed with the composition law induced by the multiplication of S 1 . Let Disk 1 / S 1 be the full subcategory of Fun(∆ 1 , Mfld 1 ) which consists of those functors h : ∆ 1 → Mfld 1 such that h(0) ∈ Disk 1 and h(1) ∈ S 1 . In other words, where the functor from Fun(∆ 1 , Mfld 1 ) to the left Mfld 1 (resp. the right Mfld 1 ) is induced by the restriction to the source (resp. the target). The projection where the left categorical equivalence follows from [25, 4.2.1.5]. Lemma 7.3. Let Disk † 1 be the full subcategory of Disk 1 spanned by nonempty spaces (namely, the empty space is omitted from Disk 1 ). We set Disk † 1 / S 1 = Disk † 1 × Disk 1 (Disk 1 / S 1 ). Let Λ be the cyclic category of Connes [9, section 2]. There is an equivalence of categories Λ op ≃ Disk † 1 / S 1 . Proof. This is a comparison between definitions which look different. We first recall that objects of Λ are (p) for p ≥ 0, which is denoted by Λ p in [9]. Let (S 1 , p) be the circle S 1 = R/Z equipped with the set of torsion points 1 p+1 Z/Z. The hom set Hom Λ ((p), (q)) is defined to be the set of homotopy classes of monotone degree one maps φ : Given (p) ∈ Λ, we think of j p : (R/Z)\( 1 p+1 Z/Z) = (I 0 p ⊔ . . . ⊔ I p p ) ֒→ R/Z = S 1 as an object of Disk † 1 / S 1 . We fix I i p ≃ (0, 1) such that I i p ֒→ R/Z is equivalent to (0, 1) ֒→ R → R/Z. We write J(p) for it. We note that every object of Disk † 1 / S 1 is equivalent to J(p) for some p ≥ 0. Since each component of Map Mfld 1 ((0, 1) ⊔p+1 , S 1 ) is naturally equivalent to S 1 , the computation of mapping spaces shows that Disk † 1 / S 1 is equivalent to the nerve of a 1-category. We may and will abuse notation by identifying Disk † 1 / S 1 with its homotopy category. Suppose that we are given a monotone degree one map φ : . Consider a rectilinear embedding ) in (0, 1) (we here abuse notation: for two subsets S, T ⊂ (0, 1), S < T if s < t for any pair (s, t) ∈ S × T ). When p = 0, we define ι i,φ by replacing I . Such a rectilinear embedding is unique up to equivalences. Given φ ∈ Hom Λ ((p), (q)), we define the class of a map φ * : J(q) → J(p) in Disk † 1 / S 1 such that the fiber of the induced morphism I 0 q ⊔ . . . ⊔ I q q → I 0 p ⊔ . . . ⊔ I p p over the connected component is not a one-point space, and if otherwise there is no component which maps to I i p . Notice that such a class is unique. It is routine to check that the assignments (p) → J(p) and φ → φ * determine a categorical equivalence Λ op ∼ → Disk † 1 / S 1 where the target is identified with the homotopy category. ✷ Lemma 7.4. Let π : Disk † 1 / S 1 → S 1 be the projection. It is a groupoid completion of Disk † 1 / S 1 .
Proof. From Lemma 6.9 (i) and Lemma 7.3, there is a groupoid completion c : Λ op ≃ Disk † 1 / S 1 → BS 1 . Thus, by the universal property, there is a canonical morphism from c : Λ op → BS 1 to π : Λ op ≃ Disk † 1 / S 1 → S 1 in (Cat ∞ ) Λ op / . It will suffice to show that the induced morphism g : BS 1 → S 1 ≃ BS 1 is an equivalence, equivalently, it is induced by an equivalence S 1 → S 1 as E 1 -monoid spaces. To this end, assume that g : BS 1 → S 1 ≃ BS 1 is induced by a map S 1 → S 1 of degree n where |n| = p + 1, p > 0. We will show that this gives rise to a contradiction. The automorphism group of (p) in Λ op ≃ Disk † 1 / S 1 is Z/(p + 1)Z so that there is the functor h : BZ/(p + 1)Z → BS 1 induced by π. By the factorization Λ op c → BS 1 g → S 1 ≃ BS 1 and our assumption, h : BZ/(p + 1)Z → BS 1 factors through the canonical morphism ∆ 0 → BS 1 . Thus, the fiber product of BZ/(p On the other hand, the space/∞-groupoid in (Disk 1 ) /S 1 ≃ Disk 1 / S 1 × S 1 ∆ 0 spanned by J(p) : (0, 1) ⊔p+1 → S 1 (obtained by discarding non-invertible morphisms) is equivalent to BZ. It gives rise to a contradiction B(Z × Z/(p + 1)Z) ≃ BZ. ✷ Remark 7.5. There is another category relevant to the cyclic category: the paracyclic category Λ ∞ . Let us recall the definition of the paracyclic category. We follow [13]. The set of objects of Λ ∞ is {(0) ∞ , (1) ∞ , . . . , (p) ∞ , . . .} p≥0 . The hom set Hom Λ∞ ((p) ∞ , (q) ∞ ) is defined to be the set of monotonically increasing maps f : Z → Z such that f (i + k(p + 1)) = f (i) + k(q + 1) for any k ∈ Z. We define a functor Λ ∞ → Λ which carries (p) ∞ to (p). The map Hom Λ∞ ((p) ∞ , (q) ∞ ) → Hom Λ ((p), (q)) carries f to Here, we regard f (i) as belonging to Z/(q + 1)Z. This determines a functor Λ ∞ → Λ. Unwinding the definition of Λ ∞ → Λ, we see that it is a (homotopy) quotient morphism Λ ∞ → Λ ∞ /BZ ≃ Λ that comes from a free action of BZ on Λ ∞ . This free action of BZ is determined by the natural equivalence from the identity functor id Λ∞ to itself such that for any p ≥ 0, the induced map (p) ∞ → (p) ∞ is the map i → i + p + 1 (see [13] for details). The paracyclic category also has a geometric description. From the proof of Lemma 7.6 below, ∆ op → Λ ∞ is (left) cofinal so that it induces an equivalence between their groupoid completions. Since the groupoid completion of ∆ op is a contractible (note that it's sifted), the groupoid completion Λ ∞ of Λ ∞ is a contractible space. It follows that the geometric realization of Λ is equivalent to BBZ = BS 1 (see also Lemma 6.9 (i)). The composition with the opposite functor Consequently, we have the induced functor Λ op ∞ → (Disk † 1 ) /S 1 ≃ Disk † 1 / S 1 × S 1 ∆ 0 . This is an equivalence. Clearly, it is essentially surjective. The map Hom Λ op ∞ ((q) ∞ , (p) ∞ ) → Hom Λ op ((q), (p)) is a homotopy quotient map that comes from a free action of Z. We see that 6. Let C be a presentable ∞-category. Let Λ ∞ be the paracyclic category, see [13] or Remark 7.5. Let Λ op ∞ ≃ (Disk † 1 ) /S 1 → Disk † 1 / S 1 ≃ Λ op be the natural functor. Let f : Disk † 1 / S 1 → C and g : S 1 → C be functors and let f → π • g be a natural transformation. Then g is a left Kan extension of f along we suppose that f : Disk † 1 / S 1 → C is the restriction of a functorf : Disk 1 / S 1 → C, the above condition that g is a left Kan extension of f is also equivalent to the condition that ∆ 0 → BS 1 → C determines a colimit of the composite Proof. There is a faithful functor ∆ op → Λ op ∞ that is (left) cofinal [27, 4.2.8]: it is the same as the functor m in Remark 7.7. Thus, for any paracyclic object F : Λ op ∞ → C, the canonical morphism is an equivalence. Our first assertion now follows from this fact and Lemma 6.9 (ii).
To prove the second assertion, it will suffice to prove that ( weakly contractible where e is the map e : φ → S 1 from the empty space to S 1 . Since e is an initial object in (Disk 1 ) /S 1 , we are reduced to proving that (Disk † 1 ) /S 1 ≃ Λ op ∞ is weakly contractible. By Quillen's theorem A, it is clear because ∆ op is weakly contractible and ∆ op → Λ op ∞ is (left) cofinal (it follows also from the fact that (Disk † 1 ) /S 1 is sifted). ✷ Remark 7.7. We have the following commutative diagram of categories: It is straightforward to observe that the composite ((Disk † 1 ) /S 1 ) (0,1)→S 1 / → Λ op is a faithful and essentially surjective functor whose image is ∆ op contained in Λ op .
Let Mfld 1 be the simplicial nerve of (Mfld ic 1 ) ∆ . The simplicial nerves of the above diagrams give rise to a commutative diagram which lies over ∧ : Γ × Γ → Γ. We abuse notation by writing ρ for the associated map.
Given an ∞-operad O ⊗ → Γ, there exist a symmetric monoidal ∞-category Env(O ⊗ ) → Γ and a map O ⊗ → Env(O ⊗ ) of ∞-operads such that for any symmetric monoidal ∞-category D ⊗ , the composition induces a categorical equivalence [26, 2.2.4]. Here Fun ⊗ (Env(O ⊗ ), D ⊗ ) denotes the ∞-category of symmetric monoidal functors. We shall refer to Env(O ⊗ ) as the symmetric monoidal envelope of O ⊗ (in loc. cit., it is referred to as the Γ-monoidal envelope). Through the categorical equivalence, for a map of ∞-operads f : O ⊗ → D ⊗ , there exists a symmetric monoidal fucntorf : Env(O ⊗ ) → D ⊗ which is unique up to a contractible space of choices. We refer tof as a symmetric monoidal functor that corresponds to f . Let Oper ∞ be the ∞-category of (small) ∞-operads [26, 2.1.4] and let Cat ⊗ ∞ be the ∞-category of (small) symmetric monoidal ∞-categories whose morphisms are symmetric monoidal functors. Then the construction of symmetric monoidal envelopes gives a left adjoint Oper ∞ → Cat ⊗ ∞ of the canonical functor Cat ⊗ ∞ → Oper ∞ . Here are some examples. The symmetric monoidal envelope E ⊗ 1 of E ⊗ 1 is equivalent to Disk ⊗ 1 as symmetric monoidal ∞-categories. Similarly, a symmetric monoidal envelope E ⊗ 2 of E ⊗ 2 is equivalent to the symmetric monoidal ∞-category Disk ⊗ 2 of (possibly empty) finite disjoint unions of (0, 1) 2 defined as in the case of Disk ⊗ 1 : mapping spaces are spaces of rectilinear embeddings, and the tensor product is again given by disjoint union. Another quick example of symmetric monoidal envelopes is Mfld 1 → Mfld ⊗ 1 . Construction 7.9. Let q : C ⊗ → Γ be a symmetric monoidal ∞-category. Let p : P ⊗ → Γ be a symmetric monoidal ∞-category (resp. an ∞-operad). We construct a symmetric monoidal structure on the ∞-category Fun ⊗ (P ⊗ , C ⊗ ) of symmetric monoidal functors (resp. the ∞-category Alg P ⊗ (C ⊗ ) of algebra objects), see [26, 3.2.4] for more details of the case of Alg P ⊗ (C ⊗ ). We define a map Fun ⊗ (P ⊗ , C ⊗ ) ⊗ → Γ (resp. Alg ⊗ P ⊗ (C ⊗ ) → Γ) by the universal property that for any α : K → Γ, the set of morphisms K → Fun ⊗ (P ⊗ , C ⊗ ) ⊗ over Γ (resp. K → Alg ⊗ P ⊗ (C ⊗ ) over Γ) is defined to be the set of morphisms f : K × P ⊗ → C ⊗ such that (i) the diagram / / Γ commutes where the lower horizontal arrow is induced by ∧ : Γ × Γ → Γ, (ii) for any vertex k of K and any p-coCartesian edge φ in P ⊗ , f (k, φ) is a q-coCartesian edge (resp. for any vertex k of K and any inert morphism φ in P ⊗ , f (k, φ) is an inert morphism in C ⊗ ). The morphism Alg ⊗ P ⊗ (C ⊗ ) → Γ is a symmetric monoidal ∞-category whose underlying ∞-category is based on the theory of categorical patterns can also be applied to Fun ⊗ (P ⊗ , C ⊗ ) ⊗ . An edge ∆ 1 → Fun ⊗ (P ⊗ , C ⊗ ) ⊗ is a coCartesian edge if and only if for any X ∈ P, the composite ∆ 1 × {X} ⊂ ∆ 1 × P ⊗ → C ⊗ determines a q-coCartesian edge (this means that the tensor product F ⊗ G of two symmetric monoidal functors F : P ⊗ → C ⊗ and G : P ⊗ → C ⊗ is informally given by objectwise tensor products (F ⊗ G)(X) = F (X) ⊗ G(X)).
Let O ⊗ → Γ be an ∞-operad and let O ⊗ → Γ be the symmetric monoidal envelope. The composition with the inclusion O ⊗ → O ⊗ induces a map over Γ O ⊗ (C ⊗ ) which preserves coCartesian edges, namely, it is a symmetric monoidal functor. Since the underlying functor is an equivalence [26, 2.2.4.9], it gives rise to a symmetric monoidal equivalence. That is, the categorical equivalence Fun ⊗ ( O ⊗ , C ⊗ ) ≃ Alg O ⊗ (C ⊗ ) is promoted to a symmetric monoidal equivalence in the natural way.
Let A be an E 2 -algebra in Mod R . By definition, it is a map of ∞-operads A : the operadic left Kan extension of A along the inclusion i : E ⊗ 2 ֒→ DCyl. If we think of the color S 1 as an object in the fiber (Mfld 1 ) 1 of Mfld 1 → Γ over 1 , the full subcategory S 1 spanned by S 1 determines the inclusion ι : BS 1 ≃ S 1 ֒→ (Mfld 1 ) 1 ⊂ Mfld 1 . Then we have the following diagram R can be viewed as the factorization homology C A in Mod R in this context, cf. [1], [26].
We continue to suppose that A is an E 2 -algebra in Mod R . Let us consider the Hochschild homology R-module spectrum of A defined as follows. Let Alg As (Sp Σ (R) c ) be the category of associative algebra objects of Sp Σ (R) c , where R be a (cofibrant) commutative symmetric ring spectrum that represents R, and Sp Σ (R) c is the full subcategory of Sp Σ (R) spanned by cofibrant objects (cf. Section 6). The ordinary category Alg As (Sp Σ (R) c ) admits a symmetric monoidal structure given by A ⊗ B = A ∧ R B. Define a symmetric monoidal functor Alg As (Sp Σ (R) c ) → Cat pc R which carries A to BA, where BA is the R-spectral category having one object * with the morphism spectrum A = BA( * , * ). We define HH • (A) to be the Hochschild homology R-module spectrum of BA. Namely, we use canonical symmetric monoidal functors see Corollary 6.12. By inverting weak equivalences we obtain symmetric monoidal functors , see Example 6.4 for the first symmetric equivalence. This functor sends A ∈ Alg As (Mod R ) to HH • (A). Note that there is a canonical categorical equivalence Alg E2 (Mod R ) ≃ Alg As Alg As (Mod R ) which follows from the trivial fibration E ⊗ 1 → As ⊗ and the equivalence E ⊗ 2 ≃ E ⊗ 1 ⊗ E ⊗ 1 (Dunn additivity theorem). Thus, we have the induced functor Alg E2 (Mod R ) ≃ Alg As Alg As (Mod R ) → Alg As (Fun(BS 1 , Mod R )).
Given A ∈ Alg E2 (Mod R ), we define HH • (A) to be the image of A in Alg As (Fun(BS 1 , Mod R )). (Fun(BS 1 , Mod R )). We write HH Λ • (−) for the composite. Let As ⊗ be the symmetric monoidal envelope of As ⊗ . The

Let us consider Alg
There is a canonical symmetric monoidal equivalence , see Construction 7.9. We write Disk 1 ֒→ Disk ⊗ 1 for the inclusion of the fiber of the coCartesin fibration Disk ⊗ 1 → Γ over 1 . Using Lemma 7.3, we have The composition with ξ induces g : Alg which is a symmetric monoidal functor.
in the natural way. In particular, induced by the composition with ξ.
Proof. We use the notation in Lemma 7.3. Let φ p−1,i : S 1 → S 1 be a monotone degree one map which we think of as a morphism (p − 1) where the final map is a symmetric monoidal functor corresponding to A. By inspection, if {φ p−1,i } 0≤i≤p are regarded as morphisms (p) → (p − 1) in Λ op , their images in Sp Σ (R) c define (p + 1) degeneracy maps A ∧p+1 → A ∧p given by the multiplication A ∧ A → A. Let ψ p,i : S 1 → S 1 be a monotone degree one map which we think of as a morphism (p) → (p− 1) such that ψ p,i (x k p ) = x k p−1 for k < i + 1, and ψ p,i (x k p ) = x k−1 p−1 for k ≥ i + 1. As in the case of {φ p−1,i } 0≤i≤p , these maps give rise to p face maps A ∧p → A ∧p+1 given by the unit R → A. Consider the rotation r p : S 1 → S 1 which sends x k p to x k+1 p for k ∈ Z/(p + 1)Z. We regard r p as an isomorphism (p) → (p). It yields the action of Z/(p + 1)Z on A ∧p+1 given by the cyclic permutation of factors. It is straightforward to check that these maps constitute a cyclic object that coincides with the cyclic object obtained from BA in Definition 6.8. ✷ Proof of Proposition 7.11. Taking into account Lemma 7.12 and where the right functor is induced by the composition with ξ : Λ op → As ⊗ . We abuse notation by writing HH Λ • (A) for the image of A under h. In the following discussion, we will use the canonical identification Alg ⊗ E1 (−) ≃ Alg ⊗ As (−) which comes from the canonical equivalence E ⊗ 1 ≃ As ⊗ of ∞-operads. Let us consider The second equivalence follows from Construction 7.9, and the third functor is induced by ξ : The composition is identified with h via Alg E1 (Fun(Λ op , Mod R )) ≃ Fun(Λ op , Alg E1 (Mod R )). Let A ♭ : As ⊗ → Alg ⊗ E1 (Mod R ) be a map of ∞-operads that corresponds to A ∈ Alg E2 (Mod R ) ≃ Alg As (Alg E1 (Mod ⊗ R )). We let A ♭ : As ⊗ → Alg ⊗ E1 (Mod R ) be a symmetric monoidal functor from the symmetric monoidal envelope As ⊗ that corresponds to A ♭ (namely, the composite As ⊗ → As ). Note that HH • (A) is defined to be the image of HH Λ • (A) under the functor Alg As (Fun(Λ op , Mod R )) → Alg As (Fun(BS 1 , Mod R )) induced by the symmetric monoidal functor L : Fun(Λ op , Mod R )) → Fun(BS 1 , Mod R ) in Lemma 6.10.
Here L is a left adjoint of the symmetric monoidal functor Fun(BS 1 , Mod R ) → Fun(Λ op , Mod R ) induced by the composition with Λ op → BS 1 . Thus, HH • (A) can also be regarded as the image under the (left adjoint) functor Fun(Λ op , Alg E1 (Mod R )) → Fun(BS 1 , Alg E1 (Mod R )) given by left Kan extensions along Λ op → BS 1 . Consequently, HH • (A) : . Consider the diagram of ∞-categories: The upper left horizontal arrow is induced by the restriction to the source. The left vertical arrow is induced by the restriction to the target. The upper right arrow is the underlying functor of A ♭ . The arrow The right triangle commutes whereas the left square does not commute (but it admits a canonical natural transformation induced by the evaluation map Lemma 7.4, it follows that the composite BS 1 ≃ S 1 → Alg E1 (Mod R ) is equivalent to HH • (A). In other words, HH • (A) is equivalent to BS 1 ≃ . For this purpose, we consider the following setting. Let DCyl → DCyl be a symmetric monoidal envelope of DCyl. Composing with maps into symmetric monoidal envelopes, we have the left diagram lying over ∧ : Γ×Γ → Γ. Then by the universal property of the tensor product of ∞-operads, it induces the right commutative diagram consisting of maps of ∞-operads over Γ, where Alg ⊗ E1 ( E ⊗ 2 ) and Alg ⊗ E1 ( DCyl) are symmetric monoidal ∞-categories (defined over Γ), and the right vertical arrow is a symmetric monoidal (fully faithful) functor. In the following discussion, we replace Mod ⊗ R by an arbitrary symmetric monoidal presentable ∞-category M ⊗ whose tensor product M × M → M preserves small colimits separately in each variable. The example of M ⊗ we keep in mind is Mod ⊗ R . Let A be an E 2 -algebra object in M ⊗ , that is, a map A : As before, we let A ♭ : E ⊗ 1 → Alg E1 (M ⊗ ) be the composite of top horizontal arrows, which amounts to . The universal property [26, 3.1.3.2] induces a canonical morphismÂ ♭ →Â ♯ . It suffices to prove that the restrictionÂ ♭ | S 1 →Â ♯ | S 1 to S 1 is an equivalence. (It gives rise to an equivalence HH • (A) ≃ i ! (A) C in Fun(BS 1 , Alg E1 (Mod R )).) To this end, it is enough to show the following Lemma, which completes the proof of Proposition 7.11. Lemma 7.13. The induced morphismÂ ♭ (S 1 ) →Â ♯ (S 1 ) is an equivalence in the ∞-category Alg E1 (M ⊗ ).
Let Disj rec (C) be the full subcategory (poset) of Disj(C) spanned by those open sets V ⊂ C such that V is the image of a rectilinear embedding, and the composite V ֒→ C = (0, 1) × S 1 pr → S 1 is not surjective. By applying the argument of [26, 5.5.2.13] to Disj rec (C) → (Disk 2 ) /C , we see that Disj rec (C) → (Disk 2 ) /C is left cofinal. It follows that there is a canonical equivalence colim Unwinding the definition, this morphism is the composite of where the right arrow is induced by the universal property of the colimit, and the left arrow is an equivalence because A ♭ (U ) ≃ A((0, 1) × U ). To see that the right arrow is an equivalence, it will suffice to prove that Disj(S 1 ) → Disj rec (C) that sends U to (0, 1) × U is left cofinal: for any V ∈ Disj rec (C), the category Disj(S 1 ) × Disj rec (C) Disj rec (C) V / is weakly contractible. Consider the image W of V under the projection (0, 1) × S 1 → S 1 . Then W belongs to Disj(S 1 ) since V ֒→ C → S 1 is not surjective. It follows that Disj(S 1 ) × Disj rec (C) Disj rec (C) V / has an initial object so that the opposite category is filtered. Thus, by [25, 5.5.8.7], Disj(S 1 ) × Disj rec (C) Disj rec (C) V / is weakly contractible as desired.
Theorem 7.14 means that we can obtain an object of Alg KS (Mod R ) which "lies over" (HH • (C), The proof proceeds in Construction 7.16, Proposition 7.17 and Construction 7.18. (Pr L R ). Let RPerf HH • (C) be the full subcategory of RMod HH • (C) spanned by compact objects. This subcategory is the smallest stable subcategory which contains HH • (C) (regarded as a right module) and is closed under retracts. Hence RPerf HH • (C) inherits an associative monoidal structure from the structure on RMod HH • (C) . We denote by RPerf ⊗ HH • (C) the resulting associative monoidal small R-linear stable idempotent-complete ∞-category which we regard as an object of Alg As (St R ).
Proposition 7.17. We continue to assume that C is a small R-linear stable idempotent-complete ∞category. If we think of D = Ind(C) as the left RMod ⊗ HH • (C) -module (as above), the restriction exhibits C as a left RPerf ⊗ HH • (C) -module, that is, an object of LMod RPerf ⊗ HH • (C) (St R ). In particular, C is promoted to Proof. We may and will suppose that C is the full subcategory of compact objects in D. The tensor product functor RMod HH • (C) × RMod HH • (C) → RMod HH • (C) sends RPerf HH • (C) × RPerf HH • (C) to RPerf HH • (C) ⊂ RMod HH • (C) . It will suffice to prove that the action functor m : RMod HH • (C) ×D → D sends RPerf HH • (C) ×C to C. Let P be the full subcategory of RMod HH • (C) spanned by those objects P such that the essential image of {P } × C is contained in C. Note that m preserves the shift functors (Σ or Ω) and small colimits separately in each variable. Moreover, the stable subcategory C ⊂ D is closed under retracts. Thus, we see that P is a stable subcategory which is closed under retracts. Since HH • (C) is a unit object, HH • (C) lies in P. Keep in mind that RPerf HH • (C) is the smallest stable subcategory which contains HH • (C) and is closed under retracts. It follows that RPerf HH • (C) ⊂ P. ✷ Construction 7.18. We take O to be LM in Proposition 6.14. We then have where the vertical arrows are given by the restriction along As ⊗ ֒→ LM. By Proposition 7.17, we think of C as an object of LMod RPerf ⊗ HH • (C) (Mod PerfR (St)). Applying the above functor to C, we obtain HH • (C) which belongs to LMod HH•(RPerf HH • (C) ) (Fun(BS 1 , Mod R )). The lower horizontal arrow carries the associative monoidal ∞-category RPerf ⊗ HH • (C) to an object of Alg As (Fun(BS 1 , Mod R )). That is, HH • (RPerf HH • (C) ) is an associative algebra object in Fun(BS 1 , Mod R ). Consequently, the Hochschild homology R-module spectrum HH • (C) is a left HH • (RPerf HH • (C) )-module object in Fun(BS 1 , Mod R ). Next, we set A = HH • (C) in Alg E2 (Mod R ). By the invariance of Hochschild homology under Morita equivalences (cf. Lemma 6.11), ) be a sequence of functors such that the first one is induced by left Kan extensions along i : E ⊗ 2 ֒→ DCyl (cf. the discussion before Proposition 4.19), the second one is the restriction along Cyl → DCyl, and the third functor (equivalence) comes from Corollary 4.18. By definition, the image of A in Alg E1 (Fun(BS 1 , Mod R )) is i ! (A) C defined in the discussion before Proposition 7.11. According to Proposition 7.11, we have the canonical equivalence HH • (A) ≃ i ! (A) C in Alg E1 (Fun(BS 1 , Mod R )) ≃ Alg As (Fun(BS 1 , Mod R )). Therefore, HH • (C) = A ∈ Alg E2 (Mod R ) and the left HH • (A)-module HH • (C) together with i ! (A) C ≃ HH • (A) determines an object of where the equivalence comes from the canonical equivalences in Corollary 4.21. In other words, it defines an object of Alg D DCyl (Mod R ) ⊂ Alg DCyl (Mod R ) which induces a KS-algebra via the restriction. Thus, we obtain the desired object of Alg KS (Mod R ).

The action
In this Section, we study the maps induced by the action of Hochschild cohomology spectrum HH • (C) on HH • (C), constructed in Theorem 7.14.
8.1. Let R be a commutative ring spectrum. We let C be a small stable R-linear ∞-category. We let A ∈ Alg As (Mod R ) and suppose that C = RPerf A . In other words, we assume that Ind(C) admits a single compact generator. In this setting, we can describe morphisms induced by module actions by means of concrete algebraic constructions. For ease of notation, we write HH • (A) for HH • (RPerf A ) = HH • (RMod A ). We can safely confuse HH • (RMod A ) with the Hochschild homology R-module spectrum HH • (A) of A because of the invariance under Morita equivalences. We do not distinguish between the notation HH • (RMod A ) and HH • (A) : we write HH • (A) for HH • (RMod A ) as well. Write A e for A op ⊗ R A. As before, by ⊗ we mean the tensor product over R when we treat the tensor products of objects in Mod R or Alg As (Mod R ).
We define a morphism HH • (A) ⊗ HH • (A) → HH • (A) which we refer to as the contraction morphism: which is informally given by the two-sided bar construction (P, Q) → P ⊗ A e Q. Note that RMod A e is lefttensored over Mod ⊗ R . If we regard HH • (A) as an object of Mod R , there is a morphism HH • (A)⊗ A → A in RMod A e , which exhibits HH • (A) as a morphism object from A to itself in RMod A e (i.e., hom R-module), see Corollary 8.6. Let (HH We shall refer to it as the contraction morphism. We denote by (HH • (C), HH • (C)) the pair endowed with the KS-algebra structure constructed in Theorem 7.14: we will think that the pair is promoted to an object of Alg KS (Mod R ). Let D and C M be colors in the colored operad KS. There is a class of an active morphism f j : ( 2 , D, C M ) → ( 1 , C M ) in KS lying over the active morphism ρ : 2 → 1 (with ρ −1 ( * ) = * ). Such a morphism f j is unique up to equivalences. This is induced by an open embedding j : (0, 1) 2 ⊔ (0, 1) × S 1 → (0, 1) × S 1 such that j 1 : (0, 1) 2 → (0, 1) × S 1 is rectilinear and j 2 : (0, 1) × S 1 → (0, 1) × S 1 is a shrinking embedding, cf. Definition 4.1. If h : KS → Mod ⊗ R denotes a map of ∞-operads that encodes (HH • (C), HH • (C)), passing to Mod R via a coCartesian natural transformation, the image of f j induces a morphism in Mod R : where the right arrow is the canonical morphism, and the middle arrow is induced by the counit map RMod HH • (RModA) → Mor R (RMod A , RMod A ) of the adjunction. Here, RMod HH • (A) is endowed with p HH • (A) : Mod R → RMod HH • (A) which carries R to HH • (A). The morphisms from Mod R are omitted from the notation. Recall that I : Alg As (Mod R ) → (Pr L R ) ModR / that sends A to p A : Mod R → RMod A with p A (R) = A is fully faithful so that the full subcategory of (Pr L R ) ModR / spanned by objects of the form p A : Mod R → RMod A is equivalent to Alg As (Mod R ). Thus, the composite RMod HH • (A)⊗A → RMod A in (Pr L R ) ModR / gives rise to a morphism of associative algebras α : that is, a morphism in Alg As (Mod R )). Since is naturally equivalent to the identity functor, we have a homotopy from the composite A → HH • (A)⊗A α → A to the identity morphism of A, where A → HH • (A) ⊗ A is induced by the morphism from the unit algebra R → HH • (A).
We can make the following observation: We will prove our claim. Namely, we show that θ B is an equivalence. For ease of notation, we set p B : where A is the right A-module determined by the multiplication of A, and Map Pr L the map determined by the identity functor M A and the identity morphism of A. We have a canonical equivalence Using this equivalence, we deduce that Proof. By Lemma 8.5, HH • (A) (endowed with α and an identification σ between A → HH • (A) ⊗ A → A and the identity morphism) is a center of A. According to [26, 5.3.1.30], the morphism HH • (A)⊗ A → A of the right A e modules, that is obtained from the center, exhibits HH • (A) as a morphism object from A to A. Thus, our claim follows. ✷ We describe the bar construction P ⊗ A e A by means of symmetric spectra. Let R be a cofibrant commutative symmetric ring spectrum. Let A be a cofibrant associative symmetric ring R-module spectrum which represents A, cf. Example 6.4. We write ∧ for the wedge/tensor product ∧ R over R. Let B • (A, A, A) be a simplicial diagram ∆ op → Sp Σ (R) c of symmetric spectra (called the bar construction), which is given by [p] → A ∧ . . . ∧ A = A ∧ A ∧p ∧ A. We refer to [30, 4.1.8] for the explicit formula of B • (A, A, A). The degeneracy maps A ∧p+2 → A ∧p+1 is induced by the multiplication of A ∧ A → A, and face maps A ∧p+2 → A ∧p+3 is induced by the unit map R → A. Each term A ∧ A ∧p ∧ A is a free left A e := A op ∧ Amodule generated by A ∧p = R ∧ A ∧p ∧ R. In addition, B • (A, A, A) can be thought of as a simplicial diagram of left A e -modules. The homotopy colimit of B • (A, A, A) is naturally equivalent to A with respect to stable equivalences [30, 4.1.9] so that the colimit of the induced diagram in LMod A e is A. Let P be a right A e -module which is cofibrant as an R-module. Let P ∧ A e B • (A, A, A) be a simplicial diagram induced by B • (A, A, A) which carries [p] to P ∧ A e (A ∧ A ∧p ∧ A) ≃ P ∧ A ∧p . Consider the composition where the final morphism is determined by the restriction ∆ op ⊂ Λ op . We write HH ∆ • (−) for the composite. Note that HH ∆ • (A) gives rise to a simplicial diagram in Mod R whose colimit is HH • (A). The standard computation shows that A ∧ A e B • (A, A, A) can be identified with HH ∆ • (A). Lemma 8.7. Let P be a right A e -module symmetric spectrum which is cofibrant as an R-module. We write P for the image of P in RMod A e . Then P ⊗ A e A can be identified with a colimit of the simplicial diagram induced by P ∧ A e B • (A, A, A). In particular, HH • (A) can be identified with A ⊗ A e A in Mod R .
Proof. Note that the two-sided bar construction preserves colimits in each variable. Moreover, the colimit of B • (A, A, A) is A after passing to LMod A e , and each P ∧ A e (A ∧ A ∧p ∧ A) ≃ P ∧ A ∧p computes P ⊗ A ⊗p ≃ P ⊗ A e (A ⊗ A ⊗p ⊗ A). Therefore, Lemma follows.
Next let us consider h ′ (C)⊗h ′ (C M ) → h ′ (C M ). According to Construction 7.18, its underlying morphism in Mod R can naturally be identified with HH • (Z) ⊗ HH • (A) ≃ HH • (Z ⊗ A) → HH • (A) induced by α : Z ⊗ A → A. Let Z and A be cofibrant and fibrant associative ring symmetric R-module spectra that represent Z and A, respectively (namely, they are objects in Alg As (Sp Σ (R)) which are both cofibrant and fibrant with respect to the projective model structure). Letᾱ : Z∧A → A be a morphism in Alg As (Sp Σ (R) c ) which represents α. The composite A ≃ R∧A → Z∧A → A induced by R → Z is equivalent to the identity morphism of A.

8.2.
Let k be a field. We suppose that R is the Eilenberg-MacLane spectrum of k. We write k for R. In this context, we will give a concrete model of the contraction morphism σ : HH • (A) ⊗ HH • (A) → HH • (A) as a morphism of chain complexes of k-vector spaces. Let Comp ⊗ (k) be the symmetric monoidal category of chain complexes of k-vector spaces, whose tensor product is given by the standard tensor product of chain complexes. There is a symmetric monoidal (projective) model structure on Comp(k) such that a morphism is a weak equivalence (resp. a fibration) if it is a quasi-isomorphism (resp. a termwise in H 1 (Hom k (HH • (A) ⊗ HH • (A), HH • (A))). Here "•" indicates the composition. This relation is known as Cartan homotopy/magic formula. In the dg setting over k, the shifted complex HH • (A) [1] inherits the structure of an L ∞ -algebra, i.e., an algebra over a (cofibrant) Lie operad in Comp(k), from the E 2 -algebra structure on HH induced by L appears as the Lie algebra action morphism on HH • (A) (see e.g. [11], [33]). Since u and B can be explicitly described (B is equivalent to Connes' operator), thus L also has an explicit presentation.

Equivariant context
Our construction in Theorem 7.14 can easily be generalized to an equivariant setting: Let G be a group object in S and BG ∈ S the classifying space. Let C ∈ St R , that is, a small stable R-lienar idempotentcomplete ∞-category. Suppose that G acts on C, i.e., an left action on C. Namely, C is an object of Fun(BG, St R ) whose image under the forgetful functor Fun(BG, St R ) → St R is C. In this setting, we have Theorem 9.1. The pair (HH • (C), HH • (C)) of Hochschild cohomology and homology R-module spectra has the structure of a KS-algebra in Fun(BG, Mod R ). In other words, (HH • (C), HH • (C)) is promoted to Alg KS (Fun(BG, Mod R )).
Remark 9.2. The forgetful functor Alg KS (Fun(BG, Mod R )) → Alg KS (Mod R ) sends the KS-algebra in Theorem 9.1 to a KS-algebra equivalent to the KS-algebra constructed in Theorem 7.14.
Theorem 9.1 follows from the following: Construction 9.3. The construction is almost the same as that of Theorem 7.14. Thus we point out necessary modifications.
(i) Let D := Ind(C) be the Ind-category which is an R-linear compactly generated ∞-category. In particular, D belongs to Pr L R . Since C → Ind(C) is functorial, the left action of G on C induces a left action on D. Namely, D is promoted to Fun(BG, Pr L R ). The functor category Fun(BG, Pr L R ) inherits a (pointwise) symmetric monoidal structure from that of Pr L R . Let Mor G R (D, D) be an internal hom object in the symmetric monoidal ∞-category Fun(BG, Pr L R ). This is explicitly described as follows: The internal hom object Mor R (D, D) (Lemma 5.1) in Pr L R has the left action of G op × G induced by the functoriality of the internal hom object and the action of G on D (here G op denotes the opposite group). The homomorphism G → G op × G informally given by g → (g −1 , g) determines a left action of G on Mor R (D, D). By the universal property of Mor R (D, D), Mor R (D, D) endowed with the G-action is an internal hom object from D to D in Fun(BG, Pr L R ). As in Lemma 5.1, Mor G R (D, D) is promoted to Alg As (Fun(BG, Mod R )) ≃ Fun(BG, Alg As (Mod R )). Here the equivalence follows from the defintion of the pointwise symmetric monoidal strucutre on Fun(BG, Mod R ) [26, 2.1.3.4].