Enriched Monoidal Categories I: Centers

Home , Center (category theory), Monoidal functor

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6 Enriched symmetric monoidal categories 39 6.1 Deﬁnitions and canonical construction ...... 39 6.2 The E2-centers of enriched braided monoidal categories in ECat ...... 39

References 41

1 Introduction

Enriched categories were introduced long ago [EK66], and have been studied intensively (see a classical review [Kel82]). A category enriched in a symmetric monoidal category A is also A called an A-enriched category or simply an A-category, and is denoted by |L. The category A is called the base category of the enriched category [Kel82]. In this work, we choose to call A A A A the background category of |L. The L in |L denotes the underlying category of |L. The classical setup of the theory of enriched category is to consider a fixed symmetric monoidal category A and develop the theory within the 2-category of A-categories, A-functors and A-natural transformations [Kel82]. Although it is quite obvious to consider the change of background category by a symmetric monoidal functor A → A′, the classical setup is natural, convenient and sufficient for many purposes as Kelly wrote in his classical book [Kel82]: Closely connected to this is our decision not to discuss the “change of base category” given by a symmetric monoidal functor V→V′, ...The general change of base, important though it is, is yet logically secondary to the basic V-category theory it acts on. In recently years, however, there are strong motivations to go beyond the classical setting. First, mathematicians start to explore enriched monoidal categories with the background categories being only braided (instead of being symmetric) [JS93, For04, BM12, MP17, KZ18b, MPP18, JMPP19]. Secondly, in physics, recent progress in the study of gapless boundaries of 2+1D topological orders [KZ20, KZ21] and topological phase transitions [CJKYZ20] demands us to consider enriched categories, enriched monoidal categories and enriched braided (or symmetric) monoidal categories such that the background categories vary from monoidal categories to braided monoidal categories and to symmetric monoidal categories, and to consider functors between enriched categories with different background categories, and to develop the represen- A tation theory of an A-enriched (braided) monoidal category |L in the 2-category of enriched (monoidal) categories with different background categories [Zhe17, KZ20, KZ21]. In other words, physics demands us to develop a generalized theory within the 2-category of enriched categories (with arbitrary background categories), generalized enriched functors that can change the background categories (see Definition 3.4) and generalized enriched natural transformations (see Definition 3.7). This work is the first one in a series to develop this generalized theory. More precisely, in this work, we study enriched (braided) monoidal categories, and study the E0-center of an enriched category, the E1-center (or the Drinfeld center or the monoidal center) of an enriched monoidal category and the E2-center (or the M üger center) of an enriched braided monoidal category via their universal properties. We introduce the canonical constructions of enriched (braided/symmetric monoidal) categories, and compute their centers (see Theorem 4.37, 5.26, 6.9 and Corollary 4.39, 5.27, 6.10). These results generalize the main results in [KZ18b], where two of the authors introduced the notion of the Drinfeld center of an enriched monoidal category without studying its universal property. These centers play important roles in the (higher) representation theory of enriched (braided) monoidal categories. We will develop the representation theory in the second work in the series.

2 The layout of this paper is given as follows. In Section 2, we review some basic notions and set our notations, and for i = 0, 1, 2, we review the notion of the Ei-center of an Ei-algebra in a symmetric monoidal 2-category. In Section 3, we review the notion of an enriched category, and introduce the notions of an enriched functor and an enriched natural transformation. We also study the canonical construction of enriched categories as a locally isomorphic 2-functor. In Sec- tion 4, we study enriched monoidal categories and the canonical construction, and compute the E0-centers of enriched categories. In Section 5, we study enriched braided monoidal categories and the canonical construction, and compute the E1-centers of enriched monoidal categories. In Section 6, we study enriched symmetric monoidal categories and the canonical construction, and compute the E2-centers of enriched braided monoidal categories. Throughout the paper, we choose a ground field k of characteristic zero when we mention a finite semisimple (or a multi-fusion) category. We use k to denote the symmetric monoidal category of finite-dimensional vector spaces over k.

Acknowledgement: LK and HZ are supported by Guangdong Provincial Key Laboratory (Grant No.2019B121203002). LK and ZHZ are supported by NSFC under Grant No. 11971219 and by Guangdong Basic and Applied Basic Research Foundation under Grant No. 2020B1515120100. WY is supported by the NSFC under grant No. 11971463,11871303,11871127. ZHZ is supported by Wu Wen-Tsun Key Laboratory of Mathematics at USTC of Chinese Academy of Sciences. HZ is also supported by NSFC under Grant No. 11871078.

2 2-categories

In this section, we recall some basic notions in 2-categories and examples, and set the notations along the way. We refer the reader to [JY20] for a detailed introduction to 2-categories and bicategories.

2.1 Examples of 2-categories A bicategory is called a 2-category if the associators and the unitors of 1-morphisms are identities. We introduce simple notations for objects, 1-morphisms and 2-morphisms in a 2-category C as x, y ∈ C,( f, g : x → y) ∈ C and (ξ : f ⇒ g) ∈ C, respectively.

Deﬁnition 2.1. A 2-functor F : C → D is called locally equivalent (or fully faithful) if Fx,y : C(x, y) → D(F(x), F(y)) is an equivalence for all x, y ∈ C; it is called locally isomorphic if Fx,y is an isomorphism for all x, y ∈ C.

Example 2.2. We give a few examples of 2-categories. • Cat: the 2-category Cat of categories, functors and natural transformations. It is a symmetric monoidal 2-category with the tensor product given by the Cartesian product × and the tensor unit given by ∗, which consists of a single object and a single morphism. • Alg (Cat): the 2-category of monoidal categories, monoidal functors and monoidal natu- E1 ral transformations. We choose this notation because a monoidal category can be viewed as an E1-algebra in Cat (see for example [Lur]). It is a symmetric monoidal 2-category with the tensor product × and the tensor unit ∗. • Alg (Cat): the 2-category of braided monoidal categories, braided monoidal functors and E2 monoidal natural transformations. It is a symmetric monoidal 2-category with the tensor product × and the tensor unit ∗. • Alg (Cat): the 2-category of symmetric monoidal categories, braided monoidal functors E3 and monoidal natural transformations. It is a symmetric monoidal 2-category with the tensor product × and the tensor unit ∗. q

In the rest of this subsection, we recall some basic notions and give a few more examples of 2-categories that are useful in this work. Let A and B be monoidal categories.

3 Deﬁnition 2.3. A lax-monoidal functor F : A → B is a functor equipped with a morphism 1B → F(1A) and a natural transformation F(x) ⊗ F(y) → F(x ⊗ y) for x, y ∈ A satisfying the following conditions:

1. (lax associativity): for all x, y, z ∈ A, the following diagram commutes:

F(x) ⊗ F(y) ⊗ F(z) / F(x) ⊗ F(y ⊗ z) (2.1)

F(x ⊗ y) ⊗ F(z) / F(x ⊗ y ⊗ z)

2. (lax unitality): for all x ∈ A, the following diagrams commute:

1B ⊗ F(x) / F(1A) ⊗ F(x) , F(x) ⊗ 1B / F(x) ⊗ F(1A) (2.2)

F(x) o F(1A ⊗ x) F(x) o F(x ⊗ 1A)

An oplax-monoidal functor G : A → B is a lax-monoidal functor Aop → Bop.

Deﬁnition 2.4. A lax-monoidal natural transformation between two lax-monoidal functor F, G : A → B is a natural transformation ηa : F(a) → G(a) such that the following diagrams commute:

1B / F(1A) F(a ⊗ F(b) / F(a ⊗ b) (2.3) ▲▲ ▲▲ η1 ⊗ η ⊗ ▲▲ ηa ηb a b ▲% G(1A) G(a) ⊗ G(b) / G(a ⊗ b)

An oplax-monoidal natural transformation between two oplax-monoidal functors is deﬁned similarly (by ﬂipping non-vertical arrows in (2.3)).

Using above two notions, we obtain some new symmetric monoidal 2-categories (all with the tensor product × and the tensor unit ∗):

• Alglax(Cat): the 2-category of monoidal categories, lax-monoidal functors and lax-monoidal E1 natural transformations. • Algoplax(Cat): the 2-category of monoidal categories, oplax-monoidal functors and oplax- E1 monoidal natural transformations. • Algoplax(Cat): the 2-category of braided monoidal categories, braided oplax-monoidal E2 functors and oplax-monoidal natural transformations. Deﬁnition 2.5. A left A-oplaxmodule is a category L equipped with an oplax-monoidal functor A → Fun(L, L), or equivalently, an action functor ⊙ : A × M → M equipped with • an oplax-associator: a natural transformation (−⊗−) ⊙−→−⊙ (−⊙−) rendering the following diagram commutative.

((a ⊗ b) ⊗ c) ⊙ x / (a ⊗ b) ⊙ (c ⊙ x) (2.4)

(a ⊗ (b ⊗ c)) ⊙ x / a ⊙ ((b ⊗ c) ⊙ x) / a ⊙ (b ⊙ (c ⊙ x))

• an oplax-unitor: a natural transformation 1⊙− → 1M rendering the following two diagrams commutative.

(1 ⊗ b) ⊙ x / 1 ⊙ (b ⊙ x) , (b ⊗ 1) ⊙ x / b ⊙ (1 ⊙ x) (2.5) ▼ ▼ ▼▼▼ qqq ▼▼▼ qqq ▼▼& xqqq ▼▼& xqqq b ⊙ x b ⊙ x

4 If the oplax-associator (resp. the oplax-unitor) is a natural isomorphism, then the left A- oplaxmodule is called strongly associative (resp. strongly unital), and is called a left A-module if it is both strongly associative and strongly unital. A left A-laxmodule is deﬁned by ﬂipping the arrows of the oplax-associator and the oplax- unitor to give the lax-associator and the lax-unitor.

Deﬁnition 2.6. For two left A-oplaxmodules M and N,a lax A-oplaxmodule functor from M to N is a functor equipped with a natural transformation

αb,m : b ⊙ F(m) → F(b ⊙ m) such that the following diagrams commute for all a, b ∈ A and m ∈ M.

a ⊙ (b ⊙ F(m)) / a ⊙ F(b ⊙ m) 1 ⊙ F(m) / F(1 ⊙ m) O ❃ ❃❃ ⊗ ⊙ ❃❃ (a b) F(m) ❃❃ (2.6) ❃❃ F((a ⊗ b) ⊙ m) / F(a ⊙ (b ⊙ m)) F(m)

If α is a natural isomorphism, then F is called an A-module functor. oplax An oplax A- module functor is defined by flipping the arrow αb,m. Similarly, a lax A- laxmodule functor and an oplax A-laxmodule functor can both be defined by flipping certain arrows.

Deﬁnition 2.7. For two lax A-oplaxmodule functors F, G : M → N, a lax A-oplaxmodule natural transformation ξ : F ⇒ G is a natural transformation rendering the following diagram

1⊙ξ a ⊙ F(m) m / a ⊙ G(m)

ξ ⊙ F(a ⊙ m) a m / G(a ⊙ m) commutative for all a ∈ A and m ∈ M. An oplax A-oplaxmodule (or oplax A-laxmodule, lax A-laxmodule) natural transformation can be deﬁned similarly.

2.2 Centers in 2-categories A contractible groupoid is a non-empty category that has a unique morphism between every two objects. An object x in a bicategory C is called a terminal object if the hom category C(y, x) is a contractible groupoid for every y ∈ C. In the rest of this section, we use C to denote a monoidal bicategory (i.e., a tricategory with one object) equipped with a tensor product ×˙ and a tensor unit ∗. By the coherence theorem of tricategories (see [GPS95, Gur13]), without loss of generality, we further assume that C is a semistrict monoidal 2-category, which is also called a Gray monoid.

Deﬁnition 2.8. An E0-algebra in C is a pair (A, u), where A is an object in C and u : ∗ → A is a 1-morphism.

Deﬁnition 2.9. A left unital (A, u)-action on an object x ∈ C is a 1-morphism g : A ×˙ x → x, together with an invertible 2-morphism α as depicted in the following diagram:

A ×˙ x × 6 ❑ g u ˙ 1x ♥♥♥ ❑❑ ♥♥ ✤✤ α ❑❑ ♥♥ ❑❑ ∗ ×˙ x = x /% x. 1x

Let x ∈ C. We deﬁne the 2-category of left unital actions on x as follows: • Objects are left unital actions on x.

5 • For i = 1, 2, let ((Ai, ui), gi, αi) be a left unital (Ai, ui)-action on x. A 1-morphism from ((A1, u1), g1, α1) to ((A2, u2), g2, α2) is a triple (p, σ, ρ), where p : A1 → A2 is a 1-morphism in C and σ and ρ are two invertible 2-morphism in C (as depicted in the following diagrams) such that the following identity of 2-morphisms holds.

A2 ×˙ x 5 O × u2 ×˙ 1x p ×˙ 1x g2 A2 ˙ x ✈; ❉❉ ✤✤ ✤✤ u2 ×˙ 1x ✈ ❉ g2 ✤✤ σ ×˙ 1 × ✤✤ ρ = ✈✈ ✤✤ ❉❉ . A1 ˙ x ✤✤ α2 ❉ 6 ❖ ✈✈ ❉ u1 ×˙ 1x ♠♠ ❖❖ g1 ✈✈ ❉❉ ♠♠♠ ✤✤ ❖❖ " ♠♠ α1 ❖❖❖ ∗ ×˙ x / x ♠♠♠ ❖❖❖ ∗ ×˙ x /' x

• Let (p, σ, ρ) and (q,β,ϕ):((A1, u1), g1, α1) → ((A2, u2), g2, α2) be two 1-morphisms, a 2- morphism (p, σ, ρ) ⇒ (q,β,ϕ) is a 2-morphism ξ : p ⇒ q such that

×˙ ×˙ 2 A2 2 A2 A2 x g2 A2 x g2 u2 F X u2 O C [ O ξ ξ ×˙ 1 p ❴❴ q q p ×˙ 1x ❴❴ q ×˙ 1x ✤✤ p ×˙ 1x ✤✤ ✤✤ σ ❴❴ +3 = ✤✤ β , ❴❴ +3 ✤✤ ϕ = ✤✤ ρ . ∗ / A1 ∗ / A1 A1 ×˙ x / x A1 ×˙ x / x u1 u1 g1 g1

Deﬁnition 2.10. Let x be an object in C. The E0-center Z0(x) of x is the terminal object in the 2-category of left unital actions on x. Remark 2.11. The terminal object in a 2-category C is unique in the sense that for any two ′ ′ terminal objects x, x ∈ C the hom category C(x, x ) is a contractible groupoid. Hence the E0- center is unique. ♦

Remark 2.12. In general, the notion of an E0-center can be deﬁned for an E0-algebra. Indeed, a left unital action of an E0-algebra on another E0-algebra (x, ∗ → x) is a left unital action on the object x “preserving” the E0-algebra structures, and the E0-center of (x, ∗ → x) is terminal among all left unital actions. One can show that the E0-center is independent of the E0-algebra structure ∗ → x, thus in practice we only use the notion of the E0-center of an object. ♦

Deﬁnition 2.13. An E1-algebra (or a pseudomonoid, see Section 3 in [DS97]) in C is a sextuple (A, u, m, α, λ, ρ), where A ∈ Ob(C), u : ∗ → A and m : A ×˙ A → A are 1-morphisms, and α, λ, ρ are invertible 2-morphisms (i.e. the associator, the left unitor and the right unitor, respectively) as depicted in the following diagrams

m ×˙ 1 u ×˙ 1 1 ×˙ u A ×˙ A ×˙ A / A ×˙ A ∗ ×˙ A / A ×˙ A o A ×˙ ∗ ✤✤ ✍ ✵✵✵✵ ρ 1 ×˙ m α m Ó ✍✍✍λ m m A ×˙ A / A 1 / A o 1 such that the following equations of pasting diagrams hold:

1 ×˙ 1 ×˙ m 1 ×˙ 1 ×˙ m A×˙ 4 / A×˙ 3 A×˙ 4 / A×˙ 3 ❇ 1 ×˙ α ❇ 1 ×˙ m ❇ 1 ×˙ m 1 ×˙ m ×˙ 1 ❴❴ ❇❇ ❇❇ ❇ ❴❴ +3 ❇ ❇ ×˙ ×˙ ×˙ m ×˙ 1 ×˙ 1 A 3 / A 2 m ×˙ 1 ×˙ 1 ❴❴ +3 m ×˙ 1 A 2 α ×˙ 1 1 ×˙ m α ❴❴ +3 = ❴❴ +3 α 1 ×˙ m A×˙ 3 m ×˙ 1 ❴❴ +3 m A×˙ 3 / a×˙ 2 m ❇❇ ❇❇ α ❇❇ ❇ ❇ ❴❴ +3 m❇ m ×˙ 1 ❇ m ×˙ 1 ❇ ❇ ×˙ 2 ×˙ 2 A m / A A m / A

A ×˙ ∗ ×˙ A A ×˙ ∗ ×˙ A 1 ×˙ u ×˙ 1 ♥ λ ▼▼ 1 1 ×˙ u ×˙ 1 ♥ ▼▼ 1 ♥♥♥ ❴❴ +3 ▼▼ ♥♥♥ ▼▼ w♥♥ 1 ×˙ m ▼& w♥♥ ρ ▼& A ×˙ A ×˙ A / A ×˙ A = A ×˙ A ×˙ A ❴❴ +3 A ×˙ A α ❴❴ m ×˙ 1 ❴❴ m m ×˙ 1 m ❴❴ +3 1 ×˙ ×˙ r A A m / A A A m / A

6 where A×˙ 4 is an abbreviation for A ×˙ A ×˙ A ×˙ A, and the unlabeled 2-morphism is induced by the monoidal structure of C. We often abbreviate an E1-algebra to (A, u, m) or A. It is not hard to check the Gray monoid structure of C induces a Gray monoid structure on the 2-category of left unital actions on x ∈ C. Then the following proposition is a simple corollary of the fact that a terminal object of a monoidal 2-category admits an E1-algebra structure.

Proposition 2.14. The E0-center Z0(x) of x ∈ C is an E1-algebra. Remark 2.15. According to the general theory of higher algebras by Lurie [Lur], if C is a symmetric monoidal higher category, the En-center of an En-algebra A in C can be deﬁned as the E -center of A in the symmetric monoidal higher category Alg (C) of E -algebras in C. We 0 En n expect that Alg (Alg (C)) ≃ Alg (C). Then an E -center is automatically an E + -algebra. ♦ E1 En En+1 n n 1 Example 2.16. Consider the symmetric monoidal 2-category (Cat, ×, ∗) of categories, functors and natural transformations. The E1-algebras in Cat are monoidal categories. 1. The following diagram represents

A × L (2.7) 1A×1 ♦7 ▲▲ ⊙ ♦♦♦ ✤✤ ▲▲▲ ♦♦♦ α ▲▲▲ ∗× L &/ L

a left unital (A, 1A)-action on L in Cat, where ⊙ : A × L → L is a functor and α is a natural isomorphism. The functor from A to the category Fun(L, L) of endofunctors deﬁned by a 7→ a ⊙− is monoidal. We see that Z0(L) = Fun(L, L).

2. For (A, 1A), (L, 1L) ∈ Alg (Cat), let the diagram (2.7) depict a left unital (A, 1A)-action on E1 L in Alg (Cat), i.e., ⊙ is a monoidal functor and α is a monoidal natural isomorphism. E1 Then ϕ ≔ −⊙ 1L : A → Z1(L) is a well-deﬁned monoidal functor, and the half-braiding of ϕ(a) is deﬁned by:

α−1⊗1 ∼ x ⊗ ϕ(a) −−−−→ (1A ⊙ x) ⊗ ϕ(a) −→ (1A ⊗ a) ⊙ (x ⊗ 1L) ∼ ∼ 1⊗α −→ (a ⊗ 1A) ⊙ (1L ⊗ x) −→ ϕ(a) ⊗ (1A ⊙ x) −−→ ϕ(a) ⊗ x. Then it is straightforward to check that the E -center of L in Alg (Cat) is given by the 0 E1 Drinfeld center Z1(L) of L. Recall that the objects of the Drinfeld center Z1(L) are pairs (x, γ−,x), where γ−,x : −⊗ x → x ⊗− is a half braiding satisfying certain compatibility conditions (see for example section 7.13 in [EGNO15]).

3. For A, L ∈ Alg (Cat), the diagram (2.7) depicts a left unital (A, 1A)-action on L in E2 Alg (Cat). Since 1A ⊙ x ≃ x and ⊙ is a braided monoidal functor, ϕ(−) ≔ −⊙ 1L de- E2 fines a braided monoidal functor from A to the M üger center Z2(L) of L. Then it is straightforward to check that the E -center of L in Alg (Cat) is given by the M üger center 0 E2 Z2(L) of L. 4. For n ≥ 3, let Alg (Cat) ≔ Alg (Alg (Cat)) be the symmetric monoidal 2-category of En E1 En−1 E -algebras in Cat. It is known that Alg (Cat) is biequivalent to the symmetric monoidal n En 2-category of symmetric monoidal categories, symmetric monoidal functors and monoidal natural transformations. The E -center of a symmetric monoidal category L in Alg (Cat) 0 En is given by L itself. q

Example 2.17. Let C be a monoidal category and x ∈ C. We can view C as a monoidal 2-category with only identity 2-morphisms and denote it by C. Then the The E0-center of x in C can be defined as the E0-center of x in C. In this sense, the theory of centers in 2-categories can be viewed as a direct generalization of that of centers in 1-categories [Ost03, Dav10, KYZ21]. q Remark 2.18. Suppose C is a monoidal 2-category and M is a C-module. We can similarly define the E0-center of an object x ∈ M in C. More generally, if C is an En+1-monoidal higher category and M is an E -module over C, then Alg (M) is an Alg (C)-module, and the E -center of an n En En n E -algebra A ∈ Alg (M) in C can be defined as the E -center of A in Alg (C). This should be a n En 0 En generalization of [KYZ21, Section 3.2]. ♦

7 3 Enriched categories

3.1 Enriched categories, functors and natural transformations

Let (A, ⊗, 1A) be a monoidal category.

A Deﬁnition 3.1 ([Kel82]). An A-enriched category |L, or a category enriched in A, consists of the following data.

A A A 1. A set of objects Ob( |L). If x ∈ Ob( |L), we also write x ∈ |L.

A A 2. An object |L(x, y) in A for every pair x, y ∈ |L.

A| A| 3. A distinguished morphism 1x : 1A → L(x, x) in A for every x ∈ L.

A A ◦ A A 4. A composition morphism |L(y, z) ⊗ |L(x, y) −→ |L(x, z) in A for every triple x, y, z ∈ |L.

A They are required to make the following diagrams commutative for x, y, z, w ∈ |L.

A A A 1⊗◦ A A |L(y, z) ⊗ |L(x, y) ⊗ |L(w, x) / |L(y, z) ⊗ |L(w, y) (3.1)

◦⊗1 ◦ A A ◦ A |L(x, z) ⊗ |L(w, x) / |L(w, z)

A| ≃ A| ≃ A| 1A ⊗ L(x, y) / L(x, y) o L(x, y) ⊗ 1A (3.2) ◦ ❦5 i❙❙ ◦ ⊗ ❦❦ ❙❙ 1y 1 ❦❦ ❙❙ 1⊗1x ❦❦❦ ❙❙❙ A A ❦ A A |L(y, y) ⊗ |L(x, y) |L(x, y) ⊗ |L(x, x)

In the rest of this work, the monoidal category A is referred to as the background category of A A |L. The underlying category of |L, denoted by L, is deﬁned by

A| A| A| Ob(L) ≔ Ob( L) and L(x, y) ≔ A(1A, L(x, y)), ∀x, y ∈ L;

g A| f A| and the composition g ◦ f of morphisms 1A −→ L(y, z) and 1A −→ L(x, y) is deﬁned by

g⊗ f A| A| ◦ A| 1A → 1A ⊗ 1A −−→ L(y, z) ⊗ L(x, y) −→ L(x, z); (3.3)

A| the identity morphism of x in L is precisely 1x : 1A → L(x, x). A Remark 3.2. Let |L be an enriched category. A morphism f ∈ L(x, y) in the underlying category A A A A L induces a morphism |L(w, f ): |L(w, x) → |L(w, y) in A for w ∈ |L, deﬁned by

A| A| f ⊗1 A| A| ◦ A| L(w, x) ≃ 1A ⊗ L(w, x) −−→ L(x, y) ⊗ L(w, x) −→ L(w, y). (3.4)

A Arev Remark 3.3. For an A-enriched category |L, we deﬁne an Arev-enriched category |Lop as follows:

Arev A • Ob( |Lop) ≔ Ob( |L);

Arev Arev A • For x, y ∈ |Lop, |Lop(x, y) ≔ |L(y, x);

A| Arev| op • The identity is given by 1x : 1A → L(x, x) = L (x, x);

8 • the composition is deﬁned by

Arev Arev A A ◦ A Arev |Lop(y, z) ⊗rev |Lop(x, y) = |L(y, x) ⊗ |L(z, y) −→ |L(z, x) = |Lop(x, z),

where ⊗rev is the reversed tensor product of A.

Arev A A Arev We call |Lop the opposite category of |L,i.e. ( |L)op ≔ |Lop. We have

rev Arev| op A| op A (1A, L (x, y)) = A(1A, L(y, x)) = L(y, x) = L (x, y).

Arev It means that the underlying category of |Lop is Lop. This explains our notation. ♦

A B Deﬁnition 3.4. An enriched functor |F : |L → |M between two enriched categories consists of the following data: 1. A lax-monoidal functor Fˆ : A → B, which is called the background changing functor; 2. amap F : Ob(L) → Ob(M);

| A| B| 3. a family of morphisms Fx,y : Fˆ( L(x, y)) → M(F(x), F(y)) in B for x, y ∈ L; satisfying the following two axioms:

ˆ | 1 1 F(1x) A| Fx,x B| 1. 1F(x) = ( B → Fˆ( A) −−−→ Fˆ( L(x, x)) −−→ M(F(x), F(x))); 2. The diagram

in B is commutative.

Remark 3.5. We want to emphasize that if A is the category Set of sets, the notion of an enriched functor defined in Definition 3.4 does not coincide with an ordinary functor unless we require Fˆ = 1A (see also Definition 3.10). In other words, even in the case A = Set, this work provides a non-trivial generalization of the usual category theory. ♦

A| For every f : 1A → L(x, y), we deﬁne

ˆ | F( f ) A| Fx,y B| F( f ) ≔ 1B → Fˆ(1A) −−→ Fˆ( L(x, y)) −−→ M(F(x), F(y)). (3.6)

The commutativity of the diagram (3.5) implies that F(g f ) = F(g)F( f ) for every f ∈ L(x, y) and g ∈ L(y, z). As a consequence, we obtain a functor F : L → M, which is called the underlying functor of |F. A B Remark 3.6. Given an enriched functor |F : |L → |M, for f ∈ L(x, y) in the underlying category L, the commutativity of the diagram (3.5) implies that of the following diagram.

| A Fw,x B Fˆ( |L(w, x)) / |M(F(w), F(x))

| A| B| In other words, Fw,− : Fˆ( L(w, −)) ⇒ M(F(w), F(−)) is a natural transformation. Similarly, | A| B| F−,w : Fˆ( L(−, w)) ⇒ M(F(−), F(w)) is also a natural transformation. Combining them together, | A| B| we obtain a natural transformation F−,− : Fˆ( L(−, −)) ⇒ M(F(−), F(−)). ♦

9 Deﬁnition 3.7. An enriched natural transformation |ξ : |F ⇒ |G between two enriched functors A B |F, |G : |L → |M consists of a lax-monoidal natrual transformation ξˆ : Fˆ → Gˆ, which is called the background changing natural transformation of |ξ, and a family of morphisms

B| A| ξx : 1B → M(F(x), G(x)), ∀x ∈ L, rendering the following diagram commutative.

ξ ⊗|F A| A| y x,y B| B| Fˆ( L(x, y)) / 1B ⊗ Fˆ( L(x, y)) / M(F(y), G(y)) ⊗ M(F(x), F(y)) (3.7)

ξˆ ◦ |G ⊗ξ A| x,y x B| B| ◦ B| Gˆ( L(x, y)) ⊗ 1B / M(G(x), G(y)) ⊗ M(F(x), G(x)) / M(F(x), G(y))

The family of morphisms {ξx} automatically deﬁnes a natural transformation ξ : F ⇒ G, which is called the underlying natural transformation of |ξ.

Remark 3.8. Two enriched natural transformations |ξ and |η are equal in lax |ECat if and only if ξˆ = ηˆ and ξ = η. ♦

Remark 3.9. The diagram (3.7) can be rewritten as follows:

| A ξˆ A Fx,y B Gˆ( |L(x, y)) o Fˆ( |L(x, y)) / |M(F(x), F(y)) (3.8)

The commutativity of (3.8) guarantees that {ξx} gives a well-deﬁned natural transformation ξ : F ⇒ G. ♦

A A Deﬁnition 3.10. An enriched functor |F : |L → |M is called an A-functor if Fˆ is the identity | | | functor 1A of A. An enriched natural transformation ξ : F ⇒ G between two A-functors is A ˆ ˆ called an -natural transformation if ξ is the identity natural transformation of 1A, i.e. ξ = 11A [Kel82, Chapter 1].

Example 3.11. An A-enriched category with only one object ∗ is determined by an algebra (A, mA : A ⊗ A → A,ιA : 1 → A) in A with the identity morphism 1∗ : 1 → A defined by ιA and the composition morphism ◦ : A ⊗ A → A defined by m. We use ∗A to denote this enriched category. If A is the monoidal category with only one object and one morphism, we simply denote ∗1 by ∗. Let B be an algebra in a monoidal category B. Then an enriched functor |F : ∗A → ∗B is | defined by a lax monoindal functor Fˆ : A → B and an algebra homomorphism F∗,∗ : Fˆ(A) → B. Given another enriched functor |G : ∗A → ∗B, an enriched natural transformations |ξ : |F ⇒ |G is defined by a lax-monoidal natural transformation ξˆ : Fˆ → Gˆ and a morphsim ξ : 1B → B rendering the following diagram commutative.

| ξ⊗ F∗,∗ Fˆ(A) / 1B ⊗ Fˆ(A) / B ⊗ B

mB (3.9) | G∗,∗⊗ξ mB Gˆ(A) ⊗ 1B / B ⊗ B / B

| | For example, when ξ = ιB and ξˆ satisfies the condition G∗,∗ ◦ ξÂ = F∗,∗, they define an enriched natural transformation. q

A A Example 3.12. Let |L be an A-enriched cateogry. For every x ∈ |L, we use |x to denote the A| | | enriched functor ∗ → L deﬁned by xˆ = 1A, ∗ 7→ x and x∗,∗ = 1x. The underlying functor of x, i.e. ∗ 7→ x, is denoted by x. q

10 3.2 2-categories of enriched categories Proposition 3.13. Enriched categories (as objects), enriched functors (as 1-morphisms) and enriched natural transformations (as 2-morphisms) form a 2-category lax |ECat. A B B C 1. The composition |G ◦ |F of 1-morphisms |F : |L → |M and |G : |M → |N is the enriched functor deﬁned by the the composition of lax-monoidal functors GˆFˆ and the underlying functor GF induced by the family of morphisms

2. The horizontal composition |η ◦ |ξ : |H ◦ |F ⇒ |L ◦ |G of 2-morphisms |ξ : |F ⇒ |G and A B B C |η : |H ⇒ |L, where |F, |G : |L → |M and |H, |L : |M → |N are 1-morphisms, is the enriched natural transformation defined by the horizontal compositions of natural transformations ηˆ ◦ ξˆ : Hˆ ◦ Fˆ → Lˆ ◦ Gˆ and η ◦ ξ : H ◦ F → L ◦ G. A 3. The vertical composition of 2-morphisms |ξ : |F ⇒ |G and |β : |G ⇒ |K, where |F, |G, |K : |L → B |M are 1-morphisms, is defined by the vertical composition of natural transformations βˆξˆ : Fˆ → Kˆ and βξ : F → K, We denote the sub-2-category of lax |ECat consisting of enriched functors |F such that Fˆ is monoidal by ECat. Fix a monoidal category A, all A-enriched categories, A-functors and A- A natural transformations form a sub-2-category of lax |ECat, denoted by |ECat. Remark 3.14. An enriched natural transformations is invertible in lax |ECat (i.e. an enriched natural isomorphism) if and only if bothα ˆ and α are natural isomorphisms. ♦ C D Definition 3.15. Let |M and |N be enriched categories. We define their Cartesian product C D |M × |N asa(C × D)-enriched category as follows: C D • The objects of |M × |N are the same as M × N.

• For any m1, m2 ∈ M and n1, n2 ∈ N, C| D| C| D| ( M × N)((m1, n1), (m2, n2)) ≔ ( M(m1, m2), N(n1, n2)) ∈ C × D.

• The composition is deﬁned by C| D| C| D| ( M × N)((m2, n2), (m3, n3)) ⊗ ( M × N)((m1, n1), (m2, n2)) C| D| C| D| = ( M(m2, m3), N(n2, n3)) ⊗ ( M(m1, m2), N(n1, n2)) C| C| D| D| = ( M(m2, m3) ⊗ M(m1, m2), N(n2, n3) ⊗ N(n1, n2)) (◦,◦) C| D| C| D| −−−→ ( M(m1, m3), N(n1, n3)) = ( M × N)((m1, n1), (m3, n3)). • The identity morphism 1(m,n) deﬁned by (1m, 1n):

≔ 1 1 (1m,1n) C|M D|N C|M D|N 1(m,n) ( C, D) −−−−→ ( (m, m), (n, n)) = ( × )((m, n), (m, n)).

11 3.3 Pushforward 2-functors Let A and B be monoidal categories and R : A → B a lax-monoidal functor. Given an enriched A| A| category L, there is a B-enriched category R∗( L) deﬁned as follows:

A| A| 1. Ob(R∗( L)) ≔ Ob( L);

A| A| 2. R∗( L)(x, y) ≔ R( L(x, y));

≔ 1 1 R(1x) A|L A|L 3. the identity morphism: 1x B → R( A) −−−→ R( (x, x)) = R∗( )(x, x); 4. the composition of morphisms is deﬁned by the following composed morphisms in B:

The two deﬁning conditions (3.1) and (3.2) hold due to the lax-monoidalness of R. A| A| Given an A-functor F : L → M, i.e. a map F : Ob(L) → Ob(M) together with Fx,y : A A |L(x, y) → |M(F(x), F(y)) for x, y ∈ L. Then the same map F : Ob(L) → Ob(M) together with

A| A| R(Fx,y): R( L(x, y)) → R( M(F(x), F(y)))

A| A| for x, y ∈ L deﬁnes a B-functor R∗(F): R∗( L) → R∗( M). A A Given an A-natural transformation ξ : F ⇒ G between two A-functors F, G : |L → |M. For x ∈ L, we deﬁne a family of morphisms R∗(ξx) in B:

≔ 1 1 R(ξx) A|M R∗(ξx) B → R( A) −−−→ R( (F(x), G(x))).

It is straightforward to check that R∗(ξx) defines an A-natural transformation R∗(ξ). Theorem 3.19. Given a lax-monoidal functor R : A → B, there is a well-defined 2-functor A| B| A| A| R∗ : ECat → ECat (called the pushforward of R) defined by L 7→ R∗( L), F 7→ R∗(F), ξ 7→ R∗(ξ).

Example 3.20. We give a few examples of the pushforward 2-functor R∗.

(1) Given a monoidal category A, the functor A(1A, −) has a lax-monoidal structure defined by A(1A, x) × A(1A, y) → A(1A ⊗ 1A, x ⊗ y) ≃ A(1A, x ⊗ y). Then the pushforward 2-functor A(1A, −)∗ maps an A-enriched category to its underlying category, an A-functor to its underlying functor, and an A-natural transformation to its underlying natural transformation. (2) Let A be a braided monoidal category. The tensor product ⊗ : A × A → A is monoidal. A × A| A| Then ⊗∗ : ECat → ECat is a well-defined 2-functor. The Cartesian product × defines a A A A A functor × : |ECat × |ECat → × |ECat. Then we obtain a composed functor

A A × A A ⊗∗ A ×˙ : |ECat × |ECat −→ × |ECat −→ |ECat.

A This functor ×˙ , together with the tensor unit ∗, endows a monoidal structure on |ECat [KZ18b]. (3) Given a braided monoidal category A, let B be a left monoidal A-module (i.e. B is equipped φ×1 with a braided monoidal functor φ : A → Z1(B)). Let ⊙ be the composed functor A × B −−−→ Z1(B) × B → B. It deﬁnes a left unital A-action on B. Moreover, ⊙ is monoidal, thus the pushforward ⊙∗ is well-deﬁned. Therefore, we obtain a composed 2-functor

A B × A B ⊙∗ B |ECat × |ECat −→ × |ECat −→ |ECat,

B A which endows |ECat with a structure of a left |ECat-module. This clariﬁes Remark 3.21 in [KZ21]. q

We obtain a new characterization of an enriched functor.

12 A B Proposition 3.21. An enriched functor |F : |L → |M is precisely a pair (Fˆ, Fˇ), where Fˆ : A → B A| B| is a lax-monoidal functor and Fˇ : Fˆ∗( L) → M is a B-functor.

Let R, R′ : A → B be two lax-monoidal functors and φ : R ⇒ R′ be a lax-monoidal nat- A|L ′ A|L ural transformation. This φ induces an enriched functor φ∗ : R∗( ) → R∗( ), called the pushforward of φ. More precisely, φ∗ is a B-functor deﬁned as follows:

1. φ∗ is the identity map on Ob(L); A|L ′ A|L 2. on morphisms: (φ∗)x,y : R∗( )(x, y) → R∗( )(x, y) is deﬁned by

φA|L ≔ A|L (x,y) ′ A|L (φ∗)x,y R∗( (x, y)) −−−−−→ R∗( (x, y)).

The lax-monoidalness and the naturalness of φ imply that such deﬁned φ∗ preserves the identities and the compositions. Therefore, φ∗ is a well-deﬁned B-functor. We obtain a new characterization of an enriched natural transformation. A B Proposition 3.22. Let |F, |G : |L → |M be two enriched functors. An enriched natural transformation |ξ : |F → |G is precisely a pair (ξ,ˆ ξˇ), where ξˆ : Fˆ → Gˆ is a lax-monoidal functor and ξˇ : Fˇ ⇒ Gˇ ◦ ξˆ∗ is a B-natural transformation.

3.4 Canonical construction Throughout this subsection, A and B are monoidal categories, L is a left A-oplaxmodule (recall Deﬁnition 2.5), and M is a left B-oplaxmodule.

Recall that L is called enriched in A if the internal hom [x, y]A exists in A for all x, y ∈ L. We sometimes abbreviate [x, y]A to [x, y] for simplicity. We use evx and coevx to denote the counit [x, y] ⊙ x → y and unit a → [x, a ⊙ x] of the adjunction

L(a ⊙ x, y) ≃ A(a, [x, y]).

′ ′ The pair ([x, y], evx) is terminal among all such pairs. For f : x → x, g : y → y , there exists a unique morphism [ f, g]:[x, y] → [x′, y′] rendering the diagram commutative.

[x, y] ⊙ x′ / [x′, y′] ⊙ x′

1⊙ f evx′ ev g [x, y] ⊙ x x / y / y′

As a consequence, the internal homs deﬁne a bifunctor [−, −]: Lop × L → A.

Remark 3.23. Let (L : B → A, R : A → B, η : 1B ⇒ RL, ε : LR ⇒ 1A) be an adjunction such that R is lax-monoidal. The functor L is automatically oplax-monoidal. The left A-oplaxmodule L pulls back to a left B-oplaxmodule with the B-action deﬁned by L(−) ⊙− : B × L → L. If L is enriched in A, then L is enriched in B with [x, y]B = R([x, y]A). The evaluation [x, y]B ⊙ x → y ε evx is deﬁned by the composed morphism: LR([x, y]A) ⊙ x −→ [x, y]A ⊙ x −−→ y. ♦

A If L is enriched in A, then L can be promoted to an A-enriched category, denoted by L. A A More explicitly, the enriched category L has the same objects as L and L(x, y) = [x, y]. The identity 1x : 1 → [x, x] and composition ◦ :[y, z] ⊗ [x, y] → [x, z] are the unique morphisms rendering the following diagrams commutative:

1x⊙1 1⊙evx 1A ⊙ x / [x, x] ⊙ x ([y, z] ⊗ [x, y]) ⊙ x / [y, z] ⊙ ([x, y] ⊙ x) / [y, z] ⊙ y (3.11) ❖❖ ❖❖❖ evx ◦⊙1 evy ux ❖❖ ❖' evx x [x, z] ⊙ x / z

A This construction of L is called the canonical construction. If L is a strongly unital, i.e., ux : 1A ⊙ x → x is invertible for every x ∈ L, then L can be canonically identiﬁed with the underlying

13 A category of L via an isomorphism of categories deﬁned by the identity map on objects and ′ ′ the map ( f : y → y ) 7→ ( f : 1A → [y, y ]) on morphisms, where f is deﬁned by the following commutative diagram.

f ⊙1 ′ 1A ⊙ y / [y, y ] ⊙ y (3.12) uy evy f ′ y / y

An enriched functor between two enriched categories from the canonical construction has a nice characterization (see Proposition 3.27). We explain that now. In the rest of this subsection, L and M are assumed to be enriched in A and B, respectively. Let L : B → A be an oplax-monoidal functor. Then the left A-oplaxmodule L pulls back along L to a left B-oplaxmodule with the B-action defined by L(−) ⊙− : B × L → L. For a functor F : L → M, a lax B-oplaxmodule functor structure on F is defined by a natural transformation α = {αb,x : b ⊙ F(x) → F(L(b) ⊙ x)}a∈A,x∈L satisfying some natural axioms (recall Definition 2.6). We also call this natural transformation α as an L-oplax structure of F. We introduce a new structure on F that is in some sense dual to the L-oplax structure of F. Definition 3.24. Given a lax-monoidal functor R : A → B and a functor F : L → M. An R-lax structure on F is defined by a natural transformation: β = {βa,x : R(a) ⊙ F(x) → F(a ⊙ x)}a∈A,x∈L rendering the following diagrams commutative.

βa⊗b,x (R(a) ⊗ R(b)) ⊙ F(x) / R(a ⊗ b) ⊙ F(x) / F((a ⊗ b) ⊙ x) 1B ⊙ F(x) / R(1A) ⊙ F(x) (3.13)

β1A,x 1⊙βb,x βa,b⊙x R(a) ⊙ (R(b) ⊙ F(x)) / R(a) ⊙ F(b ⊙ x) / F(a ⊙ (b ⊙ x)) F(x) o F(1A ⊙ x)

The functor F equipped with an R-lax structure is called an R-lax functor.

Remark 3.25. When A = B, R = 1A, an R-lax functor becomes a lax B-oplaxmodule functor. In general, an R-lax functor F is not a “module functor” in any sense because A → B → Fun(M, M) (as the composition of a lax-monoidal functor and an oplax-monoidal functor) is neither lax- monoidal nor oplax-monoidal. ♦

Remark 3.26. We explain the duality between an L-oplax structure and an R-lax structure. Let (L : B → A, R : A → B, η :1B ⇒ RL, ε : LR ⇒ 1A) be an adjunction such that R is lax-monoidal. Then L is automatically oplax-monoidal with the oplax structure maps deﬁned by

ε L(η⊗η) ε L(1B) → LR(1A) −→ 1A, L(a ⊗ b) −−−−→ L(RL(a) ⊗ RL(b)) → LR(L(a) ⊗ L(b)) −→ L(a) ⊗ L(b).

Given an R-lax structure on F {βa,x : R(a) ⊙ F(x) → F(a ⊙ x)}a∈A,x∈L, then

≔ η⊙1 βL(b),x B L αb,x b ⊙ F(x) −−→ RL(b) ⊙ F(x) −−−→ F(L(b) ⊙ x), ∀b ∈ , x ∈ deﬁne an L-oplax structure on F. Conversely, given an L-oplax structure on F {αb,x : b ⊙ F(x) → F(L(b) ⊙ x}b∈B,x∈L, then

≔ αR(a),x F(εa⊙1) A L βa,x R(a) ⊙ F(x) −−−−→ F(LR(a) ⊙ x) −−−−−→ F(a ⊙ x), ∀a ∈ , x ∈ deﬁne an R-lax structure on F. These two constructions are mutually inverse. In other words, there is a bijection between the set of R-lax structures on F and that of L-oplax structures on F.♦

The R-lax structure gives a characterization of an enriched functor between two enriched categories from the canonical construction as explained in the following proposition.

14 A B Proposition 3.27. Let |F : L → M be an enriched functor. The underlying functor F : L → M with the natural transformation

ˆ | ≔ ˆ F(coevx) ˆ Fx,a⊙x evF(x) βa,x F(a) ⊙ F(x) −−−−−→ F([x, a ⊙ x]) ⊙ F(x) −−−−→ [F(x), F(a ⊙ x)] ⊙ F(x) −−−→ F(a ⊙ x) (3.14) is an Fˆ-lax functor. Conversely, given a lax-monoidal functor Fˆ : A → B and an Fˆ-lax functor F : L → M with the Fˆ-lax structure βa,x : Fˆ(a) ⊙ F(x) → F(a ⊙ x), the morphisms

| ≔ coevF(x) [F(x),β[x,y],x] [F(x),F(evx)] Fx,y Fˆ([x, y]) −−−−−→ [F(x), Fˆ([x, y]) ⊙ F(x)] −−−−−−−−→ [F(x), F([x, y] ⊙ x)] −−−−−−−−→ [F(x), F(y)],

A B together with Fˆ and F, deﬁne an enriched functor |F : L → M. Moreover, these two constructions are mutually inverse if L and M as left oplaxmodules are strongly unital.

| A B Proof. Given an enriched functor F : L → M, we need to show the natural transformation βa,x defined in (3.14) satisfies two diagrams in (3.13). The first diagram is the outer diagram of the following commutative diagram

Fˆ21 Fˆ(coevx)1 Fˆ(a)Fˆ(b)F(x) / Fˆ(ab)F(x) / Fˆ([x, abx])F(x) ❯ ❯❯❯ ˆ O ❯❯F❯(coevbx coevx) ˆ ❯❯❯ ˆ ◦ 1F(coevx)1 ❯❯❯ F( )1 ❯❯❯❯ Fˆ(coev )11 ˆ2 ❯* bx F 1 | Fˆ(a)Fˆ([x, bx])F(x) / Fˆ([bx, abx])Fˆ([x, bx])F(x) / Fˆ([bx, abx][x, bx])F(x) Fx,abx1

| | | 1 Fx,bx1 Fbx,abx Fx,bx1 ⋆

◦1 v Fˆ(a)[F(x), F(bx)]F(x) [F(bx), F(abx)][F(x), F(bx)]F(x) / [F(x), F(abx)]F(x) ❯❯❯❯ ❯❯❯❯1evF(x) 1evF(x) ❯❯❯❯ ❯❯❯❯ ❯❯❯* ˆ ˆ evF(x) F(a)F(bx) / F([bx, abx])F(bx) | / [F(bx), F(abx)]F(bx) Fˆ(coevbx)1 Fbx,abx1

evbx v F(abx) where the pentagon ⋆ commutes because |F is an enriched functor. The second diagram is the outer diagram of the following commutative diagram

Fˆ01 1F(x) / Fˆ(1)F(x) ❁ ◗◗ ❁ ◗◗ Fˆ(coevx)1 ❁❁ ⋆ ˆ ◗◗◗ ❁ F(1x)1 ◗◗ ❁ ◗◗◗ ❁❁ ◗( ❁❁ ˆ ˆ 1 1 1 ❁ F([x, x])F(x) o F([x, x])F(x) F(x) ❁❁ ❁ | | ❁ 1 ❁❁ Fx,x1 Fx, x1 ❁ [F(x), F(x)]F(x) o [F(x), F(1x)]F(x) ♣ evF(x) ♣♣ ♣♣ evF(x) ♣♣♣ x♣♣♣ F(x) o F(1x) where the subdiagram ⋆ commutes because |F is an enriched functor. | Conversely, suppose βa,x is an Fˆ-lax structure of F, we need to show that the morphisms Fx,y, A B together with Fˆ and F, deﬁne an enriched functor |F : L → M. That |F preserves the identity morphisms follows from the commutativity of the outer diagram of the following commutative

15 diagram

0 ˆ Fˆ F(1x) 1 / Fˆ(1) / Fˆ([x, x])

coevF(x) coevF(x) coevF(x) 1F(x) [F(x), 1 ⊙ F(x)] / [F(x), Fˆ(1) ⊙ F(x)] / [F(x), Fˆ([x, x]) ⊙ F(x)]

⋆ [1,β[x,x],x] $ [F(x), F(x)][o F(x), F([x, x] ⊙ x)] , where ⋆ commutes by the deﬁnition of Fˆ-lax structure. Using the adjunction (−⊙ x) ⊣ [x, −], the condition (3.5) is equivalent to the commutativity of the outer square of the following diagram:

Fˆ21 Fˆ(◦)1 Fˆ([y, z])Fˆ([x, y])F(x) / Fˆ([y, z][x, y])F(x) / Fˆ([x, z])F(x)

1β[x,y],x ⋆ β[y,z][x,y],x β[x,z],x

β[y,z],[x,y]x F(◦1) Fˆ([y, z])F([x, y]x) / F([y, z][x, y]x) / F([x, z]x)

1F(evx) F(1 evx) F(evx)

β[y,z],y F(evy) Fˆ([y, z])F(y) / F([y, z]y) / F(z), where the square ⋆ commutes because β is an Fˆ-lax structure of F. It is easy to verify that these two constructions are mutually inverse.

Deﬁnition 3.28. For i = 1, 2, let Ri : A → B be a lax-monoidal functor and Fi : L → M an Ri-lax functor. Given a lax-monoidal natural transformation ξˆ : R1 ⇒ R2, a natural transformation ξ : F1 ⇒ F2 is ξˆ-lax or called a ξˆ-lax natural transformation if the following diagram commutes

R1(a) ⊙ F1(x) / F1(a ⊙ x) (3.15)

ξˆa⊙ξx ξa⊙x R2(a) ⊙ F2(x) / F2(a ⊙ x) , where the unlabeled arrows are given by the Ri-lax structure on Fi. Remark 3.29. The deﬁnition 3.28 is motivated by the following fact. Let (L : B → A, R : A → B, η :1B ⇒ RL, ε : LR ⇒ 1A) be an adjunction such that R is lax-monoidal and F1, F2 : L → M be R-lax functors. By Remark 3.26, Fi is equipped with an L-oplax structure induced by the R-lax structure on Fi. Then a natural transformation ξ : F1 ⇒ F2 is 1R-lax if and only if ξ is a B-module natural tranformation. ♦

Proposition 3.30. For i = 1, 2, let Fî : A → B be a lax-monoidal functor and Fi : L → M an Fî-lax | | A B functor. Then we have two enriched functors F1, F2 : L → M. Given a lax-monoidal natural transformation ξˆ : Fˆ1 → Fˆ2, a natural transformation ξ : F1 → F2 is ξˆ-lax if and only if (ξ,ˆ ξ) | | | defines an enriched natural transformation ξ : F1 → F2. | | | Proof. Given an enriched natural transformation ξ : F1 ⇒ F2, we need to show that (ξ,ˆ ξ) satisfies the diagram (3.15). This is the outer diagram of the following commutative diagram

ˆ | F1(coevx)1 ( F1)x,ax1 evF1(x) Fˆ1(a)F1(x) / Fˆ1([x, ax])F1(x) / [F1(x), F1(ax)]F1(x) / F1(ax) ❚❚❚ ❚❚ [1,ξax]1 ˆ ❚❚ ξ[x,ax]1 ⋆ ❚❚ ❚❚❚❚ | ❚❚ ( F2)x,ax1 [ξx,1]1 ❚) ˆ ξaξx Fˆ2([x, ax])F1(x) / [F2(x), F2(ax)]F1(x) / [F1(x), F2(ax)]F1(x) ξax

1ξx 1ξx evF1(x) ˆ | F2(coevx)1 ( F2)x,ax1 evF2(x) x Fˆ2(a)F2(x) / Fˆ2([x, ax])F2(x) / [F2(x), F2(ax)]F2(x) / F2(ax)

16 where the pentagon ⋆ commutes because |ξ is an enriched natural transformation. Conversely, suppose the natural transformation ξ : F1 ⇒ F2 is ξˆ-lax. Then the naturality of |ξ is the outer diagram of the following commutative diagram

coevF1(x) Fˆ1([x, y]) / [F1(x), Fˆ1([x, y])F1(x)] / [F1(x), F1([x, y]x)] ◗◗ ◗◗ [1,evx] ˆ ˆ ◗◗ ξ[x,y] [1,ξ[x,y]ξx] ⋆ [1,ξ[x,y]x] ◗◗ ◗◗◗ ◗◗( Fˆ2([x, y]) [F1(x), Fˆ2([x, y])F2(x)] / [F1(x), F2([x, y]x)] [F1(x), F1(y)] ❥5 ❦5 ◗◗ [ξx,1] ❥❥ [ξx,1] ❦❦ ◗◗ [1,F2(evx)] coev ❥❥❥ ❦❦ ◗ F2(x) ❥❥ ❦❦ ◗◗ [1,ξy] ❥❥❥ ❦❦❦ ◗◗◗ ❥❥❥ ❦❦❦ ◗◗◗ ❥ ❦❦ [1,F2(evx)] [ξx,1] ( [F2(x), Fˆ2([x, y])F2(x)] / [F2(x), F2([x, y]x)] / [F2(x), F2(y)] / [F1(x), F2(y)] where the square ⋆ commutes due to the diagram (3.15).

Let LMod be the 2-category deﬁned as follows. • The objects are pairs (A, L), where A is a monoidal category and L is a strongly unital left A-oplaxmodule that is enriched in A. • A 1-morphism (A, L) → (B, M)is apair(Fˆ, F), where Fˆ : A → B is a lax-monoidal functor and F : L → M is an Fˆ-lax functor. • A 2-morphism (Fˆ, F) ⇒ (Gˆ, G) is a pair (ξ,ˆ ξ), where ξˆ : Fˆ ⇒ Gˆ is a lax-monoidal natural transformation and ξ : F ⇒ G is a ξˆ-lax natural transformation. The horizontal/vertical composition is induced by the horizontal/vertical composition of functors and natural transformations. The 2-category LMod is symmetric monoidal with the tensor product deﬁned by the Cartesian product and the tensor unit given by (∗, ∗).

By Proposition 3.27 and Proposition 3.30, we immediately obtain the following result. Theorem 3.31. The canonical construction can be promoted to a symmetric monoidal locally isomorphic 2-functor from LMod to lax |ECat defined as follows. A • The image of (A, L) is the A-enriched category L defined by the canonical construction. A B • The image of a 1-morphism (Fˆ, F):(A, L) → (B, M) is the enriched functor L → M defined by Proposition 3.27.

• The image of a 2-morphism (ξ,ˆ ξ) is the enriched natural transformation |ξ deﬁned by the background changing natural transformation ξˆ and the underlying natural transformation ξ.

In physical applications, A and L are often finite semisimple. In this case, we obtain a stronger result of the canonical construction. A finite category over a ground field k is a k-linear category C that is equivalent to the category of finite-dimensional modules over a finite-dimensional k- algebra A (see [EGNO15, Definition 1.8.6] for an intrinsic definition). We say that C is semisimple if A is semisimple. Weuse fsCat to denote the 2-cateogory of finite semisimple categories, k-linear functors and natural transformations. A Definition 3.32. An enriched category |L is called finite (semisimple) if A and L are both finite (semisimple) categories and the tensor product ⊗ : A × A → A is k-bilinear.

A Remark 3.33. Let |L be an enriched category. If the background category A is a k-linear category such that the tensor product ⊗ : A × A → A is k-bilinear, then underlying category L is also a k-linear category. ♦

Example 3.34. Let A be a multi-fusion category and L a ﬁnite semisimple left A-module. Then A the canonical construction L is a ﬁnite semisimple enriched category. q

17 B| Ai| Deﬁnition 3.35. Let M and Li be ﬁnite enriched categories, i = 1,..., n. An enriched functor | A1| An| B| F : L1 ×···× Ln → M is called multi-k-linear (or k-linear if n = 1) if the background changing functor Fˆ : A1 ×···× An → B is multi-k-linear.

| A1| An| B| Remark 3.36. Let F : L1 ×···× Ln → M be a multi-k-linear enriched functor. It is easy to see from (3.6) that the underlying functor F : L1 ×···× Ln → M is also multi-k-linear. ♦

We denote the 2-category consisting of ﬁnite semisimple enriched categories, k-linear enriched functors and enriched natural transformations by lax |fsECat. Let fsLMod be the 2-category deﬁned as follows.

• An object is a pair (A, L), where A is a ﬁnite semisimple monoidal category and L is a strongly unital ﬁnite semisimple left A-oplaxmodule that is enriched in A (i.e. the A-action functor ⊙ : A × L → L is k-bilinear).

• A 1-morphism (A, L) → (B, M)is apair(Fˆ, F), where Fˆ : A → B is a lax-monoidal k-linear functor and F : L → M is an Fˆ-lax k-linear functor.

• A 2-morphism (Fˆ, F) ⇒ (Gˆ, G) is a pair (ξ,ˆ ξ), where ξˆ : Fˆ ⇒ Gˆ is a lax-monoidal natural transformation and ξ : F ⇒ G is a ξˆ-lax natural transformation. Theorem 3.37. The canonical construction deﬁnes a 2-equivalence from fsLMod to lax |fsECat. Moreover, this 2-equivalence is locally isomorphic.

Proof. Any k-linear functor between two ﬁnite semisimple categories is exact. In particular, L is enriched in A because the functor a 7→ L(a ⊙ x, y) : A → k is exact thus admits a right adjoint. A Moreover, |L(x, −) : L → A is k-linear, thus admits a left adjoint. By [MPP18, Theorem 1.4] A (see also [Lin81]), an enriched categoy |L is equivalent to an enriched category obtained by the A canonical construction of a strongly unital oplax module if and only if every functor |L(x, −): L → A admits a left adjoint. Therefore, the 2-functor induced by canonoical construction is essentially surjective on objects. By Proposition 3.27, Proposition 3.30, it is also fully faithful on both 1-mrophisms and 2-morphisms. Then the Whitehead theorem for 2-categories (see Theorem 7.5.8 in [JY20]) implies that this 2-functor is a 2-equivalence.

Remark 3.38. We use fsECat to denote the sub-2-category of lax |fsECat consisting of those k- linear enriched functors F such that Fˆ is monoidal. We will show in the second work in our series that fsECat has a monoidal structure with the tensor product given by an analogue of Deligne’s tensor product. ♦

4 Enriched monoidal categoies

An enriched monoidal category can deﬁned as an algebraic object in either ECat or lax |ECat. In this section, we restrict ourselves to the ECat case only. Namely, the background changing functor is always monoidal in this section. For a study of the lax |ECat case, see [For04, BM12].

4.1 Deﬁnitions and examples An enriched monoidal category should be an algebra (a pseudomonoid in [DS97]) in ECat. An A algebra in ECat is a collection ( |L, |⊗, |1, |α, |λ, |ρ) such that (recall Remark 3.18)

1. the background category (A, ⊗ˆ , 1ˆ, α,ˆ λ,ˆ ρˆ) is an algebra in the symmetric monoidal 2- category Alg (Cat) of monoidal categories; E1 2. the underlying category (L, ⊗, 1, α, λ, ρ) is an algebra in the symmetric monoidal 2-category Cat of categories, i.e. a monoidal category.

18 However, in this work, we propose a slightly different definition of an enriched monoidal category for simplicity and convenience (see Remark 4.3). It is equivalent to [KZ18b, Definition 2.3] (see also [MP17] for a strict version). Note that the notion of an algebra in Alg (Cat) is E1 equivalent to that of a braided monoidal category. Definition 4.1. Let A be a braided monoidal category. An enriched monoidal category consists of the following data: A • an enriched category |L;

| A| A| A| • a tensor product enriched functor ⊗ : L × L → L in ECat such that ⊗ˆ = ⊗A, where the monoidal structure on ⊗A is given by Covention 4.2;

A| • a distinguished object 1L ∈ L called the unit object, or equivalently, an enriched functor | A| 1L : ∗ → L in ECat (see Example 3.12);

• an associator: an enriched natural isomorphism |α : |⊗ ◦ (|⊗× 1) ⇒ |⊗ ◦ (1 × |⊗) withα ˆ given by the associator of the monoidal category A (i.e.α ˆ = αA);

| | | | | • two unitors: two enriched natural isomorphisms λ : ⊗ ◦ ( 1L × 1) ⇒ 1A|L and ρ : ⊗ ◦ | (1 × 1L) ⇒ 1A|L such that λˆ andρ ˆ are given by the left and right unitors of the monoidal category A, respectively (i.e. λˆ = λA, ρˆ = ρA); such that (L, ⊗, 1L, α, λ, ρ) is a monoidal category, which is called the underlying monoidal category A| A| | | | | | A| of L = ( L, ⊗, 1L, α, λ, ρ). The enriched monoidal category L is called strict if the underlying monoidal category L is strict.

Convention 4.2. Let A a braided monoidal category with the braiding ca,b : a ⊗ b → b ⊗ a. In the following, the monoidal structure on the tensor product functor (a, b) 7→ a ⊗ b : A × A → A is always understood as the lax-monoidal structure induced by the braiding:

⊗ ⊗ 1 cb1,a2 1 ⊗(a1, b1) ⊗⊗(a2, b2) = a1 ⊗ b1 ⊗ a2 ⊗ b2 −−−−−−−→ a1 ⊗ a2 ⊗ b1 ⊗ b2 = ⊗((a1, b1) ⊗ (a2, b2)), or equivalently, as the oplax-monoidal structure of ⊗ is deﬁned by the anti-braiding.

A Remark 4.3. Given an algebra ( |L, |⊗, |1, |α, |λ, |ρ) in ECat, there is a canonical isomorphism φ : ≃ ⊗ˆ −→⊗A,andα, ˆ λ,ˆ ρˆ are compatible with αA, λA, ρA (through φ), respectively. For convenience, we require ⊗ˆ = ⊗A andα ˆ = αA, λˆ = λA, ρˆ = ρA in Definition 4.1. It turns out that these additional requirements also endow Definition 4.1 with a new interpretation as an algebra in the monoidal A category |ECat with the tensor product ×˙ (recall Example 3.20). This new interpretation is how the notion was defined in [KZ18b, Definition 2.3]. ♦

Remark 4.4. We use A to denote the same monoidal category A equipped with the anti-braiding. By Convention 4.2, the lax/oplax-monoidal structure of ⊗ : A × A → A is induced by the anti- braiding/braiding of A. ♦

A Remark 4.5. Let |L be an enriched monoidal category. We use Lrev to denote the monoidal category obtained from L by reversing the tensor product. One can canonically construct an A A-enriched monoidal category |Lrev, whose underlying monoidal category is Lrev, as follows. A A A A As an enriched category |Lrev = |L. The tensor product enriched functor |⊗rev : |Lrev × |Lrev → A |Lrev is deﬁned by the tensor product functor A × A → A and the family of morphisms

| cA|L A|L ⊗ A| A| (x1,x2), (y1,y2) A| A| (y1,x1),(y2,x2) A| rev rev L(x1, x2) ⊗ L(y1, y2) −−−−−−−−−−−−→ L(y1, y2) ⊗ L(x1, x2) −−−−−−−−→ L(x1 ⊗ y1, x2 ⊗ y2).

|F2 : |⊗ ◦ (|F × |F) =⇒ |F ◦ |⊗ with the background changing natural transformation Fˆ2 given by the monoidal structure of Fˆ; • an enriched natural isomorphism

| 0 | | | F : 1M =⇒ F ◦ 1L with the background changing natural transformation Fˆ0 given by the monoidal structure of Fˆ; satisfying the following conditions: 1. the background changing functor Fˆ = (Fˆ, Fˆ2, Fˆ0): A → B is a braided monoidal functor; 2. the underlying functor F : L → M, together with the underlying natural transformations F2 and F0, deﬁnes a monoidal functor.

A B A B Deﬁnition 4.8. Let |L, |M be enriched monoidal categories and |F, |G : |L → |M enriched monoidal functors. An enriched monoidal natural transformation is an enriched natural transformation |ξ : |F ⇒ |G such that the underlying natural transformation ξ : F ⇒ G is a monoidal natural transformation. Remark 4.9. The symmetic monoidal 2-category of enriched monoidal categories, enriched monoidal functors and enriched monoidal natural transformations is a sub-2-category of the 2-category Alg (ECat) of algebras in ECat, and the inclusion is a biequivalence. ♦ E1

Example 4.10. Let (A, mA,ιA) be a commutative algebra in a braided monoidal category A. The A-enriched category ∗A introduced in Example 3.11 is a strict enriched monoidal category with the tensor product functor induced by mA. Conversely, every strict A-enriched monoidal category with one object arises in this way (see also [KZ18b, Example 3.5]). | A B Let (B, mB,ιB) be a commutative algebra in a braided monoidal category B and F : ∗ → ∗ be an enriched monoidal functor. The enriched natural isomorphisms |F2, |F0 are determined by 2 0 2 0 2 0 the underlying natural isomorphisms F , F , and any morphism (F )∗, (F )∗ together with Fˆ , Fˆ form an enriched natural transformation respectively since B is commutative. The condition 2 0 2 0 that (F, F , F ) is a monoidal functor is equivalent to say that (F )∗ is the inverse of (F )∗ in the underlying category of ∗B. Hence an enriched monoidal functor |F : ∗A → ∗B is deﬁned by a | braided monoidal functor Fˆ : A → B, an algebra homomorphism F∗,∗ : Fˆ(A) → B (see Example 0 B 3.11) and an isomorphism (F )∗ in the underlying category of ∗ . Let |G : ∗A → ∗B be another enriched monoidal functor deﬁned by a braided monoidal functor | 0 Gˆ, an algebra homomorphism G∗,∗ : Gˆ(A) → B and an isomorphism (G )∗. Then an enriched natural transformation |ξ : |F ⇒ |G is an enriched monoidal natural transformation if and only if 0 0 B (G )∗ = ξ∗ ◦ (F )∗ in the underlying category of ∗ , which further implies that ξ∗ is invertible. If ξ∗ is invertible, the commutativity of B implies that the diagram 3.9 is equivalent to

| | F∗,∗ ξˆA G∗,∗ Fˆ(A) −−→ B = Fˆ(A) −→ Gˆ(A) −−→ B.

As a conclusion, an enriched natural transformation |ξ : |F ⇒ |G is deﬁned by a monoidal natural | | transformation ξˆ such that F∗,∗ = G∗,∗ ◦ ξˆA and the underlying natural transformation is given 0 0 −1 q by ξ∗ = (G )∗ ◦ (F )∗ .

20 4.2 The canonical construction Let A be a braided monoidal category viewed as an algebra in Algoplax(Cat) by Convention 4.2. E1

Deﬁnition 4.11. A monoidal left A-oplaxmodule is a left A-oplaxmodule in Algoplax(Cat).A monoidal E1 left A-module is a left A-module in Alg (Cat). E1

Remark 4.12. More explicitly, a monoidal left A-oplaxmodule L consists of the following data:

• a monoidal category L; • a left A-action given by an oplax-monoidal functor ⊙ : A × L → L (i.e. the functor ⊙ is equipped with a natural transformation

(a ⊗ b) ⊙ (x ⊗ y) → (a ⊙ x) ⊗ (b ⊙ y) ∀a, b ∈ A, x, y ∈ L (4.1)

and a morphism 1A ⊙ 1L → 1L rendering the following diagrams commutative);

[(a ⊗ b) ⊗ c] ⊙ [(x ⊗ y) ⊗ z] / [a ⊗ (b ⊗ c)] ⊙ [x ⊗ (y ⊗ z)] [(a ⊗ b) ⊙ (x ⊗ y)] ⊗ (c ⊙ z) (a ⊙ x) ⊗ [(b ⊗ c) ⊙ (y ⊗ z)] (4.2) [(a ⊙ x) ⊗ (b ⊙ y)] ⊗ (c ⊙ z) / (a ⊙ x) ⊗ [(b ⊙ y) ⊗ (c ⊙ z)]

a ⊙ x o (1A ⊗ a) ⊙ (1L ⊗ x) a ⊙ x o (a ⊗ 1A) ⊙ (x ⊗ 1L) O O (4.3) (4.4) 1L ⊗ (a ⊙ x) o (1A ⊙ 1L) ⊗ (a ⊙ x) (a ⊙ x) ⊗ 1L o (a ⊙ x) ⊗ (1A ⊙ 1L)

• two oplax-monoidal natural transformation: the associator(a ⊗ b) ⊙ x → a ⊙ (b ⊙ x) and the unitor 1A ⊙ x → x (i.e. the following diagrams are commutative2,

1 1 1 [(a1 ⊗ a2) ⊗ (b1 ⊗ b2)] ⊙ (x ⊗ y) / (a1 ⊗ a2) ⊙ [(b1 ⊗ b2) ⊙ (x ⊗ y)] A ⊙ (x ⊗ y) / ( A ⊗ A) ⊙ (x ⊗ y)

(4.5) [(a1 ⊗ b1) ⊗ (a2 ⊗ b2)] ⊙ (x ⊗ y) (a1 ⊗ a2) ⊙ [(b1 ⊙ x) ⊙ (b2 ⊗ y)]

[(a1 ⊗ b1) ⊙ x] ⊗ [(a2 ⊗ b2) ⊙ y] / [a1 ⊙ (b1 ⊙ x)] ⊗ [a2 ⊙ (b2 ⊙ y)] x ⊗ y o (1A ⊙ x) ⊗ (1A ⊙ y)

(1A ⊗ 1A) ⊙ 1L / 1A ⊙ (1A ⊙ 1L) 1A ⊙ 1L / 1L ▲ unitor ▲▲▲ oplax ▲▲ (4.6) oplax ▲▲▲ oplax oplax ▲& 1A ⊙ 1L / 1L o 1A ⊙ 1L 1L

where the the arrow [(a1 ⊗ a2) ⊗ (b1 ⊗ b2)] ⊙ (x ⊗ y) → [(a1 ⊗ b1) ⊗ (a2 ⊗ b2)] ⊙ (x ⊗ y) is deﬁned −1 by the anti-braiding c : a2 ⊗ b1 → b1 ⊗ a2 (recall Convention 4.2)); b1,a2 such that L, together with the left A-action functor ⊙, the associator and the unitor deﬁned above, is a left A-oplaxmodule. If, in addition, the left A-action functor ⊙ is a monoidal functor, and the associator and the unitors are natural isomorphisms, then L is a monoidal left A-module. ♦

Remark 4.13. If, in addition, L is a strongly unital left A-module, then the commutativity of the second diagram in (4.6) follows from those of diagram (4.3), (4.4) and (4.5). Also, the commutativity of the first diagram in (4.6) is a consequence of that of the second diagram in (4.6) and the fact that L is an A-oplaxmodule. ♦ 2The first diagrams in both (4.5) and (4.6) are the defining properties of the associator as an oplax-monoidal natural transformation. The other two are those of the unitors.

21 Remark 4.14. By Deﬁnition 4.11, a monoidal structure on a left A-oplaxmodule L consists of a monoidal structure on L and a natural transformation (4.1) satisfying proper axioms. All the rest data can be included in the deﬁning data of a left A-oplaxmodule structure on L. ♦

In the following, we use I to denote the forgetful functor Z1(L) → L for any monoidal c category L. If ϕ : A → Z1(L) is a braided (oplax) monoidal functor, then ϕ ≔ I ◦ ϕ is an (oplax) central functor [Bez04].

Example 4.15. Let A be a braided monoidal cateogry and L a monoidal category. If ϕ : A → oplax Z1(L) is a braided oplax-monoidal functor, then L is a monoidal left A- module with the module action ϕc(−) ⊗− : A × L → L. The oplax-monoidal structure ϕ(a ⊗ b) ⊗ (x ⊗ y) → (ϕ(a) ⊗ x) ⊗ (ϕ(b) ⊗ y) is given by

ϕc(a ⊗ b) ⊗ (x ⊗ y) → (ϕc(a) ⊗ ϕc(b)) ⊗ (x ⊗ y) → (ϕc(a) ⊗ (ϕc(b) ⊗ x)) ⊗ y → (ϕc(a) ⊗ (x ⊗ ϕc(b))) ⊗ y → (ϕc(a) ⊗ x) ⊗ (ϕc(b) ⊗ y), where the third arrow is deﬁned by the half-braiding of ϕ(b). q

Remark 4.16. When L is a monoidal left A-module, it reduces to the usual definition (see for example [KZ18a, Definition 2.6.1]). Indeed, in this case, ϕ ≔ (−⊙ 1L) defines a braided monoidal functor ϕ : A → Z1(L) (recall Example 2.16). Two monoidal left A-modules (L, ⊙) and (L,ϕc(−) ⊗−) are isomorphic. Hence, a monoidal left A-module L can be equivalently defined by a monoidal category L equipped with a braided monoidal functor A → Z1(L). It is important to note that our convention is different from [KZ18a, Definition 2.6.1]. When L is a monoidal left A-oplaxmodule, the morphism (4.1) is not necessarily invertible, thus does not induce a half-braiding. ♦

A Proposition 4.17. Given a pair (A, L) ∈ LMod, let L be the enriched category obtained from the pair (A, L) via the canonical construction. There is a one-to-one correspondence between the A monoidal structures on L and the monoidal structures on the left A-oplaxmodule L as shown by the following two mutually inverse constructions:

A A | | | | | ⇒ Given a monoidal structure on L, i.e. a sextuple ( L, ⊗, 1L, α, λ, ρ), then the sextuple (L, ⊗, 1L, α, λ, ρ) automatically defines a monoidal structure on L by definition. A monoidal structure on the left A-oplaxmodule L is determined (recall Remark 4.14) if we further define the morphisms (4.1) to be the one induced from the composed morphism:

| coevx ⊗ coevy ⊗ (a ⊗ b) −−−−−−−−−→ [x, a ⊙ x] ⊗ [y, b ⊙ y] −→ [x ⊗ y, (a ⊙ x) ⊗ (b ⊙ y)].

A ⇐ Given a monoidal structure on the left A-oplaxmodule L. A monoidal structure on L is determined by deﬁning the morphism [x1, x2] ⊗ [y1, y2] → [x1 ⊗ y1, x2 ⊗ y2] to be the one induced by the following composed morphism

⊗ evx1 evy1 ([x1, x2] ⊗ [y1, y2]) ⊙ (x1 ⊗ y1) → ([x1, x2] ⊙ x1) ⊗ ([y1, y2] ⊙ y1) −−−−−−−→ x2 ⊗ y2,

where the ﬁrst arrow is given by the oplax-monoidal structure of ⊙ : A × L → L.

Proof. It is clear that the diagram (4.5) for a monoidal left A-oplaxmodule L commutes if and only if ⊗L is a ⊗A-lax functor with the ⊗A-lax structure given by the oplax-monoidal structure of ⊙, and the diagram (4.2), (4.3), (4.4) commutes if and only if (α, ˆ α), (λ,ˆ λ), (ˆρ, ρ) is a 2-morphism in LMod, respectively. Then the assertion follows from Theorem 3.31.

Example 4.18 (Construction 5.1 in [KZ18b]). Let ϕ : A → Z1(L) be a strongly unital braided oplax-monoidal functor. Then L equipped with the module action ϕc(−) ⊗− : A × L → L is a strongly unital monoidal left A-oplaxmodule (see Example 4.15). If L is also enriched in A, then A L is an A-enriched monoidal category. q

22 Example 4.19. Let L be a left A-module. We use FunA(L, L) to denote the category of left A-module functors and left A-module natural transformations. There is an obvious braided monoidal functor Z1(A) → Z1(FunA(L, L)) deﬁned by a 7→ I(a) ⊙−. The left A-module functor I(a) ⊙− is equipped with a half-braiding

γF,I(a) : F(I(a) ⊙−) ⇒ I(a) ⊙ F(−), ∀F ∈ FunA(L, L), deﬁned by the left A-module functor structure on F. It follows that FunA(L, L) is a monoidal left Z1(A)-module. q

Deﬁnition 4.20. Let A, B be two braided monoidal categories. Let L be a monoidal left A- oplaxmodule and M a monoidal left B-oplaxmodule. Given a braided monoidal functor Fˆ : A → B, a monoidal F-laxˆ functor F : L → M is a monoidal functor and also an Fˆ-lax functor such that the Fˆ-lax structure Fˆ(a) ⊙ F(x) → F(a ⊙ x) is an oplax-monoidal natural transformation (i.e. the following diagrams commute).

Fˆ(a ⊗ b) ⊙ F(x ⊗ y) / F((a ⊗ b) ⊙ (x ⊗ y))

(Fˆ(a) ⊗ Fˆ(b)) ⊙ (F(x) ⊗ F(y)) F((a ⊙ x) ⊗ (b ⊙ y)) (4.7)

(Fˆ(a) ⊙ F(x)) ⊗ (Fˆ(b) ⊙ F(y)) / F(a ⊙ x) ⊗ F(b ⊙ y)

Fˆ(1) ⊙ F(1) / F(1 ⊙ 1) (4.8) 1 ⊙ 1 / 1 o F(1)

Remark 4.21. Let ϕi : Ai → Z1(Li) be a braided oplax-monoidal functor, i = 1, 2. Then Li oplax is a monoidal left Ai- module. Let Fˆ : A1 → A2 be a braided monoidal functor and F : L → L c ˆ → c 1 2 be a monoidal functor. Suppose θa : ϕ2(F(a)) F(ϕ1(a)) is an oplax-monoidal natural transformation rendering the following diagram commutative,

θ ⊗1 c ˆ ⊗ a c ⊗ ∼ c ⊗ ϕ2(F(a)) F(x) / F(ϕ1(a)) F(x) / F(ϕ1(a) x) (4.9) 1⊗ ⊗ c ˆ θa ⊗ c ∼ ⊗ c F(x) ϕ2(F(a)) / F(x) F(ϕ1(a)) / F(x ϕ1(a)) where two vertical arrows are the half-braidings of ϕ2(Fˆ(a)) and ϕ1(a), respectively. Then F equipped with the natural transformation

≔ c ˆ θa⊗1 c ∼ c βa,x ϕ2(F(a)) ⊗ F(x) −−−→ F(ϕ1(a)) ⊗ F(x) −→ F(ϕ1(a) ⊗ x) is a monoidal Fˆ-lax functor. We claim that every monoidal Fˆ-lax structure on F arises in this way. Indeed, suppose c ˆ ⊗ → c ⊗ ˆ βa,x : ϕ2(F(a)) F(x) F(ϕ1(a) x) is a monoidal F-lax structure on F. Then

βa,1L c ∼ c 1 c ∼ c ≔ ˆ −→ ˆ ⊗ 1L −−−−→ ⊗ 1L −→ θa ϕ2(F(a)) ϕ2(F(a)) F( 1 ) F(ϕ1(a) 1 ) F(ϕ1(a)) 1 1 is an oplax-monoidal natural transformation. Taking b = A1 and x = L2 in the diagram (4.7), then we have ⊗ ∼ = c ˆ θa 1 c c βa,y ϕ2(F(a)) ⊗ F(y) −−−→ F(ϕ1(a)) ⊗ F(y) −→ F(ϕ1(a) ⊗ y). 1 1 By taking a = A1 and y = L2 in the diagram (4.7), we can show that the oplax-monoidal natural A A ˆ transformation θa renders the diagram (4.9) commutative. Hence, when 1 = 2 and F = 1A1 , Deﬁnition 4.20 is precisely [KZ18a, Deﬁnition 2.6.6]. ♦

23 Proposition 4.22. Let A, B be braided monoidal categories. Suppose L is a strongly unital monoidal left A-oplaxmodule that is enriched in A, and M is a strongly unital monoidal left B-oplaxmodule that is enriched in B. Then the canonical construction gives enriched monoidal A B categories L and M. Suppose Fˆ : A → B is a braided monoidal functor and F : L → M is both a monoidal functor and an Fˆ-lax functor. Then F is a monoidal Fˆ-lax functor if and only if Fˆ and A B F define an enriched monoidal functor |F : L → M. A B Proof. By Theorem 3.31, the Fˆ-lax functor F defines an enriched functor |F : L → M. Then we only need to show that F is a monoidal Fˆ-lax functor if and only if (Fˆ2, F2) and (Fˆ0, F0) are 2-morphisms in LMod. It is routine to check that (Fˆ2, F2) is a 2-morphism in LMod if and only if the diagram (4.7) commutes, and (Fˆ0, F0) is a 2-morphism in LMod if and only if the diagram (4.8) commutes. Definition 4.23. Let A, B be two braided monoidal categories. Let L be a monoidal left A- oplax oplax module and M be a monoidal left B- module. Suppose Fî : A → B is a braided monoidal functor and Fi : L → M is a monoidal Fî-lax functor, i = 1, 2. Given a monoidal natural transformation ξˆ : Fˆ1 ⇒ Fˆ2,a monoidal ξˆ-lax natural transformation ξ : F1 ⇒ F2 is a ξˆ-lax natural transformation and also a monoidal natural transformation. oplax Remark 4.24. For i = 1, 2, let Li be a monoidal left Ai- module induced by a braided oplax- monoidal functor ϕi : Ai → Z1(Li). Assume that Fî : A1 → A2 is a braided monoidal functor and Fi : L1 → L2 is a monoidal Fî-lax functor. Let ξˆ : Fˆ1 → Fˆ2 and ξ : F1 → F2 be two monoidal natural transformations. Then ξ is a monoidal ξˆ-lax natural transformation if and only if the following diagram commutes,

c ˆ θ1 c ϕ2(F1(a)) / F1(ϕ1(a)) ξˆ ξ c ˆ θ2 c ϕ2(F2(a)) / F2(ϕ1(a)) c ◦ ˆ ⇒ ◦ c where θi : ϕ2 Fi Fi ϕ1 is the oplax-monoidal natural transformation that induces the monoidal Fˆi-lax module structure on Fi (see Remark 4.21). ♦ The proof of the following proposition is clear.

Proposition 4.25. Under the assumption of Proposition 4.22, let Fî : A → B be a braided monoidal functor and Fi : L → M be a monoidal Fî-lax functor, i = 1, 2. Then (Fî, Fi) defines an | A B enriched monoidal functor Fi : L → M for i = 1, 2. Suppose ξˆ : Fˆ1 ⇒ Fˆ2 is a monoidal natural transformation and ξ : F1 ⇒ F2 is a ξˆ-lax natural transformation. Then ξ is a monoidal ξˆ-lax natural transformation if and only if (ξ,ˆ ξ) defines an enriched monoidal natural transformation. We use ECat⊗ to denote the 2-category of enriched monoidal categories, enriched monoidal functors and enriched monoidal natural transformations. It is symmetric monoidal with the tensor product given by the Cartesian product × and the tensor unit given by ∗. We define the 2-category LMod⊗ as follows: • The objects are pairs (A, L), where A is a braided monoidal category and L is a strongly unital monoidal left A-oplaxmodule that is enriched in A. • A 1-morphism (A, L) → (B, M) is a pair (Fˆ, F), where Fˆ : A → B is a braided monoidal functor and F : L → M is a monoidal Fˆ-lax functor. • A 2-morphism (Fˆ, F) ⇒ (Gˆ, G) is a pair (ξ,ˆ ξ), where ξˆ : Fˆ ⇒ Gˆ is a monoidal natural transformation and ξ : F ⇒ G is a monoidal ξˆ-lax natural transformation. The horizontal/vertical composition is induced by the horizontal/vertical composition of functors and natural transformations. The 2-category LMod⊗ is symmetric monoidal with the tensor product defined by the Cartesian product and the tensor unit given by (∗, ∗). By Proposition 4.17, Proposition 4.22 and Proposition 4.25 we obtain the following result. Theorem 4.26. The canonical construction defines a symmetric monoidal 2-functor from LMod⊗ to ECat⊗. Moreover, this 2-functor is locally isomorphic.

24 4.3 The category of enriched endo-functors B B B Let |M be a B-enriched category. We use Fun( |M, |M) to denote the category of B-functors A from |M to itself and B-natural transformations between them (recall Deﬁnition 3.10). For every | | B|M B|M P F, G ∈ Fun( , ), we deﬁne a category (|F,|G) as follows: B| • The objects are pairs (a, {ax}x∈M), where a ∈ Z1(B) and ax : I(a) → M(F(x), G(x)) are morphisms in B such that the following diagram

commutes, where the unlabeled arrow is given by the half-braiding of a.

• A morphism f :(a, {ax}) → (b, {bx}) is a morphism in Z1(B) such that the follwoing diagram

I( f ) I(a) / I(b) (4.11) ◗◗ ♠ ◗◗◗ ♠♠♠ a ◗◗ ♠♠ x ◗◗ ♠♠ bx B ( v♠ |M(F(x), G(x))

commutes for every x ∈ M. B Deﬁnition 4.27. We say |M satisﬁes the condition (∗) if: | | B|M B|M P for every F, G ∈ Fun( , ), the category (|F,|G) has a terminal object. (∗) P | | | | We denote the terminal object of (|F,|G) by ([ F, G], {[ F, G]x}x∈M).

B| Remark 4.28. A family of morphisms {ξx : 1B → M(F(x), G(x))} renders the diagram (4.10) commutative (with I(a), ax replaced by 1B, ξx, respectively) if and only if the family of morphisms | | | B| {ξx} defines a B-natural transformation ξ : F ⇒ G. In particular, if M satifies the condition (∗), | | B| B| | | then the hom set Z1(B)(1B, [ F, G]) is isomorphic to the hom set Fun( M, M)( F, G). ♦ Remark 4.29. Let (A, m,ι) be an algebra in a monoidal category A. By Example 3.11, Fun(∗A, ∗A) is the category defined as follows: • The objects are algebra homomorphisms f : A → A. • A morphism f → g is a morphism ξ ∈ A(1, A) such that the following diagram commutes.

∼ ξ⊗ f A / 1 ⊙ A / A ⊗ A ∼ m g⊗ξ m A ⊗ 1 / A ⊗ A / A

It is also clear that the object [1∗A , 1∗A ] (if exists) is the full center of A (see [Dav10] for the deﬁnition of the full center of an algebra). ♦

Lemma 4.30. There are two well-defined functors. P P P (1) (|G,|H) × (|F,|G) → (|F,|H) is defined by ((b, {bx}), (a, {ax})) 7→ (b ⊗ a, {bx}◦{ax}), where {bx}◦{ax} is defined by the composed morphism:

25 P P P ′ ′ ′ ′ ′ (2) (|F,|G) × (|F′,|G′) → (|FF′,|GG′) deﬁned by (a, {ax}) ⊗ (a , {ax}) 7→ (a ⊗ a , {ax}•{ax}), where {ax}•{ax} is deﬁned by the composed morphism:

B B B Proposition 4.31. If |M satisﬁes the condition (∗), then Fun( |M, |M) can be promoted to a strict Z1(B)| B| B| Z1(B)-enriched monoidal category Fun( M, M). More explicitly, B B B • the hom objects are Z1( )|Fun( |M, |M)(|F, |G) ≔ [|F, |G];

| | • the identity morphism 1B → [ F, F] is induced by the identity enriched natural transformation of F via the universal property of [|F, |F];

• the composition morphism [|G, |H] ⊗ [|F, |G] → [|F, |H] is deﬁned Lemma4.30 (1). ′ ′ • the tensor product morphism [|F, |G] ⊗ [|F′, |G′] → [|F|F , |G|G ] is deﬁned by Lemma 4.30 (2).

Z1(B)| B| B| Proof. It is routine to check that Fun( M, M) is an Z1(B)-enriched category. Note that B B B B B the underlying category of Z1( )|Fun( |M, |M) is precisely Fun( |M, |M) by Remark 4.28. This explains the notation. B B B It remains to show that Z1( )|Fun( |M, |M) is monoidal. We claim the the following equation holds

| | | | | | | | | | | | | | | | [ F2, F3] ⊗ [ G2, G3] ⊗ [ F1, F2] ⊗ [ G1, G2] → [ F2, F3] ⊗ [ F1, F2] ⊗ [ G2, G3] ⊗ [ G1, G2] ◦⊗◦ | | | | | | | | −−→ [ F1, F3] ⊗ [ G1, G3] → [ F1 G1, F3 G3] | | | | [ F1 G1, F3 G3]x B|M −−−−−−−−−→ (F1G1(x), F3G3(x)) | | | | | | | | | | | | | | | | ◦ | | | | = [ F2, F3] ⊗ [ G2, G3] ⊗ [ F1, F2] ⊗ [ G1, G2] → [ F2 G2, F3 G3] ⊗ [ F1 G1, F2 G2] −→ [ F1 G1, F3 G3] | | | | [ F1 G1, F3 G3]x B|M −−−−−−−−−→ (F1G1(x), F3G3(x)),

| | where the ﬁrst unlabeled arrow is induced by the half braiding of [ F1, F2]. Indeed, by the deﬁnition of [|G, |H] ⊗ [|F, |G] → [|F, |H] and [|F, |G] ⊗ [|F′, |G′] → [|F|F′, |G|G′], it not hard to check that both sides of the above equation are equal to

| | | | | | | | | | | | 1⊗[ G2, G3]x⊗1⊗[ G1, G2]x [ F2, F3] ⊗ [ G2, G3] ⊗ [ F1, F2] ⊗ [ G1, G2] −−−−−−−−−−−−−−−−−→ [|F ,|F ] ⊗(|F ) ⊗[|F ,|F ] ⊗(|F ) | | B| | | B| 2 3 G3(x) 2 G2(x),G3(x) 1 2 G2(x) 1 G1(x),G2(x) [ F2, F3] ⊗ M(G2(x), G3(x)) ⊗ [ F1, F2] ⊗ M(G1(x), G2(x)) −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ B| B| B| B| M(F2G3(x), F3G3(x)) ⊗ M(F2G2(x), F2G3(x)) ⊗ M(F1G2(x), F2G2(x)) ⊗ M(F1G1(x), F1G2(x)) B| → M(F1G1(x), F3G3(x)).

Similar, we can check that

Therefore the morphisms [|F, |G] ⊗ [|F′, |G′] → [|F|F′, |G|G′] deﬁne an enriched functor. Clearly the B B B B B underlying monoidal category of Z1( )|Fun( |M, |M) is Fun( |M, |M).

B B B B In the rest of this subsection, we discuss the construction of Z1( )|Fun( |M, |M) when |M = B M for (B, M) ∈ LMod and M is a left B-module. In particular, in this case, we give an easy-to-check condition that is equivalent to the condition(∗) (see Lemma 4.32). B B lax By Theorem 3.31, we can identify Fun( M, M) with FunB (M, M), i.e. the category of lax B- module functors and B-module natural transformations. There is an obvious monoidal functor lax from Z1(B) into FunB (M, M) which maps a to the B-module functor I(a) ⊙−. It is not hard to

26 lax check that FunB (M, M) is a strongly unital monoidal left Z1(B)-module with the module action lax induced by the monoidal functor Z1(B) → FunB (M, M) and the natural transformation

∼ I(a ⊗ b) ⊙ FG(−) −→ I(a) ⊙ [I(b) ⊙ FG(−)] → I(a) ⊙ F(I(b) ⊙ G(−)), (4.12) where the ﬁrst arrow is induced by the monoidal structure of I and the left B-module structure of M and the second arrow is induced by the lax B-module structure of F. B Lemma 4.32. When (B, M) ∈ LMod and M is a left B-module, M satisﬁes the condition (∗) lax | | if and only if FunB (M, M) is enriched in Z1(B). In this case, the object [ F, G] is given by the | | | | B internal hom [F, G] ∈ Z1(B), and the morphism [ F, G]x :[ F, G] → M(F(x), G(x)) is induced by (ev ) I([F, G]) ⊙ F(x) = ([F, G] ⊙ F)(x) −−−−→F x G(x).

| | | | Proof. Since both ([ F, G], {[ F, G]x}x∈M) and the internal hom ([F, G], evF = {(evF)x}x∈M) are terminal objects, it suﬃces to show that they satisfy the same universal property. Let F, G ∈ lax FunB (M, M) and a ∈ Z1(B). Given a family of morphisms {αx : I(a) → [F(x), G(x)]}x∈M), deﬁne

ev ≔ αx⊙1 F(x) α˜ x I(a) ⊙ F(x) −−−→ [F(x), G(x)] ⊙ F(x) −−−→ G(x).

Conversely, coevF(x) [1,α˜ x] αx = I(a) −−−−−→ [F(x), I(a) ⊙ F(x)] −−−−→ [F(x), G(x)].

It is routine to check that {αx} renders the diagram (4.10) commutative if and only if the following diagram commutes,

γb,a⊙1 ≃ (b ⊗ I(a)) ⊙ F(x) / (I(a) ⊗ b) ⊙ F(x) / I(a) ⊙ (b ⊙ F(x))

≃ b ⊙ (I(a) ⊙ F(x)) I(a) ⊙ F(b ⊙ x) (4.13)

1⊗α˜ x α˜ b⊙x b ⊙ G(x) / G(b ⊙ x) i.e.α ˜ : a ⊙ F ⇒ G is a B-module natural transformation.

B lax If M satisﬁes the condition (∗), by Proposition 4.31, FunB (M, M) can be promoted to a Z1(B)| lax Z1(B)-enriched monoidal category FunB (M, M). We obtain the following result.

lax Proposition 4.33. When (B, M) ∈ LMod and M is a left B-module, if FunB (M, M) is enriched in Z1(B), then we obtain a monoidal equivalence:

Z1(B)| B B Z1(B)| lax Z1(B) lax Fun( M, M) = FunB (M, M) ≃ FunB (M, M).

Proof. By Lemma4.32, we can set [|F, |G] = [F, G]. It suﬃces to verify (1) and (2).

(1) The composition [|G, |H] ⊗ [|F, |G] → [|F, |H] deﬁned in Lemma 4.30 coincides with the canonical morphism [G, H] ⊗ [F, G] → [F, H]. It is because both morphisms are induced from the same universal property according to Lemma 4.32. ′ ′ (2) The tensor product morphism [|F, |G] ⊗ [|F′, |G′] → [|F|F , |G|G ] deﬁned in Lemma 4.30 is induced by

ev ⊗ ev ′ [F, G] ⊗ [F′, G′] ⊙ FF′ → [F, G] ⊙ F ⊗ [F′, G′] ⊙ F′ −−−−−−−→F F GG′,

where the ﬁrst arrow is induced by the lax B-module functor structure of F.

27 In other words, we need to show the outer subdiagram of the following diagram

1⊗[F′,G′] ⊙1 [F, G] ⊗ [F′, G′] ⊙ FF′(x) x / [F, G] ⊗ [F′(x), G′(x)] ⊙ FF′(x) ❣❣ ❣❣❣❣ ❣❣❣❣ 1⊗F⊙1 ❣❣❣❣❣ s❣❣❣❣ [F, G] ⊙ F([F′, G′] ⊙ F′(x)) / [F, G] ⊙ F([F′(x), G′(x)] ⊙ F′(x)) [F, G] ⊗ [FF′(x), FG′(x)] ⊙ FF′(x) ❝ ❤❤❤❤ ❝❝❝❝❝❝❝❝ ⊙ ′ ❤❤ ❝❝❝❝ ′ ⊙ ′ 1⊙F(evF′ ) 1 F(evF (x)) ❝❝❝❝❝ [F,G]G (x) evFF (x) ❝❝❝❝❝⊙ ′ ❤❤❤ ❝❝❝❝❝❝ 1 evFF (x) t❤❤❤ ❝❝❝❝❝❝❝❝ ⊙ ′ q❝❝ ′ ′ ′ ⊙ ′ [F, G] FG (x) / GG (x)[o ev ′ FG (x), GG (x)] FG (x) (evF)G′(x) FG (x) commutes for each x ∈ M. The upper left quadrangle commutes due to the naturality of the lax B-module functor structure of F, and the other subdiagrams commute by the adjunctions associated to internal homs.

lax When B is rigid, we have FunB (M, M) = FunB(M, M), i.e. the category of B-module functors, because the morphism a ⊙ F(x) → F(a ⊙ x) now has an inverse given by F(a ⊙ x) → (a ⊗ aL) ⊙ F(a ⊙ x) → a ⊗ F(aL ⊙ (a ⊙ x)) → a ⊙ F(x) for a ∈ B, x ∈ M.

Corollary 4.34. When (B, M) ∈ LMod and M is a left B-module, if B is rigid and FunB(M, M) is enriched in Z1(B), we have Z (B)| B B Z (B) 1 Fun( M, M) ≃ 1 FunB(M, M).

4.4 The E0-centers of enriched categroies in ECat

Z1(B)| B| B| B| B| In this subsection, we prove that Fun( M, M) is the E0-center of M in ECat when M satisﬁes the condition (∗). A| | | A| Recall that an E0-algebra in ECat is a pair ( L, U), where U : ∗ → L is an enriched A B functor in ECat. A left unital ( |L, |U)-action on |M in ECat consists of an enriched functor A B B |⊙ : |L × |M → |M in ECat and an enriched natural isomorphism |ξ as depicted in the following diagram (recall Deﬁnition 2.9).

A B |L × |M (4.14) | 6 ❖ | U×1 ♠♠♠ ❖❖❖ ⊙ ♠♠ ✤✤ | ❖❖ ♠♠ ✤✤ ξ ❖❖ B ♠ ❖' B ∗× |M / |M 1B|M

| | | We set 1L ≔ U(∗) ∈ L. There exists a canonical enriched natural isomorphism η : 1L ⇒ U (recall Example 3.12) deﬁned by 1A → Uˆ (∗) (from the monoidal structure of Uˆ ) and the identity A| | underlying natural transformation. It deﬁnes a left unital ( L, 1L)-action as depicted in the following diagram.

A|L × B|M 5 O |1 L×1B|M 1A|L×1B|M |⊙ ✤✤ A| B| ✤✤ |η×1 L × M | 6 P U×1B|M ♠♠ P |⊙ ♠♠♠ PPP ♠♠ ✤✤ | PP ♠♠♠ ξ PPP B| ♠ P( B| ∗× M / M 1B|M

A| | A| | In particular, the left unital ( L, 1L)-action is isomorphic to that of ( L, U)-action. Therefore, Z1(B)| B| B| B| in order to show that Fun( M, M) is the E0-center of M in ECat, it is enough to only A| | B| consider the left unital ( L, 1L)-action on M in ECat as depicted in the following diagram.

A B |L × |M (4.15) |1 6 ❖ | L×1 ♠♠♠ ❖❖❖ ⊙ ♠♠ ✤✤ | ❖❖ ♠♠ ✤✤ ξ ❖❖ B| ♠ ❖' B| ∗× M / M 1B|M

28 B Example 4.35. If |M satisﬁes the condition (∗), then there exists a canonical left unital action

B B B B Z1( )|Fun( |M, |M) × |M (4.16) | 8 ❑ B| × B| q ❑ | 1 M 1 Mqqq ❑❑ ev q ✤✤ | ❑❑ qqq α ❑❑ B q ❑% B ∗× |M / |M

Z1(B)| B| B| | B| of the E0-algebra ( Fun( M, M), 1B|M) on M in ECat defined as follows: | • The enriched functor ev is defined by the monoidal functor êv ≔ ⊙ : Z1(B) × B → B, the map of objects (|F, x) 7→ F(x) and the family of morphisms

| | ⊗| | ≔ | | B|M [F,G]y Fx,y B|M B|M ◦ B|M ev(|F,x),(|G,y) I([ F, G]) ⊗ (x, y) −−−−−−−−→ (F(y), G(y)) ⊗ (F(x), F(y) −→ (F(x), G(y);

• The background changing natural transformationα ˆ of |α is given by the left unitor of B and the underlying natural transformation α of |α is given by the identity natural transformation. q It is routine to check the following fact. The proof is omitted. | | A| | B| Lemma 4.36. Let ( ⊙, ξ) be a left unital ( L, 1L)-action of the E0-algebra on M as depicted in B B the diagram (4.15). Then there exists a functor Φ : L → Fun( |M, |M) deﬁned as follows: • For each a ∈ L, Φ(a) is the B-functor deﬁned by the map x 7→ a ⊙ x and the family of morphisms

ˆ−1 |⊙ B| ξ B| 1a⊙ˆ 1 A| B| (a,x),(a,y) B| M(x, y) −−→ 1A ⊙ˆ M(x, y) −−−→ L(a, a) ⊙ˆ M(x, y) −−−−−−→ M(a ⊙ x, a ⊙ y).

A| • For each morphism f : 1A → L(a, b) in L, Φ( f ) is the B-natural transformation deﬁned by the family of morphisms

−1 | ξˆ f ⊙ˆ 1 A| B| ⊙(a,x),(b,x) B| 1B −−→ 1A ⊙ˆ 1B −−→ L(a, b) ⊙ˆ M(x, x) −−−−−−→ M(a ⊙ x, b ⊙ x).

B| Z1(B)| B| B| B| Theorem 4.37. If M satisﬁes the condition (∗), then Fun( M, M) is the E0-center of M B| Z1(B)| B| B| in ECat, i.e. Z0( M) ≃ Fun( M, M). | | A| | B| Proof. Let( ⊙, ξ) be a left unital action of an E0-algebra ( L, 1L) on M as depicted in the diagram A B B B (4.15). We ﬁrst show that there exist an enriched functor |Φ : |L → Z1( )|Fun( |M, |M) in ECat and two enriched natural isomorphisms |σ and |ρ such that the following pasting diagrams

B B B B Z1( )|Fun( |M, |M) × |M

6 O Z1(B)| B| B| B| | | Fun( M, M) × M 1B|M×1B|M Φ×1B|M |ev | 9 ❍ B B s ❍ | 1 |M×1 |M s ❍ ev ✤✤ A| B| ✤✤ | s ❍ ✤✤ | ×1 L × M ✤✤ ρ = s ✤✤ ❍ . (4.17) σ ss ✤✤ | ❍ |1 6 ◗ s α ❍❍ L×1B|M ❧❧ ◗◗ |⊙ s ❍ ❧❧ ◗ B s $ B ❧ ✤✤ | ◗◗ | | ❧❧❧ ✤✤ ξ ◗◗ ∗× M / M ❧❧❧ ◗◗◗ ∗× B|M (/ B|M are equal in ECat, where the right hand side of the equation is the canonical left unital action of Z1(B)| B| B| | B| ( Fun( M, M), 1B|M) on M deﬁned in Example 4.35. B B • Let Φ : L → Fun( |M, |M) be the functor described in Lemma 4.36. Then Φ can be pro- A B B B moted to an enriched functor |Φ : |L → Z1( )|Fun( |M, |M) in ECat with the background changing functor Φˆ : A → Z1(B) given by − ⊙ˆ 1B (see Example 2.16) and the family of | A| 1 morphisms Φa,b : L(a, b) ⊙ˆ B → [Φ(a), Φ(b)] given by the unique morphisms rendering the following diagrams commutative.

| A| I( Φa,b ) I( L(a, b) ⊙ˆ 1B) / I([Φ(a), Φ(b)])

1⊙ˆ 1x [Φ(a),Φ(b)]x | A B ⊙(a,x),(b,x) B |L(a, b) ⊙ˆ |M(x, x) / |M(a ⊙ x, b ⊙ x)

29 • The enriched natural isomorphism |σ is deﬁned by the backgound changing natural trans- − = ˆ 1 1B → 1A ⊙ˆ 1B formationσ ˆ ∗ ξ1B : and the underlying natural transformation given by the unique morphism σ∗ : 1B → [1B|M, Φ(1L)] rendering the following diagram commutative.

I(σ∗) I(1B) / I([1B|M, Φ(1L)]) ❚❚ ❚❚❚ 1 ❚❚ [1B|M,Φ( L)]x −1 ❚❚❚ ξx ❚* B| M(x, 1L ⊙ x)

• The enriched natural isomorphism |ρ is deﬁned by the backgound changing natural transformation 1⊗ξˆ−1 ∼ ∼ I(Φˆ (−)) ⊗− = (− ⊙ˆ 1B) ⊗− ====⇒ (− ⊙ˆ 1B) ⊗ (1A ⊙−ˆ ) =⇒ (−⊗ 1A) ⊙ˆ (1B ⊗−) =⇒− ⊙−ˆ

and the identity underlying natural transformation. Then it is not hard to check that the equation (4.17) holds. | | | Let( Φi, σi, ρi), i = 1, 2, be two triples such that the similar equations as depicted by(4.17)hold. | | | We only need to show that there exists a unique enriched natural isomorphism β : Φ1 ⇒ Φ2 such that B B B B B B Z1( )|Fun( |M, |M) Z1( )|Fun( |M, |M) | 6 G W | 6 O 1B|M |β 1B|M | | | Φ ❴❴ Φ Φ2 ✤✤ | 1 ❴❴ +3 2 = ✤✤ | , (4.18) σ1 σ2 ∗ A|L ∗ A|L | / | / 1L 1L

B B B B B B B B Z1( )|Fun( |M, |M) × |M Z1( )|Fun( |M, |M) × |M D Z | O | |β×1 ev ev |Φ × |Φ × |Φ ×1 1 1 ❴❴ +3 2 1 ✤✤ | = 1 ✤✤ | . (4.19) ✤✤ ρ2 ✤✤ ρ1 A B " B A B " B |L × |M / |M |L × |M / |M |⊙ |⊙ Since Z (B) is the E -center of B in Alg (Cat) (see Example 2.16), there exists a unique 1 0 E1 monoidal natural isomorphism βˆ : Φˆ 1 ⇒ Φˆ 2 such that Z B Z B Z B × B Z B × B 1 3 1( ) 4 1( ) 1( ) 1( ) B 1B ˆev F βˆ X O D βˆ×1 Z O ˆev ✤✤ Φˆ 1 ❴❴ +3 Φˆ 2 ✤✤ Φˆ 2 Φˆ 1×1 ❴❴ +3 Φˆ 2×1 ✤✤ Φˆ 1×1 ✤✤ ✤✤ σˆ ❴❴ = ✤✤ σˆ , ❴❴ ✤✤ ρˆ2 = ✤✤ ρˆ1 . 1 2 ∗ / A ∗ / A A × B / B A × B / B 1A 1A ⊙ˆ ⊙ˆ

Deﬁne βa : 1B → [Φ1(a), Φ2(a)] to be the unique morphism rendering the following diagram

I(βa) 1B = I(1B) / I([Φ1(a), Φ2(a)]) ❯❯❯ ❯❯❯ [Φ1(a),Φ2(a)]x (ρ )−1 ◦(ρ ) ❯❯ 2 (a,x) 1 (a,x) ❯❯* B| M(Φ1(a)(x), Φ2(a)(x))

| commutative. Then the enriched natural isomorphism β deﬁne by(β,ˆ {βa}) is the unique enriched natural isomorphism such that the equations (4.18) and (4.19) hold. B B B Remark 4.38. It is straightforward to check that the monoidal structure of Z1( )|Fun( |M, |M) B| coincides with the E1-algebra structure of the E0-center of M in ECat (see Proposition 2.14). ♦ The following result follows immediately from Corollary 4.34 and Theorem 4.37. Corollary 4.39. When (B, M) ∈ LMod and M is a left B-module, if B is rigid and FunB(M, M) is enriched in Z1(B), we have B Z1(B) Z0( M) ≃ FunB(M, M). Remark 4.40. Corollary 4.39 has an important application in physics. In particular, it provides a mathematical foundation to Deﬁnition 3.18 and Remark 3.19 in [KZ21]. ♦

30 5 Enriched braided monoidal categories

• an enriched natural transformation |c : |⊗ → |⊗ ◦ Σ; such that 1. The background category A is symmetric. 2. The background changing natural transformation cˆ is equal to the braiding of A. 3. The underlying monoidal category L equipped with the underlying natural transformation c of |c is a braided monoidal category (called the underlying braided monoidal category of A |L).

A B Deﬁnition 5.2. An enriched monoidal functor |F : |L → |M between two enriched braided monoidal categories is an enriched braided monoidal functor if the underlying functor F : L → M is a braided monoidal functor. A Remark 5.3. Let |L be an enriched braided monoidal category. There is an A-enriched braided A monoidal category |L whose underlying braided monoidal category equals to L, deﬁned as A A A follows. As an enriched monoidal category |L = |L, but the braiding of |L is given by the ≔ −1 L A A anti-braiding c¯x,y cy,x of and the braiding of . Since is symmetric, (cˆ, c) is an enriched natural transformation if and only if (cˆ, c¯) is. ♦

Example 5.4. Let A be a commutative algebra in a symmetric monoidal category A. Then the A-enriched monoidal category ∗A introduced in Example 4.10 is an enriched braided monoidal category with the braiding given by c∗,∗ = 1∗. q

5.2 The canonical construction Let A be a symmetric monoidal category viewed as an algebra in Algoplax(Cat). E2 Deﬁnition 5.5. A braided monoidal left A-oplaxmodule is a left A-oplaxmodule in Algoplax(Cat). E2 Remark 5.6. More explicitly,a braided monoidal left A-oplaxmodule L is both a braided monoidal category and a monoidal left A-oplaxmodule such that the action ⊙ : A × L → L is a braided oplax-monoidal functor (i.e. the following diagram commutes,

(a ⊗ b) ⊙ (x ⊗ y) / (a ⊙ x) ⊗ (b ⊙ y)

c cˆa,b⊙cx,y a⊙x,b⊙y (5.1) (b ⊗ a) ⊙ (y ⊗ x) / (b ⊙ y) ⊗ (a ⊙ x) where c is the braiding of L and cˆ is the braiding of A). If, in addition, L is a monoidal left A-module, then L is called a braided monoidal left A-module.♦

Example 5.7. Let A be a symmetric monoidal cateogry and L be a braided monoidal category. If ϕ : A → Z2(L) is a symmetric oplax-monoidal functor, then L is a braided monoidal left A-oplaxmodule with the module action ϕ(−) ⊗− : A × L → L. The monoidal left A-oplaxmodule structure of L is induced by the composite functor A → Z2(L) → Z1(L). q

31 Remark 5.8. If(L, ⊙ : A × L → L) is a braided monoidal left A-module, the functor ϕ ≔ (−⊙ 1L) is a symmetric monoidal functor ϕ : A → Z2(L) (see Example 2.16). A braided monoidal left A-module L can be equivalently deﬁned by a braided monoidal category L equipped with a symmetric monoidal functor A → Z2(L). ♦

Proposition 5.9. Let A be a symmetric monoidal category and L a braided monoidal category. Let cˆ and c be the braidings of A and L, respectively. If L is also a strongly unital monoidal left A A-oplaxmodule that is enriched in A, then the pair (cˆ, c) deﬁnes a braiding structure on L if and only if L is a braided monoidal left A-oplaxmodule.

Proof. The pair (cˆ, c) is a 2-morphism in LMod if and only if the diagram (5.1) commutes.

Example 5.10. Let ϕ : A → Z2(L) be a braided oplax-monoidal functor, where A is a symmetric monoidal category and L is a braided monoidal category. Then (L,ϕ) is a monoidal left A- oplaxmodule (see Example 5.7). If (L,ϕ) is strongly unital and L is enriched in A, then the A enriched monoidal category L constructed in Example 4.18 (note that A = A), together with the braiding in A and L, is an enriched braided monoidal category. q

Deﬁnition 5.11. Let L be a braided monoidal left A-oplaxmodule and M be a braided monoidal left B-oplaxmodule. Given a symmetric monoidal functor Fˆ : A → B, a monoidal Fˆ-lax functor F : L → M is called a braided monoidal F-laxˆ functor if F is braided monoidal.

The proof of the following proposition is clear. Proposition 5.12. Let A, B be symmetric monoidal categories. Suppose L is a strongly unital braided monoidal left A-oplaxmodule that is enriched in A, and M is a strongly unital oplax braided monoidal left B-module that is enriched in B. Then the canonical construction gives A B enriched braided monoidal categories L and M. Suppose Fˆ : A → B is a symmetric monoidal functor and F : L → M is a monoidal Fˆ-lax functor. Then F is a braided monoidal Fˆ-lax functor A B if and only if Fˆ, together with F, deﬁnes an enriched braided monoidal functor |F : L → M.

We use ECatbr to denote the 2-category of enriched braided monoidal categories, enriched braided monoidal functors and enriched monoidal natural transformations. The 2-category ECatbr is symmetric monoidal with the tensor product given by the Cartesian product and the tensor unit given by ∗. We deﬁne the 2-category LModbr as follows: • The objects are pairs (A, L), where A is a symmetric monoidal category and L is a strongly unital braided monoidal left A-oplaxmodule that is enriched in A. • A 1-morphism (A, L) → (B, M)isa pair (Fˆ, F), where Fˆ : A → B is a symmetric monoidal functor and F : L → M is a braided monoidal Fˆ-lax functor. • A 2-morphism (Fˆ, F) ⇒ (Gˆ, G) is a pair (ξ,ˆ ξ), where ξˆ : Fˆ ⇒ Gˆ is a monoidal natural transformation and ξ : F ⇒ G is a monoidal ξˆ-lax natural transformation. The horizontal/vertical composition is induced by the horizontal/vertical composition of functors and natural transformations. The 2-category LModbr is symmetric monoidal with the tensor product deﬁned by the Cartesian product and the tensor unit given by (∗, ∗). By Proposition 5.9, Proposition 5.12 and Proposition 4.25 we obtain the following result.

Theorem 5.13. The canonical construction induces a symmetric monoidal 2-functor LModbr → ECatbr. Moreover, this 2-functor is locally isomorphic.

5.3 The Drinfeld center of an enriched monoidal category A A Deﬁnition 5.14. Let |L be an enriched monoidal category. A half-braiding for an object x ∈ |L A | is an -natural isomorphism β−,x : −⊗ x → x ⊗− such that the underlying natural isomorphism β−,x is a usual half-braiding.

32 | In other words, the half-braiding β−,x is a half-braiding β−,x on the underlying monoidal category such that the following diagram commutes.

| A 1⊗1x A A ⊗ A |L(y, z) / |L(y, z) ⊗ |L(x, x) / |L(y ⊗ x, z ⊗ x)

1x⊗1

A A | | A|L L(x, x) ⊗ L(y, z) (1,βz,x)

|⊗ A|L A (βy,x,1) A |L(x ⊗ y, x ⊗ z) / |L(y ⊗ x, x ⊗ z)

A| A| | | Deﬁnition 5.15. Let L be an enriched monoidal category. Suppose x, y ∈ L and β−,x, β−,y | | are half-braidings. Deﬁne an object [(x, β−,x), (y, β−,y)] ∈ Z2(A) to be the terminal one (if exists) A| among all pairs (a,ζ), where a ∈ Z2(A) and ζ : a → L(x, y) is a morphism in A such that the diagram commutes,

| 1z⊗ζ A A ⊗ A a / |L(z, z) ⊗ |L(x, y) / |L(z ⊗ x, z ⊗ y)

A|L ζ⊗1z (1,βz,y) (5.2) A|L ⊗ A|L A|L ⊗ ⊗ A|L ⊗ ⊗ (x, y) (z, z) | / (x z, y z) A| / (z x, y z) ⊗ L(βz,x,1)

ζ A A A A (i.e. a −→ |L(x, y) equalizes two morphisms |L(x, y) ⇒ |L(z ⊗ x, y ⊗ z), for every z ∈ |L).

| | A| Remark 5.16. By the universal property, the morphism [(x, β−,x), (y, β−,y)] → L(x, y) is monic. | | A| Thus [(x, β−,x), (y, β−,y)] is the maximal subobject a of L(x, y) in Z2(A) such that the diagram (5.2) commutes. ♦

A Deﬁnition 5.17. We say that an enriched monoidal category |L satisﬁes the condition (∗∗) if:

A| | | | | For all objects x, y ∈ L and half-braidings β−,x, β−,y, the object [(x, β−,x), (y, β−,y)] exists. (∗∗)

A Definition 5.18. Let |L be an enriched monoidal category that satisfies the condition(∗∗). Then Z2(A)| A| we define an Z2(A)-enriched braided monoidal category Γ1( L) as follows:

| A| | • The objects are pairs (x, β−,x), where x ∈ L and β−,x is a half-braiding.

Z2(A)| A| | | | | • The hom objects are Γ1( L)((x, β−,x), (y, β−,y)) ≔ [(x, β−,x), (y, β−,y)] ∈ Z2(A).

A • The composition and identity morphisms are induced by those of |L.

A • The monoidal structure is induced by that of |L.

| | βx,y | | • The braiding is given by (x, β−,x) ⊗ (y, β−,y) −−→ (y, β−,y) ⊗ (x, β−,x).

Z2(A)| A| A| This enriched braided monoidal category Γ1( L) is called the Drinfeld center of L. Its A| underlying category is denoted by Γ1( L).

Remark 5.19. This construction is different from the Drinfeld center defined in [KZ18b, Defini- tion 4.2]. The hom objects defined in [KZ18b] are maximal subobjects in A, not Z2(A). In some special cases these two definitions coincide (see [KZ18b, Theorem 4.7]). ♦

Z2(A)| A| Lemma 5.20. Under the condition (∗∗), the Drinfeld center Γ1( L) is a well-deﬁned enriched braided monoidal category.

33 | Z2(A)| A| Proof. For any (x, β−,x) ∈ Γ1( L) the diagram

| 1z⊗1x A A ⊗ A 1 / |L(z, z) ⊗ |L(x, x) / |L(z ⊗ x, z ⊗ x)

A| 1x⊗1z L(1,βz,x) A|L ⊗ A|L A|L ⊗ ⊗ A|L ⊗ ⊗ (x, x) (z, z) | / (x z, x z) A| / (z x, x z) ⊗ L(βz,x,1) commutes because both two composite morphisms are equal to βz,x. Thus the identity morphism A| | | 1x : 1 → L(x, x) factors through [(x, β−,x), (x, β−,x)]. | | | Z2(A)| A| For any (x, β−,x), (y, β−,y), (z, β−,z) ∈ Γ1( L), the diagram

| | commutes: the upper left rectangle commutes by the deﬁnition of [(x, β−,x), (y, β−,y)], and the other subdiagrams commute due to the functoriality of |⊗ and the associativity of the composi- | | | | A| A| ◦ A| tion. Thus the morphism [(y, β−,y), (z, β−,z)] ⊗ [(x, β−,x), (y, β−,y)] → L(y, z) ⊗ L(x, y) −→ L(x, z) | | factors through [(x, β−,x), (z, β−,z)]. This shows that the identity and composition morphisms in A| Z2(A)| A| L induce those of Γ1( L). For the monoidal structure, consider the following diagram:

| | | | | A| A| ⊗ A| [(x, β−,x), (z, β−,z)][(y, β−,y), (w, β−,w)] / L(x, z) L(y, w) / L(xz, yw) ▲ ▲▲▲ ▲▲▲ ▲▲▲ A A ▲▲ A |L(x, z) |L(y, w) ▲▲ |L(axz, ayw) ▲▲▲ ▲▲▲ | ▲▲ A|L ⊗ ▲ (1,βa,y1) . ▲▲▲ ▲ A|L A A & (βa,x1,1) A |L(xz, yw) |L(xaz, yaw) / |L(axz, yaw)

A| A| L(1,1βa,w) L(1,1βa,w) A|L A|L A|L (xza, ywa) A| / (xaz, ywa) A| / (axz, ywa) L(1βa,z,1) L(βa,x1,1)

| | The upper quadrangle commutes by the deﬁnitions of [(x, β−,x), (z, β−,z)], the left quadrangle | | commutes by the deﬁnition of [(y, β−,y), (w, β−,w)], and the lower right rectangle is obviously commutative. It follows that the morphism

| | | | | A| A| ⊗ A| [(x, β−,x), (z, β−,z)] ⊗ [(y, β−,y), (w, β−,w)] → L(x, z) ⊗ L(y, w) −→ L(x ⊗ z, y ⊗ w)

| | factors through [(x ⊗ z, β−,x⊗z), (y ⊗ w, β−,y⊗w)]. It is not hard to see that this deﬁnes the monoidal Z2(A)| A| structure of Γ1( L). βx,y A For the braiding structure, note that the morphism 1 −−→ |L(x ⊗ y, y ⊗ x) equalizes two A| A| morphisms L(x ⊗ y, y ⊗ x) ⇒ L(a ⊗ x ⊗ y, y ⊗ x ⊗ a) because β−,x⊗y is a half-braiding in the

34 βx,y A| | | underlying category. Therefore, 1 −−→ L(x⊗y, y⊗x) factors through [(x⊗y, β−,x⊗y), (y⊗x, β−,y⊗x)] Z2(A)| A| and thus gives the braiding morphism in Γ1( L), denoted by

| βx,y | (x ⊗ y, β−,x⊗y) −−→ (y ⊗ x, β−,y⊗x).

| We need to verify that this braiding β−,− is an enriched natural isomorphism, i.e. the diagram

| | | | c | | | | [(x, β−,x), (z, β−,z)] ⊗ [(y, β−,y), (w, β−,w)] / [(y, β−,y), (w, β−,w)] ⊗ [(x, β−,x), (z, β−,z)]

|⊗ | | | ⊗ [(y ⊗ x, β−,y⊗x), (w ⊗ z, β−,w⊗z)]

[βx,y,1] | | [1,βz,w] | | [(x ⊗ y, β−,x⊗y), (z ⊗ w, β−,z⊗w)] / [(x ⊗ y, β−,x⊗y), (w ⊗ z, β−,w⊗z)]

| | commutes, where c is the braiding of A. Since the morphism [(x ⊗ y, β−,x⊗y), (w ⊗ z, β−,w⊗z)] → A |L(x ⊗ y, w ⊗ z) is monic, it suﬃces to show that the diagram

| | | | A| A| c A| A| [(x, β−,x), (z, β−,z)][(y, β−,y), (w, β−,w)] / L(x, z) L(y, w) / L(y, w) L(x, z)

|⊗

A|L −1 A|L (1,βz,w) (βx,y,1) A|L −1 A|L (βx,w,1) (1,βx,w) A| A| A| A| A|L L(xw, zw) L(xy, xw) / L(wx, zw) L(xy, wx) (βx,y,1) ❤❤❤ | ❤❤❤❤ ⊗ ◦ ❤❤❤❤ ❤❤❤❤ ◦ ❤❤❤ A|L A s❤❤❤ (1,βz,w) A |L(xy, zw) / |L(xy, wz)

| | commutes. The upper left rectangle commutes by the deﬁnition of [(y, β−,y), (w, β−,w)], the lower right pentagon and the lower middle triangle commute due to the associativity of the composition ◦, and the other two subdiagrams commtes due to the functoriality of |⊗. Hence, Z2(A)| A| we conclude that Γ1( L) is a well-deﬁned enriched braided monoidal category.

A Remark 5.21. Let |L be an enriched monoidal category that satisﬁes the condition (∗∗). The | Z2(A)| A| A| forgetful functor I : Γ1( L) → L is an enriched monoidal functor whose background ♦ .changing functor is the inclusion Z2(A) ֒→ A

A| Proposition 5.22. The underlying category Γ1( L) is a full subcategory of Z1(L).

| | Z2(A)| A| | Proof. Suppose (x, β−,x), (y, β−,y) ∈ Z1( L). By deﬁnition we have (x,β−,x), (y, β−,y) ∈ Z1(L). It suﬃces to show that

| | Z2(A)(1, [(x, β−,x), (y, β−,y)]) ≃ Z1(L)((x,β−,x), (y,β−,y)).

By the universal property, the left hand side is isomorphic to the set of morphisms f : 1 → A |L(x, y) rendering the diagram

| 1z⊗ f A A ⊗ A 1 / |L(z, z) ⊗ |L(x, y) / |L(z ⊗ x, z ⊗ y)

A| f ⊗1z L(1,βz,y) A|L ⊗ A|L A|L ⊗ ⊗ A|L ⊗ ⊗ (x, y) (z, z) | / (x z, y z) A| / (z x, y z) ⊗ L(βz,x,1)

35 A commutative for every z ∈ |L. It means that the equation

βz,y ◦ (1z ⊗ f ) = ( f ⊗ 1z) ◦ βz,x holds in the underlying category L. In other words, f is exactly a morphism f :(x,β−,x) → (y,β−,y) in the Drinfeld center Z1(L).

In the rest of this subsection, we compute the Drinfeld center of an enriched monoidal C category M obtained via the canonical construction from the pair (C, M) ∈ LModbr such that M is a monoidal left C-module deﬁned by a braided monoidal functor (recall Remark 4.16)

ϕ : C → Z1(M).

C Z2(C)| C Suppose M satisfies the condition (∗∗). Then the Drinfeld center Γ1( M) is a well-defined C enriched braided monoidal category. By Proposition 5.22, Γ1( M) is a full subcategory of Z1(M). C It is straightforward to see that Γ1( M) is in Z2(ϕ), which denotes the M üger centralizer of ϕ C in Z1(M) via ϕ. Note that the composite functor Z2(C) ֒→ C −→ Z1(M) factors through φ ((Z2(C) −→ Z2(ϕ) ֒→ Z1(M). Moreover, the image of φ is contained in the M üger center Z2(Z2(ϕ of Z2(ϕ). Thus Z2(ϕ) is a braided monoidal left Z2(C)-module with the braided module structure defined by φ : Z2(C) → Z2(Z2(ϕ)). Lemma 5.23. Let C be a braided monoidal category and M a strongly unital mononidal left | | C-module that is enriched in C. The object [(x, β−,x), (y, β−,y)] is precisely the internal hom in Z2(C).

| | Proof. It is enough to show that [(x, β−,x), (y, β−,y)] satisﬁes the same universal property as the internal hom in Z2(C). Given an object a ∈ Z2(C) and a morphism ζ : a → [x, y]C, we can deﬁne a morphism ζ˜ : ϕc(a) ⊗ x → y in M by

c ⊗ c ϕ (ζ) 1 c evx ζ˜ ≔ ϕ (a) ⊗ x −−−−−→ ϕ ([x, y]C) ⊗ x −−→ y.

Conversely, we have ˜ coevx c [1,ζ] ζ = a −−−−→ [x,ϕ (a) ⊗ x]C −−−→ [x, y]. It is routine to check that ζ renders the diagram (5.2) commutative if and only if the following diagram commutes,

γz,a⊗1 1⊗βz,x z ⊗ ϕc(a) ⊗ x / ϕc(a) ⊗ z ⊗ x / ϕc(a) ⊗ x ⊗ z

1⊗β β⊗1

βz,y z ⊗ y / y ⊗ z

| | (i.e. ζ˜ is a morphism ζ˜ : a ⊙ (x, β−,x) → (y, β−,y) in Z2(ϕ)).

Corollary 5.24. Let C be a braided monoidal category and M be a strongly unital monoidal left C Z2(C)| C Z2(C) C-module that is enriched in C. If M satisﬁes the condition (∗∗), we have Γ1( M) = Z2(ϕ).

5.4 The E1-centers of enriched monoidal categories in ECat In this subsection we prove that the Drinfeld center of an enriched monoidal category is the E1-center in ECat. C Let |M be an enriched monoidal category that satisﬁes the condition(∗∗). Then the Drinfeld Z2(C)| C center Γ1( M) is a well-deﬁned enriched braided monoidal category.

36 A A C Suppose |L is an enriched monoidal category. A left unital action of |L on |M is an enriched monoidal functor |⊙ and an enriched monoidal natural isomorphism |u as depicted in the following diagram: A|L × C|M r9 ■■ |1× rr ■ |⊙ 1 rr ■■ . (5.3) r ✤✤ | ■■ rrr u ■■ C r ■$ C ∗× |M / |M

Z2(C)| C| C| Example 5.25. There is an obvious left unital action of the Drinfeld center Γ1( M) on M:

Z2(C)| C| C| Γ1( M) × M ♥7 ◆◆ | |1×1 ♥♥ ◆◆ ⊛ ♥♥♥ ✤✤ ◆◆ ♥♥ ✤✤ | ◆◆ ♥♥ v ◆◆◆ C| ♥ ◆' C| ∗× M / M

| | | | Z2(C)| C| C| The enriched monoidal functor ⊛ ≔ ⊗ ◦ ( I × 1), where I : Γ1( M) → M is the forgetful functor (see Remark 5.21). The monoidal structure of |I is induced by the braided monoidal C| structure of Z2(C) × C → C and the monoidal structure of Z1(M) × M → M (recall that Γ1( M) is | a full subcategory of Z1(M)). The background changing natural transformation of v is given by the left unitor of C and the underlying natural transformation of |v is given by the left unitor of M. q

C| Z2(C)| C| Theorem 5.26. The Drinfeld center is the E1-center in ECat, i.e. Z1( M) ≃ Γ1( M)asenriched braided monoidal categories.

A C Proof. Suppose (|⊙, |u) is a left unital action of an enriched monoidal category |L on |M as depicted in the diagram (5.3). First we show that there is an enriched monoidal functor |P : A| Z2(C)| C| | L → Γ1( M) and an enriched monoidal natural isomorphism ρ such that the following pasting diagrams

Z2(C)| C| C| Γ1( M) × M

6 O Z2(C)| C| C| |1 | Γ ( M) × M ×1C|M P×1C|M |⊛ 1 |1 : ❋ ×1C|M ✉✉ ❋ |⊛ ✤✤ A| C| ✤✤ | ✉ ❋ ✤✤ | 0× L × M ✤✤ ρ = ✉ ✤✤ ❋ . (5.4) P 1 ✉✉ ✤✤ | ❋❋ |1 6 P ✉ v ❋ L×1C|M ♠♠ PP |⊙ ✉✉ ❋ ♠♠ PP C| # C| ♠♠ ✤✤ | P ♠♠ ✤✤ u PP ∗× M / M ♠♠♠ PPP ∗× C|M /' C|M are equal.

A • Since the underlying functor ⊙ of |⊙ is a monoidal functor, for any x ∈ |L, the natural isomorphism

−1 um ⊗1 1⊗um βm,x : m ⊗ (x ⊙ 1M) −−−−→ (1L ⊙ m) ⊗ (x ⊙ 1M) ≃ x ⊙ m ≃ (x ⊙ 1M) ⊗ (1L ⊙ m) −−−→ (x ⊙ 1M) ⊗ m

| is a half-braiding on x ⊙ 1M ∈ Z1(M) (see Example 2.16). It is clear that β−,x is a C-natural isomorphism because it is the composition of 2-isomorphisms in ECat and the background | changing natural transformation is identity. Hence x ⊙ 1M together with β−,x is an object Z2(C)| C| | A| Z2(C)| C| in Γ1( M). Then we deﬁne an enriched monoidal functor P : L → Γ1( M) as follows:

– The background changing functor Pˆ is given by − ⊙ˆ 1C : A → Z2(C), which is a braided monoidal functor (see Example 2.16).

Z2(C)| C| – The object P(x) ∈ Γ1( M) is deﬁned by the object x ⊙ 1M together with the half- | braiding β−,x.

37 | A| – The morphism Px,y : L(x, y) ⊙ˆ 1C → [P(x), P(y)] is induced by the morphism

⊙ˆ 1 | A| 1 1 M A| C| ⊙ C| L(x, y) ⊙ˆ 1C −−−−−→ L(x, y) ⊙ˆ M(1M, 1M) −→ M(x ⊙ 1M, y ⊙ 1M)

| and the universal property of [P(x), P(y)]. In other words, Px,y is the unique morphism rendering the following diagram

| A| Px,y L(x, y) ⊙ˆ 1C / [P(x), P(y)]

1⊙ˆ 11M

| A| C| ⊙ C| L(x, y) ⊙ˆ M(1M, 1M) / M(x ⊙ 1M, y ⊙ 1M)

commutative. – The monoidal structure is induced by that of |⊙.

• The enriched monoidal natural isomorphism |ρ : |⊛ ◦ (|P × 1) ⇒ |⊙ is deﬁned by the background changing natural transformation

−1 1⊗uˆc ≃ ρˆa,c :(a ⊙ˆ 1C) ⊗ˆ c −−−−→ (a ⊙ˆ 1C) ⊗ˆ (1A ⊙ˆ c) −→ a ⊙ˆ c

and the underlying natural transformation

−1 1⊗um ≃ ρx,m :(x ⊙ 1M) ⊗ m −−−−→ (x ⊙ 1M) ⊗ (1L ⊙ m) −→ x ⊙ m.

It is not hard to check that the equation (5.4) holds. | | Let ( Qi, ρi), i = 1, 2, be two pairs such that the similar equations as depicted by (5.4) hold. We only need to show that there exists a unique enriched monoidal natural isomorphism |α : | | Q1 ⇒ Q2 such that

Z2(C)| C| C| Z2(C)| C| C| Γ1( M) × M Γ1( M) × M | | D | Z ⊛ O ⊛ | α×1 | | × ❴❴ × Q1×1 Q1 1 ❴❴ +3 Q2 1 ✤✤ | = ✤✤ | . (5.5) ✤✤ ρ2 ✤✤ ρ1 A C C A C C |L × |M / |M |L × |M / |M |⊙ |⊙

Since Z (C) is the E -center of C in Alg (Cat) (see Example 2.16), there exists a unique 2 0 E2 monoidal natural isomorphismα ˆ : Qˆ 1 ⇒ Qˆ 2 such that

Z2(C) × C Z2(C) × C ⊛ˆ ⊛ˆ D αˆ×1 Z O Qˆ 1×1 ❴❴ +3 Qˆ 2×1 ✤✤ Qˆ 1×1 ✤✤ ❴❴ ✤✤ ρˆ2 = ✤✤ ρˆ1 . A × C / C A × C / C ⊙ˆ ⊙ˆ

Deﬁne αa : 1C → [Q1(a), Q2(a)] to be the unique morphism rendering the following diagram

αa 1C / [Q1(a), Q2(a)] −1 (ρ2) ◦(ρ1)(a,1) (a,1) C| ≃ C| M(Q1(a) ⊗ 1M, Q2(a) ⊗ 1M) / M(Q1(a), Q2(a))

| commutative. Then the enriched natural isomorphism α deﬁneby(ˆα, {αa}) is the unique enriched natural isomorphism such that the equation (5.5) holds. Z2(C)| C| It is routine to check that the braiding structure of Γ1( M) coincides with the E2-algebra C| structure of the E1-center of M in ECat induced from the universal property.

38 Combining Corollary 5.24 and Theorem 5.26, we obtain a corollary. Corollary 5.27. Let C be a braided monoidal category and M a strongly unital mononidal left C C-module that is enriched in C. If M satisﬁes the condition (∗∗), we have

C Z2(C) Z1( M) = Z2(ϕ).

Remark 5.28. Corollary 5.27 has important applications in physics. When C is a non-degenerate braided fusion category and M is a multi-fusion left C-module defined by a braided monoidal functor ϕ : C → Z1(M) [KZ18a], the condition (∗∗) is satisfied automatically. In this case, we C obtain Z1( M) ≃ Z2(ϕ). This recovers Corollary 5.4 in [KZ18b]. This result has important applications in the theory of gapless boundaries of 2+1D topological orders [KZ20, KZ21]. Moreover, for an indecomposable multi-fusion category A and a finite semisimple left A-module L, then we have A Z1(Z0( L)) ≃ k, i.e. the center of a center is trivial. This is the mathematical manifestation of a physical fact: “the bulk of a bulk is trivial”. ♦

6 Enriched symmetric monoidal categories

6.1 Deﬁnitions and canonical construction A Deﬁnition 6.1. An enriched braided monoidal category |L is called an enriched symmetric monoidal category if the underlying braided monoidal category L is symmetric.

Deﬁnition 6.3. Let A be a symmetric monoidal category and L be a braided monoidal left A-oplaxmodule. If L is symmetric, we say L is a symmetric monoidal left A-oplaxmodule.

Deﬁnition 6.4. A braided monoidal Fˆ-lax functor F : L → M between two symmetric monoidal categories L and M is also called a symmetric monoidal F-laxˆ functor.

Let ECatsym be a symmetric monoidal full sub-2-category of ECatbr consisting of enriched symmetric monoidal categories. Let LModsym be the symmetric monoidal full sub-2-category of LModbr consisting of pairs (A, L) such that L is strongly unital symmetric monoidal left A-oplaxmodule that is enriched in A. We obtain a corollary of Theorem 5.13. Corollary 6.5. The canonical construction induces a symmetric monoidal 2-functor LModsym → ECatsym. Moreover, this 2-functor is locally isomorphic.

A| A| A| Definition 6.6. Let L be an enriched braided monoidal category. The M üger center Γ2( L) of A A |L is defined by the subcategory of |L consisting of the transparent objects in L and the same A| A| hom spaces as those in L (i.e. Γ2( L) = Z2(L)).

Theorem 6.7. Let A be a symmetric monoidal category and L be a strongly unital braided C| C C monoidal left A-module that is enriched in A. Then we have Γ2( M) = Z2(M).

6.2 The E2-centers of enriched braided monoidal categories in ECat C| C| In this subsection we prove that the M ¨uger center Γ2( M) of an enriched braided monoidal C| category M is its E2-center in ECat. E A C Suppose |L is an enriched braided monoidal category. A left unital action of |L on |M is an enriched braided monoidal functor |⊙ and an enriched monoidal natural isomorphism |u as

39 depicted in the following diagram:

E|L × C|M r9 ■■ |1×1 rr ■■ |⊙ rr ✤✤ ■■ . (6.1) rr ✤✤ | ■■ rr u ■■ C r $ C ∗× |M / |M

C| C| C| Example 6.8. There is an obvious left unital action of the M ¨uger center Γ2( M) on M:

C| C| C| Γ2( M) × M | ♣7 ▼▼ | 1×1 ♣♣ ▼▼ ⊗ ♣♣ ✤✤ ▼▼ ♣♣ ✤✤ | ▼▼ ♣♣ v ▼▼ C ♣ ▼& C ∗× |M / |M

C The enriched braided monoidal functor |⊗ is given by the tensor product of |M. The background changing natural isomorphism of |v is given by the left unitor of C and the underlying natural isomorphism of |v is given by the left unitor of M. q

C| C| C| Theorem 6.9. The M ¨uger center Γ2( M) is the E2-center of M in ECat.

E Proof. Suppose (|⊙, |u) is a left unital action of an enriched braided monoidal category |L on C |M as depicted in the diagram (6.1). First we show that there is an enriched braided monoidal | E| C| C| | functor P : L → Γ2( M) and an enriched monoidal natural isomorphism ρ such that the following pasting diagrams

C| C| C| Γ2( M) × M 6 O C| C| C| |1 | Γ ( M) × M ×1C|M P×1C|M |⊗ 2 |1 : ❊ ×1C|M ✈✈ ❊ |⊗ ✤✤ E| C| ✤✤ | ✈ ❊ ✤✤ | 0× L × M ✤✤ ρ = ✈ ✤✤ ❊❊ . (6.2) P 1 ✈✈ ✤✤ | ❊ |1 6 P ✈ v ❊ L×1C|M ❧❧ PP |⊙ ✈✈ ❊ ❧❧ PP C| " C| ❧❧ ✤✤ | P ❧❧ ✤✤ u PP ∗× M / M ❧❧❧ PPP ∗× C|M /( C|M are equal.

| • Since the underlying functor ⊙ of ⊙ is a braided monoidal functor, x ⊙ 1M ∈ Z2(M) is E transparent for any x ∈ |L (see Example 2.16). Then we deﬁne an enriched monoidal | E| Z2(C)| C| | | functor P : L → Γ1( M) by P ≔ (−⊙ 1M). More explicitly:

– The background changing functor Pˆ is given by − ⊙ˆ 1C : E → Z2(C). C| C| – The object P(x) ∈ Γ1( M) is deﬁned by the object x ⊙ 1M ∈ Z2(M). | E| C| – The morphism Px,y : L(x, y) ⊙ˆ 1C → M(P(x), P(y)) is induced by the morphism

⊙ˆ 1 | E| 1 1 M E| C| ⊙ C| L(x, y) ⊙ˆ 1C −−−−−→ L(x, y) ⊙ˆ M(1M, 1M) −→ M(x ⊙ 1M, y ⊙ 1M).

– The braided monoidal structure is induced by that of |⊙.

• The enriched monoidal natural isomorphism |ρ : |⊗ ◦ (|P × 1) ⇒ |⊙ is deﬁned by the background changing natural transformation

−1 1⊗uˆc ≃ ρˆa,c :(a ⊙ˆ 1C) ⊗ˆ c −−−−→ (a ⊙ˆ 1C) ⊗ˆ (1A ⊙ˆ c) −→ a ⊙ˆ c

and the underlying natural transformation

−1 1⊗um ≃ ρx,m :(x ⊙ 1M) ⊗ m −−−−→ (x ⊙ 1M) ⊗ (1L ⊙ m) −→ x ⊙ m.

40 It is not hard to check that the equation (6.2) holds. | | Let ( Qi, ρi), i = 1, 2, be two pairs such that the similar equations as depicted by (6.2) hold. We only need to show that there exists a unique enriched monoidal natural isomorphism |α : | | Q1 ⇒ Q2 such that

C| C| C| C| C| C| Z1( M) × M Z1( M) × M | | D| Z ⊗ O ⊗ | α×1 | | × ❴❴ × Q1×1 Q1 1 ❴❴ +3 Q2 1 ✤✤ | = ✤✤ | . (6.3) ✤✤ ρ2 ✤✤ ρ1 E C C E C C |L × |M / |M |L × |M / |M |⊙ |⊙

Since C itself is the E -center of C in Alg (Cat) (see Example 2.16), there exists a unique 0 E3 monoidal natural isomorphismα ˆ : Qˆ 1 ⇒ Qˆ 2 such that

C × C C × C A ] ⊗ˆ O ⊗ˆ αˆ×1 Qˆ ×1 ❴❴ Qˆ ×1 Qˆ 1×1 1 ❴❴ +3 2 ✤✤ ρˆ = ✤✤ ρˆ . 2 1 E × C / C E × C / C ⊙ˆ ⊙ˆ

Deﬁne (ρ )−1 ≔ 1 (ρ1)a,1 1 2 a,1 1 αa Q1(a) ≃ Q1(a) ⊗ M −−−−→ a ⊙ −−−−→ Q2(a) ⊗ ≃ Q2(a). | Then the enriched natural isomorphism α deﬁne by (ˆα, {αa}) is the unique enriched natural isomorphism such that the equation (6.3) holds.

By Theorem 6.7, we obtain the following corollary. Corollary 6.10. Let A be a symmetric monoidal category and L be a strongly unital braided C C monoidal left A-module that is enriched in A. Then we have Z2( M) = Z2(M).

Example 6.11. Let E be a symmetric fusion category, i.e. E = Rep(G) or Rep(G, z) for a ﬁnite E group G. Then the canonical construction E is an example of enriched symmetric monoidal E E category. We have Z2( E) ≃ E. q

Remark 6.12. When A is a non-degenerate braided fusion category and L is a multi-fusion left A C-module [KZ18a], we have Z2(Z1( L)) ≃ k. ♦

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