MONOIDAL DISTRIBUTORS

Proof. For small monoidalV-categoriesM_andN_{, we define the hom-category of lax monoidal}

distributors fromM_to N_{by letting}

MonDistlaxV [M,N] =defMONCATlaxV [Nop⊗M,V].

Then, we define the functor

(−)†: MonDistlaxV [M,N]→RiglaxV [P(M), P(N)]

as the composite

MonDistlaxV [M,N]

λ

/

/_MONCATlax_{V [}M_{, P(}N_)] (−)c //_Riglax_{V [P(}M_{), P(}N_)].

The functor λis an equivalence by Proposition 2.1.5. Since (−)c is also an equivalence (being a quasi-inverse to composition with yM), it follows that (−)† is an equivalence, as required. The rest of the data necessary to have a bicategory is determined by the requirement to have a Gabriel factorization. In particular, the second part of the Gabriel factorization is then defined by mapping a small monoidalV-category to M† _{=def P(}M_{), viewed as a V-rig with the convolution monoidal}

structure. For the first part of the factorization, we need to show that ifu:M_→N_{is a lax monoidal}

V-functor, then the distributor u•:M → N is lax monoidal. But the functor u!: P(M) → P(N)

is lax monoidal, since the functor u: M _→ N _{is lax monoidal. Corollary 2.1.6 implies that the}

distributoru•is lax monoidal, since we have (u•)†∼=u!.

The composition operation of MonDistlaxV is obtained as a restriction of the composition op-

eration of DistV. Indeed, for lax monoidal distributors F: M→Nand G: N→P, the composite

distributorG◦F:M_→P_{is lax monoidal, since there is an isomorphism}

(G◦F)†∼=G†◦F†

and lax monoidalV-functors are closed under composition. Similarly, the identity morphism on a small monoidalV-category is the identity distributor IdM:M→M, which is lax monoidal since the hom-functor ofM _{is a lax monoidalV-functor. All of the above admits a restriction to the case of}

monoidal, rather than lax monoidal, V-functors. In order to make the theory work out smoothly, however, it is appropriate to define the notion of a monoidal distributor as follows.

Definition2.2.3. LetM_,N_{be small monoidalV-categories. Amonoidal distributor F:} M_→ N_{is a lax monoidal distributor such that the lax monoidal functorλF:}N_→P(M_{) is monoidal.}

Let us point out that requiring a lax monoidal distributor F:M _→ N_{to be monoidal is not}

equivalent to requiring the lax monoidalV-functorF:Nop_⊗M_{→ V to be monoidal. For example,}

consider the identity distributor IdM:M→M, which is given by the hom-functorM(−,−) :Mop⊗

M_{→ V. ThisV-functor is lax monoidal, but not monoidal. However, IdM:}M_→M_{is a monoidal}

distributor sinceλ(IdM) :M→P(M) is the Yoneda embedding yM: M→P(M), which is monoidal. Note that, by Corollary 2.1.6, a lax monoidal distributor F:M_→N_{is monoidal if and only if the}

lax monoidal functorF†:P(N_)→P(M_{) is monoidal.}

The next theorem defines the bicategory of monoidal distributors.

Theorem2.2.4. Small monoidalV-categories, monoidal distributors and monoidalV-transfor- mations form a bicategory, called the bicategory of monoidal distributors and denoted byMonDistV,

26 2. MONOIDAL DISTRIBUTORS

which fits in a Gabriel factorization diagram

MonCatV P // (−)• $ $ RigV MonDistV. (−)† < <

Proof. For small monoidalV-categoriesM_andN_{, we define the category MonDistV[}M_,N_{] as}

the full sub-category of MonDistlaxV [M,N] spanned by monoidal distributors. The rest of the proof

follows the pattern of the one of Theorem 2.2.2. In particular, the functor λused in the proof of Theorem 2.2.2 restricts to an equivalence

λ: MonDistV[M,N]→RigV[M, P(N)]

by the very definition of the notion of a monoidal distributor. Remark2.2.5. The tensor productF1⊗F2:M1⊗M2→N1⊗N2of lax monoidal (resp. monoidal)

distributorsF1: M1→N1 and F2:M2→N2 is lax monoidal (resp. monoidal). The operation de-

fines a symmetric monoidal structure on the bicategories MonDistlaxV and MonDistV. Moreover, the

homomorphisms in the Gabriel factorizations of Theorem 2.2.2 and Theorem 2.2.4 are symmetric monoidal. Let us also point out that the symmetric monoidal bicategory MonDistlaxV is compact:

the dual of a monoidalV-categoryM_{is the oppositeV-category}Mop_{. The counitε:M}op_{⊗M→I is}

given by the hom-functor Iop_⊗Mop_⊗M=Mop_{⊗M→ V and similarly for the unitη: I→M⊗M}op_.

In contrast, the symmetric monoidal bicategory MonDistV isnot compact.

We conclude this section by restricting to the monoidal case the operation of composition of a distributor with a functor, defined in (1.3.7).

Proposition2.2.6. LetRbe aV-rig. Then the composite of a monoidal distributorF:M_→N

with a monoidal functor T:N_{→ Ris a monoidal functorT ◦F:}M_{→ R.}

Proof. It suffices to show that (T◦F)c:P(M_{)→ Ris monoidal. The functorF}†_:P(M_)→ P(N_{) is monoidal, since the distributorF:} M_→N_{is monoidal, and the functorTc: P(}N_{)→ R is}

monoidal, since T is monoidal. Hence, the functor Tc◦F†:P(M)→ Ris monoidal. This proves the result, since we have (T◦F)c∼=Tc◦F† by Lemma 1.3.6.

2.3. Symmetric monoidalV-categories and symmetric V-rigs

The aim of this section is to develop the counterpart for symmetric monoidal V-categories of the material in Section 2.1. Let us recall that, for symmetric monoidalV-categoriesA_andB_{, a lax}

monoidalV-functorF:A_→B_{is said to besymmetric if, for any pair of objectsx∈}A_{andy ∈}B_,

the following square commutes

F(x)⊗F(y) σ µ(x,y) / /F(x⊗y) F(σ) F(y)⊗F(x) µ(y,x) //F(y⊗x),