2.1 Presentable linear categories

In particular, a category which has enough compact projectives in the above sense is locally finitely presentable.

In enriched category theory, it is standard to consider $\lambda$-locally presentable categories for a regular cardinal $\lambda$ (see [Reference Adámek and RosickyAR94] for details); taken together, these form the locally presentable categories. Locally finite presentable categories correspond to $\lambda =\aleph _0$, while categories with enough compact projectives in the above sense correspond to $\lambda =0$ [Reference Brandenburg, Chirvasitu and Johnson-FreydBCJ15]. For a general regular cardinal $\lambda$, $\lambda$-locally presentable categories are defined similarly to the locally finitely presentable categories, except that the cardinal $\lambda$ enters into the notion of compact objects (their Hom functors preserve only $\lambda$-filtered objects), and the notion of generation (the $\lambda$-compact objects generate under $\lambda$-filtered colimits). Since we will not require any technical arguments involving general cardinals, the reader can safely think of locally finitely presentable categories in all arguments, without loss of generality.

The 2-category of $\mathbf {k}$-linear categories is symmetric monoidal: given two $\mathbf {k}$-linear categories, their linear (a.k.a. ‘naive’) tensor product $\mathcal {C}\otimes \mathcal {D}$ has as its objects pairs of objects of $\mathcal {C}$ and $\mathcal {D}$, and morphisms defined by

\[ \operatorname{Hom}_{\mathcal{C}\otimes \mathcal{D}}((c_1,d_1),(c_2,d_2)):=\operatorname{Hom}_\mathcal{C}(c_1,c_2)\otimes_\mathbf{k} \operatorname{Hom}_{\mathcal{D}}(d_1,d_2). \]

However, the $\mathbf {k}$-linear tensor product of two categories in $\operatorname {\mathbf {Pr}}$ is not again in $\operatorname {\mathbf {Pr}}$. Each of $\operatorname {\mathbf {Pr}}$ and $\operatorname {\mathbf {Pr}}^\circ$ nevertheless admit a natural symmetric monoidal structure, extending the $\mathbf {k}$-linear tensor product, and make the following definition.

Definition 2.7 The Deligne–Kelly [Reference KellyKel82, Reference DeligneDel07, Reference FrancoFra13] tensor product of categories $\mathcal {C},\mathcal {D} \in \operatorname {\mathbf {Pr}}$ is another category $\mathcal {C}\boxtimes \mathcal {D}\in \operatorname {\mathbf {Pr}}$, equipped with a linear functor $\mathcal {C} \otimes \mathcal {D} \rightarrow \mathcal {C}\boxtimes \mathcal {D}$, cocontinuous in each variable, which is universal for this property, in the sense that we have an equivalence of groupoids,

\[ \operatorname{Hom}_{\operatorname{\mathbf{Pr}}}(\mathcal{C}\boxtimes \mathcal{D},\mathcal{E})^\times \simeq \operatorname{Lin}^{c,c}(\mathcal{C}\otimes\mathcal{D},\mathcal{E}), \]

where on the left-hand side we throw away non-invertible natural transformations and $\operatorname {Lin}^{c,c}$ denotes the groupoid whose objects are $\mathbf {k}$-linear functors which are cocontinuous in each variable, and whose morphisms are natural isomorphisms.

2.2 Dualizability

We recall the following from [Reference LurieLur]. Roughly speaking, a symmetric monoidal $n$-category is said to be $k$-dualizable for some $1\leqslant k\leqslant n$ if every object has a dual in the usual sense, and every $i$-morphism for $1\leqslant i < k$ has a left and a right adjoint. This implies, in particular, that every object is $k$-dualizable in the sense that its evaluation and coevaluation maps have left and right adjoint, the units and counits of which themselves have adjoints, and so on.

This is made more precise as follows. Let $\mathcal {C}$ be an $n$-category and let $h_2\mathcal {C}$ be its homotopy 2-category, that is, the 2-category whose objects and morphisms are the same as $\mathcal {C}$ and whose 2-morphisms are isomorphism classes of 2-morphisms of $\mathcal {C}$.

Definition 2.11 Let $\mathcal {C}$ be a 2-category and $x,y\in \mathcal {C}$. Two morphisms $L:x\rightarrow y$ and $R:y\rightarrow x$ are said to be adjoints if there exists 2-morphisms $\eta :\operatorname {id}_x\rightarrow R\circ L$ and $\epsilon :L\circ R\rightarrow \operatorname {id}_y$ such that

\[ (R\epsilon)\cdot (\eta R)=\operatorname{id}_R, \quad (\epsilon L)\cdot (L \eta)=\operatorname{id}_L\!.\]

Now let $\mathcal {S}$ be a symmetric monoidal $n$-category and define $B\mathcal {S}$ to be the $(n+1)$-category with one object $*$ and morphisms given by $\operatorname {End}(*):=\mathcal {S}$.

2.3 Tensor categories

The following definitions are standard, but we emphasize that throughout this paper we always work in $\operatorname {\mathbf {Pr}}$. For any details about what ‘the obvious diagrams’ means in some of these definitions, see [Reference Etingof, Gelaki, Nikshych and OstrikEGNO15].

Definition 2.17 A tensor category is a category $\mathcal {A}$ in $\operatorname {\mathbf {Pr}}$ equipped with a cocontinuous functor (i.e. a morphism in $\operatorname {\mathbf {Pr}}$)

\[ \otimes: \mathcal{A} \boxtimes \mathcal{A} \longrightarrow \mathcal{A}, \]

a distinguished object $\mathbf {1}_\mathcal {A}$ (the unit) and natural isomorphism

\[ \alpha_{x,y,z}:(x\otimes y)\otimes z \longrightarrow x \otimes (y\otimes z) \]

and

\[ l_x:\mathbf{1}_{\mathcal{A}}\otimes x \rightarrow x \quad r_x:x \otimes \mathbf{1}_{\mathcal{A}} \rightarrow x \]

such that the following diagram commutes:

Definition 2.22 Let $\mathcal {A},\mathcal {B}$ be tensor categories. An $(\mathcal {A}\text{--}\mathcal {B})$-bimodule is a $\operatorname {\mathbf {Pr}}$ category $\mathcal {M}$ which is simultaneously a left $\mathcal {A}$-module and a right $\mathcal {B}$-module, together with an isomorphism

\[ \Gamma:(a\otimes m)\otimes b \longrightarrow a \otimes (m \otimes b) \]

making the obvious diagram commute.

Definition 2.24 Let $\mathcal {A}$ be a tensor category and $\mathcal {M},\mathcal {N}$ be a right and a left $\mathcal {A}$-module category, respectively. An $\mathcal {A}$-balanced functor is a pair of a functor from $F:\mathcal {M} \boxtimes \mathcal {N}$ to some $\operatorname {\mathbf {Pr}}$ category $\mathcal {E}$ and a natural transformation

\[ f:F((m\otimes a)\boxtimes n)\cong F(m\boxtimes (a \otimes n)) \]

for $a\in \mathcal {A},m\in \mathcal {M},n\in \mathcal {N}$ making the obvious diagrams commute.

2.4 Braided tensor categories

Definition 2.28 A braided tensor category is a tensor category $(\mathcal {A},\otimes ,\alpha )$ together with a natural automorphism $\beta$ of $- \otimes -$ making the following diagrams commute:

Finally, let us recall the following well-known construction of a braided tensor category out of a tensor category.

The following properties of the center are straightforward and well known,

Proposition 2.34 [Reference Etingof, Gelaki, Nikshych and OstrikEGNO15, Example 8.5.2]

Let $\mathcal {C}$ be a tensor category. The assignment $(x,\beta ) \mapsto (x,\beta ^{-1})$ induces a braided tensor equivalence

\[ Z(\mathcal{C})^{{\rm mop}}\longrightarrow Z(\mathcal{C}^{{\rm mop}}). \]

Proposition 2.35 Let $\mathcal {A}$ be a braided tensor category with braiding $\beta$. Then there are braided tensor functors

\begin{align*} \mathcal{A} &\longrightarrow Z(\mathcal{A}), \quad \mathcal{A}^{{\rm bop}} \longrightarrow Z(\mathcal{A})\\ x &\longmapsto (x,\beta_{x,-}),\quad x \longmapsto (x, \beta_{-,x}^{-1}) \end{align*}

which assemble into a single braided tensor functor $\mathcal {A} \boxtimes \mathcal {A}^{{\rm bop}} \rightarrow Z(\mathcal {A})$.

It is a general fact that the category of modules over an $E_2$-algebra is an $E_1$ (i.e. monoidal) category. Specializing in the case at hand, this recovers the following well-known proposition.

3.1 Models for higher Morita theories

There are two closely related models for higher Morita theories of $E_n$-algebras. Following a proposal of Lurie in [Reference LurieLur09], Scheimbauer gave a geometric construction in [Reference ScheimbauerSch14], using the framework of locally constant factorization algebras on stratified spaces of the form $(\mathbb {R}^k\subset \mathbb {R}^{k+1}\subset \cdots \subset \mathbb {R}^n)$ to build an $(n+1)$-category of $E_n$-algebras. Independently, Haugseng [Reference HaugsengHau17] gave an alternative combinatorial/operadic construction of an $(n+1)$-category of $E_n$-algebras, building explicit simplicial sets capturing the basic structures, operations, and coherences capturing the iterative notion of bimodules for $E_n$-algebras.

While there are a number of expected relations between the two constructions, there are at present writing no theorems providing a functorial relationship between the two models, and it appears to be a formidable undertaking to construct such a functorial relation precisely. At the level of their objects only, Lurie's theorem asserts the equivalence between $E_n$-algebras and locally constant factorization algebras on $\mathbb {R}^n$ (see [Reference LurieLur09, § 5.2.4], and see [Reference GinotGin15] for an exposition). However extending Lurie's equivalence to the entire Morita theory seems to be a very difficult task.

An important distinction between the two models for higher Morita theories is the role of pointings, that is, maps from the unit of $\mathcal {S}$ to all outputs in the theory. The geometric formulation of Scheimbauer's constructions endows objects and morphisms at all levels with (strong) pointings; essentially these arise because the empty open set is initial in the category of open sets on any manifold, so it induces a canonical map to the value of the factorization algebra on any open set, compatibly with pushforward operations. These pointings are not present in Haugseng's work, except in the low degrees where they endow $E_k$-algebras in the theory with units.

Subsequently to their introduction, each of these two higher Morita theories was extended in [Reference Johnson-Freyd and ScheimbauerJS17] to so-called ‘even higher Morita theories’, in which the symmetric monoidal $1$-category $\mathcal {S}$ is replaced by a symmetric monoidal $k$-category. Correspondingly, instead of an $(n+1)$-category of $E_n$-algebras, one obtains an $(n+k)$-category of $E_n$-algebras. One of the main examples highlighted in [Reference Johnson-Freyd and ScheimbauerJS17, Example 8.9] is precisely the one we need in the present work: the $4$-category ${\mathbf {BrTens}}$, of braided tensor categories, viewed as $E_2$-algebras in the $2$-category $\operatorname {\mathbf {Pr}}$.

Our theorems establishing dualizability in ${\mathbf {Tens}}$ and ${\mathbf {BrTens}}$ hold only in the unpointed formulation of Haugseng: in the strongly pointed setting it is shown in [Reference Gwilliam and ScheimbauerGS18] that $E_n$-algebras are no more than $n$-dualizable. For this reason, we deploy Haugseng's model in our constructions in this section. Nonetheless, each structure appearing in the construction of ${\mathbf {BrTens}}$ can indeed be understood in the language of locally constant factorization algebras, and this gives useful geometric motivation for often elaborately phrased categorical definitions. For this reason, we also outline in § 3.2 the relationship to Scheimbauer's Morita theory of locally constant factorization algebras, at an informal level and simply ignoring the issue of pointings.

An important feature of Haugseng's construction is Theorem 5.49 [Reference HaugsengHau17], which states that the $\operatorname {Hom}$ $n$-categories between two $E_n$-algebras $A$ and $B$ can be identified inductively with the $n$-category of $E_{n-1}$-algebras in the $E_{n-1}$-category of $(A\text{--}B)$-bimodules. Applying this description inductively, in combination with [Reference Johnson-Freyd and ScheimbauerJS17] in the top degrees, we obtain the following description of the Morita theory of $E_2$-algebras.

• • Objects are given by of $E_2$-algebras $\mathcal {A}$, $\mathcal {B}$, ….

• • $\operatorname {Hom}(\mathcal {A},\mathcal {B})$ consists of $E_1$-algebras $\mathcal {C}$,$\mathcal {D}$, …internal to $(\mathcal {A}\text{--}\mathcal {B})$-bimodules.

• • $\operatorname {Hom}(\mathcal {C},\mathcal {D})$ consists of $(\mathcal {C}\text{--}\mathcal {D})$-bimodules $\mathcal {M}$, $\mathcal {N}$, …internal to $(\mathcal {A}\text{--}\mathcal {B})$-bimodules.

• • $\operatorname {Hom}(\mathcal {M},\mathcal {N})$ consists of structure-preserving functors $F$, $G$, ….

• • $\operatorname {Hom}(F,G)$ consists of structure-preserving natural transformations.

We unwind these definitions completely in the case of ${\mathbf {BrTens}}$ (i.e. in the case $\mathcal {S}=\operatorname {\mathbf {Pr}}$). We describe the objects, $1$-, $2$-, $3$- and $4$-morphisms in terms of more familiar constructions in the language of braided tensor categories. Many of the algebraic constructions which arise in this unwinding have appeared already in the study of defects in Witten–Reshetikhin–Turaev theory [Reference Fuchs, Schweigert and ValentinoFSV13], independently of the rigorous $(\infty ,4)$-formulation of ${\mathbf {BrTens}}$ but clearly with this motivation in mind. However, as there does not appear to be a source with complete details, which we require to formulate our main results, we provide them here.

3.2 Relation to locally constant factorization algebras

The sections which follow feature a number of purely algebraic constructions, as required by Haugseng's framework. On the other hand, it is expected that Haugseng's framework can be related to a yet-undefined framework of non-unital locally constant factorization algebras. Hence, parallel to the rigorous extraction of the algebraic structures we need from Haugseng's general construction, we outline in Remarks 3.2, 3.4, 3.10 and 3.14, and in corresponding Figures 1, 2, 3 and 4 how each structure arises also in the locally constant factorization algebra framework. This relationship is not strictly necessary to the results in the present paper, but may provide geometric intuition and motivation for our definitions.

Figure 1.

Braided tensor categories as locally constant factorization algebras on $\mathbb {R}^2$. (a) depicts a basic open set; its inclusion onto $\mathbb {R}^2$ is a retract, so it is assigned the category $\mathcal {A}$ canonically. (b) depicts an embedding $\mathbb {D}^2\sqcup \mathbb {D}^2\hookrightarrow \mathbb {D}^2$, which induces the product functor $T:\mathcal {A}\boxtimes \mathcal {A}\to \mathcal {A}$. (c) depicts an isotopy (with this choice of representatives, it is an identity) between two composite disk embeddings $\mathbb {D}\sqcup \mathbb {D}\sqcup \mathbb {D}\hookrightarrow \mathbb {D}$, which induces the associator natural isomorphism $\alpha$ on $\mathcal {A}$. (d) depicts an isotopy between two disk inclusions, which induces the braiding isomorphism $\sigma$ on $\mathcal {A}$.

Figure 2.

Central algebras as locally constant factorization algebras on $(\mathbb {R}\subset \mathbb {R}^2)$.(a) depicts the three basic open sets $\mathbb {D}_\mathcal {A}$, $\mathbb {D}_\mathcal {B}$, $\mathbb {D}_\mathcal {C}$ on $(\mathbb {R}\subset \mathbb {R}^2$): disks disjoint from $\mathbb {R}$ are governed by the braided tensor category structure on $\mathcal {A}$ and $\mathcal {B}$ in each connected region, while disks intersecting $\mathbb {R}$ are governed by the tensor structure of $\mathcal {C}$. (b) depicts a disk embedding $\mathbb {D}_\mathcal {A}\sqcup \mathbb {D}_\mathcal {B}\hookrightarrow \mathcal {D}_\mathcal {C}$; this induces a functor $F:\mathcal {A}\boxtimes \mathcal {B}\to \mathcal {C}$. (c) depicts an isotopy between two disk embeddings $\mathbb {D}_\mathcal {A}\sqcup \mathbb {D}_\mathcal {A}\sqcup \mathbb {D}_\mathcal {B}\sqcup \mathbb {D}_\mathcal {B}\to \mathbb {D}_\mathcal {C}$; this induces an isomorphism $J:F(-\otimes -) \xrightarrow {\sim } F(-)\otimes F(-)$, upgrading $F$ to a tensor functor. (d) depicts an isotopy between two disk embeddings $\mathbb {D}_\mathcal {A}\sqcup \mathbb {D}_\mathcal {C}\to \mathbb {D}_\mathcal {C}$; this induces a half-braiding on the image of $F$, and hence a lift $F:\mathcal {A}\to Z(\mathcal {C})$. The analogous half-braiding on $\mathcal {B}$ induces a lift $F:\mathcal {B}^{\textrm {bop}}\to Z(\mathcal {C})$, owing to the differing orientation of the bottom region relative to $\mathbb {R}$.

Figure 3.

(a) depicts the basic open sets on the stratified space $(\mathbb {R}\sqcup \mathbb {R} \subset \mathbb {R}^2$): a locally constant factorization algebra $\mathcal {F}$ is defined by labeling the basic opens as indicated. (b) depicts a map, $\pi : (\mathbb {R}\sqcup \mathbb {R}\subset \mathbb {R}^2) \to (\mathbb {R}\subset \mathbb {R}^2)$, of stratified spaces collapsing the region between the two lines. The composition is defined as $\mathcal {C}\circ \mathcal {D} := \pi _*(\mathcal {F})$. Excision yields an equivalence $\mathcal {C}\circ \mathcal {D}\simeq \mathcal {C}\boxtimes _{\mathcal {A}_2}\mathcal {D}$, as categories, with structure maps given as in Proposition 3.9.

Figure 4.

(a) depicts the stratified space $(\mathbb {R}^0\subset \mathbb {R} \subset \mathbb {R}^2)$. (b) depicts the five basic open sets appearing in the stratification. (c) depicts an isotopy between two composite disk inclusions; this induces the central structure on the bimodule $\mathcal {M}$, identifying the action of $\mathcal {A}$ on $\mathcal {M}$ through $\mathcal {C}$ and through $\mathcal {D}$. The analogous isotopy on $\mathcal {B}$ gives the same structure for the induced $\mathcal {B}^{\textrm {bop}}$ action.

To that end, we now briefly recall the notion of locally constant factorization algebras on topological spaces, and then its refinement to stratified spaces [Reference Ayala, Francis and TanakaAFT17, Reference Costello and GwilliamCG16, Reference GinotGin15, Reference LurieLur]. A locally constant factorization algebra $\mathcal {F}$ on a topological space $X$, with coefficients in some symmetric tensor (typically, non-discrete) $1$-category $\mathcal {S}$ is a pre-cosheaf, satisfying a local constancy condition, carrying a factorization structure, and finally satisfying a sheaf-like condition for gluing local sections.

Let us detail these notions one at a time. Firstly, to say that $\mathcal {F}$ is a pre-cosheaf means that $\mathcal {F}$ assigns an object $\mathcal {F}(U)\in \mathcal {S}$ for every open set $U\subset X$; and for every inclusion $U_1\subset U_2$, a morphism $\mathcal {F}(U_1)\to \mathcal {F}(U_2)$, compatibly with respect to composition of inclusions. Secondly, to say that $\mathcal {F}$ is ‘locally constant’ is to further require that whenever an inclusion of open sets is a retract, the induced functor is an equivalence in $\mathcal {S}$. Thirdly, factorization algebras are required to map disjoint unions of open sets to tensor products in $\mathcal {S}$. This is where the name ‘factorization’ comes from, and it may be regarded as a ‘non-commutative’ multiplicative structure on $\mathcal {F}$, in that only sections which are spatially disjoint on $X$ multiply using the symmetric structure on $\mathcal {S}$. Finally, $\mathcal {F}$ must satisfy certain sheaf-like gluing conditions, but only with respect to so-called Weiss (a.k.a. factorizing) coverings.

More generally, given a stratified space, that is, a filtration

\begin{gather*}\emptyset = X_{-1} \subset X_0 \subset X_1 \subset \cdots \subset X_n= X,\end{gather*}

of $X$ by closed submanifolds $X_i$ of dimension $i$, we may consider locally constant factorization algebras with respect to the stratification. This means the same as above, except that we only require that retracts $U_1\hookrightarrow U_2$ be equivalences, when $U_1$ and $U_2$ are good neighborhoods of some closed stratum $X_i$. Here, a neighborhood is called good if for some $i$, $U\cap X_i\neq \emptyset$, and $U$ is contained in a single component of $X\backslash X_{i-1}$.

The ordinary case of locally constant factorization algebras on topological spaces is recovered as a special case when $X_i=\emptyset$ for all but the final $X_i$.

3.3 Braided tensor categories as objects

Following [Reference HaugsengHau17, Reference Johnson-Freyd and ScheimbauerJS17], given a symmetric monoidal $m$-category $\mathcal {S}$, there is a symmetric monoidal $(m+n)$-category of $E_n$-algebras in $\mathcal {S}$. Specializing to the case $\mathcal {S}=\operatorname {\mathbf {Pr}}$, and the case $n=2$, it is well known (see, for example, [Reference FiedorowiczFie92, Reference FresseFre17, Reference LurieLur]) that $E_2$-algebras in $\operatorname {\mathbf {Pr}}$ are identified with braided tensor categories. Hence these form the objects of ${\mathbf {BrTens}}$, as desired.

This identification requires fixing some conventions which we now do. To identify $E_2$-algebras in $\mathcal {S}$ with $E_1$-algebras in $E_1$-algebras in $\mathcal {S}$, we regard the $x$-direction multiplication as giving an $E_1$-algebra structure on the $E_1$-algebra structure in the $y$-direction. In order to regard an $E_2$-algebra as a braided tensor category we use multiplication in the $x$-direction as the tensor product and define the braiding via 180-degree clockwise rotation.

3.4 Central algebras as $1$-morphisms

We now turn our attention to $1$-morphisms. Again following [Reference HaugsengHau17, Reference Johnson-Freyd and ScheimbauerJS17], a $1$-morphism $\mathcal {C}$ from an $E_2$-algebra $\mathcal {A}$ to an $E_2$-algebra $\mathcal {B}$ is an $E_1$-algebra in $(\mathcal {A}\text{--}\mathcal {B})$-bimodules with respect to the ‘inner’ multiplication on $\mathcal {A}$ and $\mathcal {B}$ thought of as $E_1$-algebras in $E_1$-algebras. Under our conventions, this means that multiplication in $\mathcal {C}$ is in the horizontal direction, and the $(\mathcal {A}\text{--}\mathcal {B})$-bimodule structure is given by vertical multiplication.Footnote ¹⁰ The goal of this section is to translate these definitions into the language of braided tensor categories.

Recall from Proposition 2.36 that for a braided tensor category $\mathcal {A}$, the 2-category $\mathcal {A}\operatorname {-mod}$ carries a monoidal structure. Given a pair $\mathcal {A}$ and $\mathcal {B}$ of braided tensor categories, we obtain, similarly, a monoidal structure on the $2$-category of $(\mathcal {A}\text{--}\mathcal {B}^\textrm {bop})$-module categories. Here we use $\textrm {bop}$ instead of $\textrm {mop}$ because we are looking at $(\mathcal {A}\text{--}\mathcal {B})$-bimodules with respect to the vertical direction.

We can capture this structure more explicitly in the notion of an $(\mathcal {A}\text{--}\mathcal {B})$-central algebra, defined below. For motivation, we recall the analogous but simpler situation in the category of vector spaces. Given a commutative algebra $A$ over $\mathbf {k}$, an $A$-algebra can be defined either as an algebra object in the monoidal category $A\operatorname {-mod}$, or equivalently as a $\mathbf {k}$-algebra $B$ together with an algebra morphism $A\longrightarrow Z(B)$. This motivates the following definition-proposition [Reference Drinfeld, Gelaki, Nikshych and OstrikDGNO10].

Conventions. (a) Given a tensor functor $F:\mathcal {A}\to \mathcal {C}$, an $\mathcal {A}$-central structure on $\mathcal {C}$ may be given by specifying half-braiding natural isomorphisms $\sigma :F(X)\otimes Y \to Y \otimes F(X)$. We will call such a half-braiding natural isomorphism a central structure on $F$, and call $F$ a central functor.

(b) For a pair $\mathcal {A}$, $\mathcal {B}$ of braided tensor categories, we will abbreviate the phrases ‘$(\mathcal {A}\boxtimes \mathcal {B}^{\textrm {bop}})$-algebra’ as ‘$(\mathcal {A}\text{--}\mathcal {B})$-algebra’ and ‘$(\mathcal {A}\boxtimes \mathcal {B}^{\textrm {bop}})$-central structure’ as ‘$(\mathcal {A}\text{--}\mathcal {B})$-central structure’.

For later use, we record the following lemma, which is a straightforward categorification of the analogous statement for algebras over commutative algebras:

Proof. The inverse functor is given as follows. Given a $\mathcal {C}$-module $\mathcal {M}$, equip it with an $\mathcal {A}$-module structure using the monoidal functor $F_\mathcal {C}:\mathcal {A}\rightarrow \mathcal {C}$. This turns $\mathcal {M}$ into a $\mathcal {C}$-module in $\mathcal {A}\operatorname {-mod}$, in such a way that forgetting this structure simply recovers $\mathcal {M}$ as a $\mathcal {C}$-module. Conversely, if $\mathcal {M}$ is a $\mathcal {C}$-module in $\mathcal {A}\operatorname {-mod}$, then by definition for all $a\in \mathcal {A}$ and $m\in \mathcal {M}$ the balanced structure on the action functor induces a natural isomorphism

\[ F_\mathcal{C}(a)\otimes m\cong a \otimes m \]

where the left-hand side uses the $\mathcal {C}$ action on $\mathcal {M}$, while the right- hand side is the given $\mathcal {A}$ action. One checks that this induces an equivalence between the given $\mathcal {A}$-module structure on $\mathcal {M}$ and the one induced by seeing $\mathcal {M}$ as a mere category and turning it into an $\mathcal {A}$-module using $F_\mathcal {C}$.

3.4.1 Composition of $1$-morphisms

By the preceding discussion, a $1$-morphism in ${\mathbf {BrTens}}$ from $\mathcal {B}$ to $\mathcal {A}$ is determined by an $(\mathcal {A}\text{--}\mathcal {B})$-algebra $\mathcal {C}$. Given braided tensor categories $\mathcal {A}_1$, $\mathcal {A}_2$, and $\mathcal {A}_3$, an $(\mathcal {A}_1\text{--}\mathcal {A}_2)$-algebra $\mathcal {C}$ and an $(\mathcal {A}_2\text{--}\mathcal {A}_3)$-algebra $\mathcal {D}$, their composition is identified as a category with the balanced Deligne–Kelly tensor product $\mathcal {C}\boxtimes _{\mathcal {A}_2}\mathcal {D}$. Let us now describe the $(\mathcal {A}_1\text{--}\mathcal {A}_3)$-central tensor structure on the composition.

Recall that if $\mathcal {C},\mathcal {D}$ are tensor categories, then so is $\mathcal {C} \boxtimes \mathcal {D}$ with multiplication given on pure tensors by

\[ (c_1\boxtimes d_1)\otimes (c_2\boxtimes d_2) := (c_1\otimes c_2)\boxtimes (d_1\otimes b_2), \]

and extended uniquely by cocontinuity.

Definition-Proposition 3.8 Fix a braided tensor category $\mathcal {A}_2$ and central functors

\[ F_\mathcal{C}:\mathcal{A}_2^{\textrm{bop}}\rightarrow \mathcal{C},\quad F_{\mathcal{D}}:\mathcal{A}_2 \rightarrow \mathcal{D}. \]

The composition,

\[ \tilde T:\mathcal{C} \boxtimes \mathcal{D} \boxtimes \mathcal{C} \boxtimes \mathcal{D} \longrightarrow \mathcal{C} \boxtimes \mathcal{D} \longrightarrow \mathcal{C} \boxtimes_{\mathcal{A}_2} \mathcal{D}, \]

of the multiplications on $\mathcal {C}$ and $\mathcal {D}$ with the canonical functor (where the $\mathcal {A}_2$-module structure on $\mathcal {C},\mathcal {D}$ is induced by $F_\mathcal {C},F_\mathcal {D}$) has a canonical balancing on the first and third tensor product. The first one is an isomorphism, for $c_i \in \mathcal {C}, d_i \in \mathcal {D}$, $a \in \mathcal {A}$ from

\[ \tilde T((c_1 \otimes a )\boxtimes d_1 \boxtimes c_2 \boxtimes d_2)=(c_1\otimes a \otimes c_2)\boxtimes_{\mathcal{A}_2} (d_1\otimes d_2) \]

to

\[ \tilde T(c_1 \boxtimes (a\otimes d_1) \boxtimes c_2 \boxtimes d_2)=(c_1 \otimes c_2)\boxtimes_{\mathcal{A}_2} (a\otimes d_1\otimes d_2) \]

given by composing the central structure on $\mathcal {C}$ applied on $a \otimes c_2$ with the balancing on $\boxtimes _{\mathcal {A}_2}$, and the second balancing is defined similarly. Therefore, $\tilde T$ factors through a functor

\[ T:(\mathcal{C} \boxtimes_{\mathcal{A}_2} \mathcal{D}) \boxtimes (\mathcal{C} \boxtimes_{\mathcal{A}_2} \mathcal{D}) \longrightarrow \mathcal{C} \boxtimes_{\mathcal{A}_2} \mathcal{D} \]

which induces a tensor structure on $\mathcal {C} \boxtimes _{\mathcal {A}_2} \mathcal {D}$, such that the projection $\mathcal {C} \boxtimes \mathcal {D} \rightarrow \mathcal {C} \boxtimes _{\mathcal {A}_2} \mathcal {D}$ is monoidal.

Proof. This is proven [Reference GreenoughGre10, Theorem 6.2] in the setting of finite categories, but the proof extends to our setting as well.

Definition-Proposition 3.9 With the same assumptions as in Definition-Proposition 3.8, assume moreover that $\mathcal {C}$ (respectively, $\mathcal {D}$) is equipped with a central functor $F$ from $\mathcal {A}_1$ (respectively, from $\mathcal {A}_3^\textrm {bop}$). Then $\mathcal {C} \boxtimes _{\mathcal {A}_2}\mathcal {D}$ is naturally an $(\mathcal {A}_1\text{--}\mathcal {A}_3)$-algebra, with central functors given by the compositions

\[ \mathcal{A}_1 \rightarrow \mathcal{C} \rightarrow \mathcal{C} \boxtimes \mathcal{D} \rightarrow \mathcal{C} \boxtimes_{\mathcal{A}_2} \mathcal{D} \]

and

\[ \mathcal{A}_3^{\textrm{bop}} \rightarrow \mathcal{D} \rightarrow \mathcal{C} \boxtimes \mathcal{D} \rightarrow \mathcal{C} \boxtimes_{\mathcal{A}_2} \mathcal{D}. \]

Hence the composition of $1$-morphisms $\mathcal {C}:\mathcal {A}_2\to \mathcal {A}_3$ and $\mathcal {D}:\mathcal {A}_1\to \mathcal {A}_2$ is given by $\mathcal {C}\circ \mathcal {D} := \mathcal {C}\boxtimes _{\mathcal {A}_2}\mathcal {D}$, regarded as and $(\mathcal {A}_3\text{--}\mathcal {A}_1)$-central tensor category as above.

Remark 3.10 In the locally constant factorization algebra framework, the balanced tensor product appearing in the composition of 2-morphisms is really a consequence of a geometric pushforward operation, together with an excision result which computes this pushforward as a balanced tensor product. This is outlined in Figure 3.

3.5 Centered bimodules as $2$-morphisms

Again following [Reference HaugsengHau17, Reference Johnson-Freyd and ScheimbauerJS17], given two $\mathcal {A}$-algebras $\mathcal {C}$ and $\mathcal {D}$, the 2-morphisms in ${\mathbf {BrTens}}$ are identified with $(\mathcal {C}\text{--}\mathcal {D})$-bimodules internal to the monoidal $2$-category $\mathcal {A}\operatorname {-mod}$. In this section we introduce the notion of a centered bimodule, which encapsulates the explicit list of functors, natural isomorphisms, and coherence conditions, which define the 2-morphisms.

Definition 3.11 Given two $\mathcal {A}$-central algebras $\mathcal {C}$ and $\mathcal {D}$, with central functors $(F_\mathcal {C},\sigma ^{\mathcal {C}})$, $(F_\mathcal {D},\sigma ^{\mathcal {D}})$ respectively, an $\mathcal {A}$-centered structure on a $(\mathcal {C}\text{--}\mathcal {D})$-bimodule $\mathcal {M}$ is the data of isomorphisms,

\[ \eta_{a,m}:F_\mathcal{C}(a) \otimes m \cong m \otimes F_{\mathcal{D}}(a), \]

natural in $m\in \mathcal {M}$ and $a\in \mathcal {A}$ and required to satisfy coherence conditions, expressed as the commutativity of the following diagrams:

(3.1)

(3.2)

(3.3)

for $a,b\in \mathcal {A},m\in \mathcal {M}, c\in \mathcal {C}$ and $d\in \mathcal {D}$. Here, for the sake of clarity we have omitted the explicit mention of the bimodule associator in all three diagrams.

Conventions. Given $(\mathcal {A}\text{--}\mathcal {B})$-central algebras $\mathcal {C}$ and $\mathcal {D}$ and a $(\mathcal {C}\text{--}\mathcal {D})$-bimodule $\mathcal {M}$, we will abbreviate the phrases ‘$(\mathcal {A}\boxtimes \mathcal {B}^{\textrm {bop}})$-centered structure on $\mathcal {M}$’ as ‘$(\mathcal {A}\text{--}\mathcal {B})$-centered structure on $\mathcal {M}$’ and ‘$(\mathcal {A}\boxtimes \mathcal {B}^{\textrm {bop}})$-centered $(\mathcal {C}\text{--}\mathcal {D})$-bimodule’ as ‘$(\mathcal {A}\text{--}\mathcal {B})$-centered $(\mathcal {C}\text{--}\mathcal {D})$-bimodule’.

The following proposition identifies the notion of centered bimodule with the 2-morphisms in ${\mathbf {BrTens}}$.

3.5.1 Vertical composition of $2$-morphisms

By the preceding discussion, the $(\mathcal {A}\text{--}\mathcal {B})$-centered $(\mathcal {C}\text{--}\mathcal {D})$-bimodules comprise the $2$-morphisms in ${\mathbf {BrTens}}$, between the parallel $1$-morphisms $\mathcal {C}, \mathcal {D}:\mathcal {A}\to \mathcal {B}$. Let us now turn to an explicit description of their vertical composition as 2-morphisms. Let $\mathcal {A}$ be a braided tensor category, let $\mathcal {C},\mathcal {D},\mathcal {E}$ be $\mathcal {A}$-algebras, and let $\mathcal {M},\mathcal {N}$ be $\mathcal {A}$-central $(\mathcal {C}\text{--}\mathcal {D})$- and $(\mathcal {D}\text{--}\mathcal {E})$-bimodules. Recall that $\mathcal {M}\boxtimes _\mathcal {D} \mathcal {N}$ has a natural $(\mathcal {C}\text{--}\mathcal {E})$-bimodule structure, where the $\mathcal {C}$ action is induced by the composition

\[ \mathcal{C}\boxtimes \mathcal{M} \boxtimes \mathcal{N}\xrightarrow{act} \mathcal{M} \boxtimes \mathcal{N} \xrightarrow{F} \mathcal{M} \boxtimes_{\mathcal{D}} \mathcal{N} \]

with balanced structure induced by that on $\boxtimes _\mathcal {D}$ (and likewise for the $\mathcal {E}$ action).

Definition-Proposition 3.15 Denote by $F_\mathcal {C},F_\mathcal {D},F_\mathcal {E}$ the central structures on $\mathcal {C},\mathcal {D},\mathcal {E}$. The $(\mathcal {C}\text{--}\mathcal {E})$-bimodule $\mathcal {M} \boxtimes _\mathcal {D} \mathcal {N}$ has a natural $\mathcal {A}$-centered structure defined on pure tensors by

\[ (F_\mathcal{C}(a)\otimes m) \boxtimes_{\mathcal{D}} n \cong (m\otimes F_{\mathcal{D}}(a))\boxtimes_{\mathcal{D}} n\cong M \boxtimes_{\mathcal{D}} (F_{\mathcal{D}}(a)\otimes n)\cong m\boxtimes_{\mathcal{D}} (n \otimes F_\mathcal{E}(a)), \]

for $m\in \mathcal {M},n\in \mathcal {N}, a\in \mathcal {A}$, and extended uniquely by bilinearity. Here the first map is the centered structure on $\mathcal {M}$, the second is the canonical balancing on $\boxtimes _\mathcal {D}$ and the third is the centered structure on $\mathcal {N}$.

Proof. Clear.

3.5.2 Horizontal composition of $2$-morphisms

Finally, we give an explicit description of the horizontal composition of 2-morphisms. We start with the following proposition.

Proposition 3.16 Let $\mathcal {A}$ be a braided tensor category, let $\mathcal {C}$, $\mathcal {D}$ be $\mathcal {A}$- and $\mathcal {A}^{{\rm bop}}$-algebras respectively, and let $\mathcal {M}$ and $\mathcal {N}$ be left $\mathcal {C}$- and $\mathcal {D}$-modules respectively. Then the natural $\mathcal {C}\boxtimes \mathcal {D}$ action on $\mathcal {M} \boxtimes \mathcal {N}$ descends to an action of $\mathcal {C}\boxtimes _\mathcal {A} \mathcal {D}$, with its tensor structure coming from Proposition 3.8, on $\mathcal {M} \boxtimes _\mathcal {A} \mathcal {N}$.

Proof. The proof is the same as for Proposition 3.8: the composition,

\[ \mathcal{C} \boxtimes\mathcal{D} \boxtimes\mathcal{M} \boxtimes \mathcal{N} \rightarrow \mathcal{M} \boxtimes \mathcal{N} \rightarrow \mathcal{M} \boxtimes_{\mathcal{A}} \mathcal{N}, \]

of the action with the canonical functors has a canonical $\mathcal {A}$-balancing on the first and third tensor product given by combining the central structure on $\mathcal {C}$ and $\mathcal {D}$ and the balancing on $\boxtimes _\mathcal {A}$.

Corollary 3.17 Let $\mathcal {A}_1,\mathcal {A}_2,\mathcal {A}_3$ be braided tensor categories, $\mathcal {C}_1,\mathcal {D}_1$ be $(\mathcal {A}_1\text{--}\mathcal {A}_2)$-algebras, $\mathcal {C}_2,\mathcal {D}_2$ be $(\mathcal {A}_2\text{--}\mathcal {A}_3)$-algebras and $\mathcal {M},\mathcal {N}$ be centered $(\mathcal {C}_1\text{--}\mathcal {D}_1)$- and $(\mathcal {C}_2\text{--}\mathcal {D}_2)$-bimodules respectively. Then the action induced from the previous proposition, together with the obvious centered structure, turns $\mathcal {M} \boxtimes _{\mathcal {A}_2} \mathcal {N}$ into an $(\mathcal {A}_1\text{--}\mathcal {A}_3)$-centered $(\mathcal {C}_1\boxtimes _{\mathcal {A}_2} \mathcal {C}_2 \text{--} \mathcal {D}_1 \boxtimes _{\mathcal {A}_2} \mathcal {D}_2)$-bimodule.

3.6 Functors and natural transformations as $3$- and $4$-morphisms

Using the strong (rather than lax or oplax) version of the construction in [Reference Johnson-Freyd and ScheimbauerJS17], we have the following definitions for the higher morphisms.

Definition 3.18 A morphism $(\mathcal {M},\eta ^\mathcal {M})\to (\mathcal {N},\eta ^\mathcal {N})$ of $\mathcal {A}$-centered $(\mathcal {C}\text{--}\mathcal {D})$-bimodules is a functor

\[ F:\mathcal{M} \longrightarrow \mathcal{N} \]

of $(\mathcal {C}\text{--}\mathcal {D})$-bimodules, such that the following diagram commutes for $a\in \mathcal {A},m\in \mathcal {M}$:

where the vertical arrows are the $\mathcal {C}$- and $\mathcal {D}$-module functor structures on $F$ respectively, and where for the sake of clarity we have suppressed the central functors.

3.7 Symmetric monoidal structure

The 4-category ${\mathbf {BrTens}}$ has a natural symmetric monoidal structure inherited from that of $\operatorname {\mathbf {Pr}}$. Let us recall here what it does on objects and morphisms in each degree.

• • The monoidal product of two objects $\mathcal {A},\mathcal {B}$ is $\mathcal {A} \boxtimes \mathcal {B}$ with its componentwise braided tensor structure.

• • Given two braided tensor categories $\mathcal {A},\mathcal {B}$ and given $\mathcal {C}$ and $\mathcal {D}$ an $\mathcal {A}$-central and a $\mathcal {B}$-central tensor category, respectively, the tensor product of the central functors induces a braided tensor functor

  \[ \mathcal{A} \boxtimes \mathcal{B} \rightarrow Z(\mathcal{C})\boxtimes Z(\mathcal{D})\simeq Z(\mathcal{C} \boxtimes \mathcal{D}) \]
  which equips $\mathcal {C}\boxtimes \mathcal {D}$ with the structure of an $(\mathcal {A}\text{--}\mathcal {B})$-central tensor category. This computes the monoidal product on the 1-morphisms.

• • Given $\mathcal {A}$-central tensor categories $\mathcal {C}_1,\mathcal {D}_1$, and $\mathcal {B}$-central tensor categories $\mathcal {C}_2,\mathcal {D}_2$, an $\mathcal {A}$-centered $(\mathcal {C}_1\text{--}\mathcal {D}_1)$-bimodule $\mathcal {M}$ and a $\mathcal {B}$-centered $(\mathcal {C}_2\text{--}\mathcal {D}_2)$-bimodule $\mathcal {N}$, the Deligne–Kelly tensor product $\mathcal {M} \boxtimes \mathcal {N}$ becomes a $(\mathcal {C}_1 \boxtimes \mathcal {C}_2\text{--}\mathcal {D}_1\boxtimes \mathcal {D}_2)$-bimodule, and the tensor product of the centered structure of $\mathcal {M}$ and $\mathcal {N}$ induces a central structure on $\mathcal {M} \boxtimes \mathcal {N}$. This defines the monoidal product on the 2-morphisms.

• • The monoidal product of functors and natural transformation is the same as in $\operatorname {\mathbf {Pr}}$.

4.1 Characterizations of rigidity

In this section we establish some basic facts about rigidity in a tensor category, which we will use to establish dualizability. First, we prove the equivalence between the different formulations given in the introduction.

We start with the following lemma, adapted from [Reference Ben-Zvi and NadlerBN09, Lemma 3.5].

Proof. Let $c=\operatorname {colim} c_i$ be an object in $\mathcal {C}$ written as a colimit of dualizable objects, let $m\in \mathcal {M},n\in \mathcal {N}$ and write $m=\operatorname {colim} m_i$ where the $m_i$ are compact projective. Since the Deligne–Kelly tensor product distributes over colimits, and since the action functor $\mathcal {C} \boxtimes \mathcal {M}\rightarrow \mathcal {M}$ is cocontinuous by assumption, we have

\[ (\operatorname{colim} c_i)\otimes m\cong \operatorname{colim} (c_i \otimes m). \]

On the other hand, if $c_i$ is dualizable we have

\[ \operatorname{Hom}_\mathcal{M}(m,c_i \otimes m')\cong \operatorname{Hom}_\mathcal{M}(c_i^*\otimes m,m'). \]

Together with the definition of an adjunction, the cocontinuity and $\mathcal {C}$-linear structure of $F$ and the properties of $\operatorname {Hom}$ it follows that

\begin{align*} \operatorname{Hom}_\mathcal{M}(m,F^{\textrm{R}}(\operatorname{colim} c_i \otimes n))&\cong \operatorname{Hom}_\mathcal{N}(\operatorname{colim} F(m_i),\operatorname{colim} c_j \otimes n)\\ &\cong \lim \operatorname{Hom}_\mathcal{N}(F(m_i),\operatorname{colim} c_j \otimes n)\\ &\cong \operatorname{colim} \lim \operatorname{Hom}_\mathcal{N}(F(m_i),c_j \otimes n)\\ &\cong \operatorname{colim} \lim \operatorname{Hom}_\mathcal{N}(c_j^* \otimes F(m_i), n)\\ &\cong \operatorname{colim} \lim \operatorname{Hom}_\mathcal{N}(F(c_j^* \otimes m_i), n)\\ &\cong \operatorname{colim} \lim \operatorname{Hom}_\mathcal{M}(c_j^* \otimes m_i, F^{\textrm{R}}( n))\\ &\cong \operatorname{colim} \lim \operatorname{Hom}_\mathcal{M}( m_i, c_i \otimes F^{\textrm{R}}( n))\\ &\cong \operatorname{Hom}_\mathcal{M}( m, \operatorname{colim} c_i \otimes F^{\textrm{R}}( n)). \end{align*}

We record for later use the following corollary.

Thus, having assumed a compact projective $P$ may be written as a small colimit of compact projective and dualizable objects, then by the lemma, it is in fact a finite direct sum of summands of dualizable objects. Dualizability is clearly closed under taking finite direct sums and taking direct summands, hence $P$ is dualizable.

We may now regard $\mathcal {M}=\mathcal {C}\boxtimes \mathcal {C}^{\textrm {mop}}$ and $\mathcal {N}=\mathcal {C}$ as modules for the cp-rigid tensor category $\mathcal {C}^e=\mathcal {C}\boxtimes \mathcal {C}^{\textrm {mop}}$. Then $T$ is a map of $\mathcal {C}^e$-modules, which we claim preserves compact projective objects. Indeed, if $x,y \in \mathcal {C}$ are compact projective, hence also dualizable by assumption, we have, for $c=\operatorname {colim} c_i \in \mathcal {C}$,

\begin{align*} \operatorname{Hom}_\mathcal{C}(x \otimes y,c)&\cong \operatorname{Hom}_\mathcal{C}(x ,(\operatorname{colim} c_i)\otimes y^*)\\ &\cong \operatorname{Hom}_\mathcal{C}(x ,\operatorname{colim} (c_i \otimes y^*))\\ &\cong \operatorname{colim}\operatorname{Hom}_\mathcal{C}(x , c_i \otimes y^*)\\ &\cong \operatorname{colim} \operatorname{Hom}_\mathcal{C}(x \otimes y,c_i). \end{align*}

Therefore, by Lemma 4.2, its right adjoint $T^{\textrm {R}}$ has a canonical $\mathcal {C}^e$-module structure.

Finally, for $3\implies 1$, we can simply define

\begin{align*} X^* &= (\operatorname{id}_\mathcal{C}\boxtimes \operatorname{Hom}(X,-))(T^{\textrm{R}}(\mathbf{1}_\mathcal{C})),\\ ^*X &= (\operatorname{Hom}(X,-)\boxtimes\operatorname{id})(T^{\textrm{R}}(\mathbf{1}_\mathcal{C})), \end{align*}

and check that they define duality functors. For instance, we have natural isomorphisms

\begin{align*} \operatorname{Hom}(\mathbf{1}_\mathcal{C}, Y\otimes X^*) &\cong \operatorname{Hom}(\mathbf{1}_\mathcal{C}\boxtimes X, (Y\boxtimes \mathbf{1}_\mathcal{C})\otimes T^{\textrm{R}}(\mathbf{1}_\mathcal{C}))\\ &\cong \operatorname{Hom}(\mathbf{1}_\mathcal{C}\boxtimes X, T^{\textrm{R}}(Y))\\ &\cong \operatorname{Hom}(X, Y), \end{align*}

from which we can extract the required evaluation and coevaluation data.

4.2 The Morita theory ${\mathbf {RigidTens}}$

This section is devoted to the proof of the following definition-proposition.

4.2.2 Closure under composition of $1$-morphisms

Suppose that $\mathcal {C}\in {\mathbf {RigidTens}}$, and that $\mathcal {M},\mathcal {N}$ are right and left $\mathcal {C}$-module categories, respectively, with enough compact projectives. Then we need to show that $\mathcal {M}\boxtimes _\mathcal {C} \mathcal {N}$ has enough compact projectives.

For this, we recall that by construction $\mathcal {M}\boxtimes _\mathcal {C} \mathcal {N}$ is generated under colimits by the image of $\mathcal {M}\boxtimes \mathcal {N}$ through the canonical functor,

\[ F: \mathcal{M}\boxtimes \mathcal{N}\to \mathcal{M}\boxtimes_\mathcal{C} \mathcal{N}. \]

Objects in the image of $F$ are called ‘pure tensors’. We may rewrite this functor as

\[ \mathcal{M}\boxtimes \mathcal{N} \simeq (\mathcal{M}\boxtimes \mathcal{N}) \boxtimes_{\mathcal{C}^e} \mathcal{C}^e \xrightarrow{\operatorname{id}\boxtimes m} (\mathcal{M} \boxtimes \mathcal{N})\boxtimes_{\mathcal{C}^e} \mathcal{C} \simeq \mathcal{M}\boxtimes_\mathcal{C} \mathcal{N}. \]

By the assumption of cp-rigidity, $m$ has a cocontinuous right adjoint $m^{\textrm {R}}$, hence it preserves compact projectives. Hence it follows that $F$ preserves compact projectives as well. Combining the two observations, the pure tensors give a collection of compact projective generators of the balanced tensor product.

4.3 The Morita theory ${\mathbf {RigidBrTens}}$

This section is devoted to the proof of the following definition-proposition.

4.3.2 Closure under composition of $1$-morphisms

Let $\mathcal {A}_1,\mathcal {A}_2,\mathcal {A}_3$ be cp-rigid braided tensor categories, and let $\mathcal {C}$ and $\mathcal {D}$ be cp-rigid $(\mathcal {A}_1\text{--}\mathcal {A}_2)$- and $(\mathcal {A}_2\text{--}\mathcal {A}_3)$-central tensor categories, respectively. We need to show that $\mathcal {C}\boxtimes _{\mathcal {A}_2} \mathcal {D}$ lies in ${\mathbf {RigidTens}}$.

Since $\mathcal {A}_2$ is cp-rigid, the arguments of the preceding section imply that $\mathcal {C} \boxtimes _{\mathcal {A}_2} \mathcal {D}$ has enough compact projectives. The canonical functor,

\[ F: \mathcal{C}\boxtimes \mathcal{D} \to \mathcal{C} \boxtimes_{\mathcal{A}_2} \mathcal{D}, \]

is a tensor functor, hence it maps dualizable objects to dualizable objects. In particular, pure tensor products of dualizable, compact and projective objects will again be dualizable, compact and projective, and will be generators for $\mathcal {C}\boxtimes _{\mathcal {A}_2} \mathcal {D}$, as desired.

4.4 The Morita theory ${\mathbf {BrFus}}$

5.1 Dualizability over the enveloping algebra

Proposition 5.3 Suppose that $\mathcal {C}\in \operatorname{\mathbf{Pr}}$ is dualizable over $\mathcal {C}^e$, and that $\mathcal {M}$ is a left or right $\mathcal {C}$-module, which is dualizable as an object of $\operatorname{\mathbf{Pr}}$. Then we have a canonical equivalence,

\[ \operatorname{Hom}_\mathcal{C}(\mathcal{M},\mathcal{C})\boxtimes_\mathcal{C} \mathcal{N} \simeq \operatorname{Hom}_\mathcal{C}(\mathcal{M},\mathcal{N}). \]

Lemma 5.4 We have an equivalence of categories,

(5.1)\begin{equation} \operatorname{Hom}_{\mathcal{C}^e}(\mathcal{C},\operatorname{Hom}(\mathcal{M},\mathcal{N})) \simeq \operatorname{Hom}_\mathcal{C}(\mathcal{M},\mathcal{N}), \end{equation}

given by $F\mapsto F(\mathbf {1}_\mathcal {C})$, where $\operatorname {Hom}(\mathcal {M},\mathcal {N})$ – that is, $\operatorname {Hom}$s in $\operatorname{\mathbf{Pr}}$ – is viewed as a $\mathcal {C}$-bimodule by pre- and postcomposition, and the $\mathcal {C}^e$-module structure on the left-hand side uniquely determines the $\mathcal {C}$-balancing on the right-hand side.

The result then follows from a sequence of equivalences: