Vertex algebras and 2 -monoidal categories
Estanislao Herscovich
Abstract
In this article we address Problem 5.12 in [8]. More precisely, we prove that the singular tensor product introduced by R. Borcherds in the previous reference is part of a2-monoidal category structure in a certain category of functors. We also complete some missing points in the previously mentioned article, most notably in the definitions of singular tensor products and of vertex algebras themselves, which are however verified in all the examples appearing in that reference. To prove our results it will be extremely useful, if not essen- tial, to frame our objects within the language of bicategories. We also introduce a slightly more general notion of (quantum) vertex algebra than the one in [8], that we call categorical (quantum) vertex algebra, enjoying all the properties mentioned by Borcherds in that article and having as particular example the definition presented by that author.
Mathematics subject classification 2020:17B69, 18M05, 18M50, 18N10.
Keywords:Duoidal categories,2-monoidal categories, vertex algebras.
1 Introduction
Vertex algebras were introduced by R. Borcherds in his seminal article [6], where he provided the axiomatic framework to deal with vertex operators, which arose in the study of dual resonance models in string theory in the 1970s but also in- dependently around the same time in the representation theory of certain infinite dimensional Lie algebras (see [13,16]). Vertex algebras have become a rather per- vasive object of study in mathematics, mainly in representation theory, finite group theory, combinatorics, algebraic geometry and number theory.
In the article [8] (see also [7]), Borcherds proposed a new formulation for the mainstay of vertex algebras, namely the use of a new tensor product, calledsingu- lar tensor product, defined in a rather convoluted category of functors (cf.[3]). The main purpose of this new tensor product is, in Borcherds’ own words, “to make the theory of vertex algebras trivial”, which should presumably be interpreted in the spirit of the structuralist approach, typically attributed to A. Grothendieck, that claims that “it is better to have a good category with bad objects than a bad category with good objects”.
The article [8] is however rather scarce in details –or even statements– concern- ing the singular tensor product, since its objective is presumably of programmatic nature. As the reader can verify from our exposition below, there are even some missing structural hypotheses in the definition of singular tensor product of [8].
However, as the reader might also expect, these missing hypotheses hold in all the examples considered in that article, which shows that the ideas laid in [8] are right.
On the other hand, we want to remark that the article [8] has not received much attention. Indeed, even though several authors studied (and generalized) some constructions on bicharacters of bialgebras and their application to vertex algebras
following [8] (seee.g.[2,24]), as far as we know the categorical constructions in [8]
and the singular tensor product have not been studied in general. However, in the interesting preprint [10] the author compares the definition of vertex algebra in [8]
for the specific choice recalled in Thm. 5.9, and the classical definition of vertex algebra.
The question posed in [8] we mainly concern us with is his Problem 5.12, where he states the need for a theory of categories with two tensor products, that applies to the singular tensor product, together with a canonical tensor product of the objects he considers. The main goal of this article is to show that the theory of 2-monoidal categories introduced in [1] (see also [9,25]) fits this situation, verify- ing all the properties mentioned by Borcherds, in particular the exchange law as well as the fact that classical tensor product of vertex algebras should naturally be vertex algebras (see the penultimate paragraph of p. 64 in [8]). More precisely, we remark thatvertex algebras, introduced in [8], Def. 3.12, are indeed particular examples of thecategorical vertex algebraswe introduce (see Definitions5.3and 5.8). Moreover, the latter satisfy all the categorical properties Borcherds mentioned in Section 3 of [8] (see Theorems4.5,4.8and4.10). To be fair, since the latter article does not contain many details on which properties these tensor products should satisfy (others than the ones we mentioned), we cannot be sure if our proposal fits exactly what Borcherds had in mind, but we believe that our results should be the closest possible.
The structure of the article is as follows. In Section2we recall the basic ter- minology on (braided) monoidal categories, (co)algebras and (co)modules over them. In Subsection 2.3we present some elementary constructions from bicate- gory theory. In particular, we introduce a notion of module over a homomorphism of bicategories and present an elementary but useful result (see Proposition2.2), both of which we were unable to find in the literature, although we are convinced they should be well known. Even though most, if not all, of the contents of the sec- tion are well known, the main reason for recalling them briefly is to establish the notation we shall later use for the proof of our results. In Section3we lay the basic objects we will study and prove the basic results we will need to define singular tensor products.
Finally, in Section4we prove the main results of this work, namely that a cer- tain category Cof modules over a commutative algebra has a natural structure of symmetric2-monoidal category (see Theorems4.5,4.8and4.10). To deal with the rather convoluted algebraic properties of the objects we are presented with, it will prove essential to make use of the constructions of bicategories recalled in Subsection 2.3. Not only do they provide a very natural framework to work in, but also they will be doing some heavy lifting. After setting a general categori- cal framework, we introduce in Subsection5.1the notion of categorical (quantum) vertex algebra, which verifies all the categorical properties Borcherds mentioned in Section 3 of [8] for his (quantum) vertex algebras. Finally, in Subsection5.2we present the notion of (quantum) vertex algebra introduced in [8], Def. 3.12, as a special case of categorical (quantum) vertex algebra. We remark that the version of quantum vertex algebra in [8] is different from others in the literature (cf.[12]).
I thank Carina Boyallian for pointing out the interesting reference [10].
2 Preliminaries
In this section we briefly review the well-known notions and results of (braided) monoidal categories as well as the basic algebraic structures within them we will use. The main purpose is to lay the notation we will use in the sequel. In Sub- section2.3, we recall some basic results coming from bicategory theory. In the last part of this section we briefly recall the not so well-known notion of2-monoidal
(or duoidal) category.
2.1 Notation
We will denote by N0 (resp.,N) the set of nonnegative (resp., positive) integers {0,1,2, . . .}(resp.,{1,2, . . .}), and givenn0, n00∈N0we denote byJn0, n00Kthe set {n∈N0:n0≤n≤n00}. Givenn∈N, letSnbe the group of bijections ofJ1, nK.
To alleviate the burden of notation, and as it is widely done in the literature, we will often commit the abuse of omitting the extra structure of an algebraic object if this does not cause any confusion (e.g. we will denote a monoidal categoryC simply by the underlying class of objects and morphisms). The reason is typically that the notation we follow for the extra structure is obtained by simply decorat- ing fixed symbols with the object one mentioned (e.g. for the case of a monoidal categoryC,⊗C will denote the tensor product andIC the unit).
2.2 Review of basic facts on monoidal categories
We assume the reader is familiar with the basic definitions of categories (see [4]), as well as the particular case of (braided)monoidal categories(see [14], Ch. XI and XIII). We will only recall some basic facts about them, which also allow us to set the notation. Moreover, to avoid set-theoretic issues –and as it is implicitly assumed in the literature– we fix an infinite universe U such that the classes of objects and morphisms of all of our categories are subsets of the universe (i.e. we deal withU-categories), except for the categoryCatof all (U-)categories. We just recall that a small category in this situation simply means that the class of objects is an element of the universe.
We remark that, if(C,⊗C,IC)is a monoidal category, we will denote theasso- ciativity,left unitandright unit constraintsby
aC(X, Y, Z) : (X⊗C Y)⊗C Z−→X⊗C (Y ⊗C Z), lC(X) :IC ⊗C X −→X and rC(X) :X⊗C IC −→X,
respectively, for objectsX,Y andZofC. We will sometimes omit the arguments in the previous natural transformations if they are clear from the context. More- over, in some expressions, specially to alleviate the notation in the composition of several morphisms of algebraic objects such as (co)algebras and (co)modules, we will sometimes omit the associativity constraint, if it is clear from the context.
We shall typically denote the monoidal category simply by its underlying category C, since we will use the previous notation to denote the rest of its structure, un- less otherwise stated. Analogously, a braided monoidal category(C,⊗C,IC, τC) will simply be denoted by C. We say that the unitIC of the monoidal category C is strict if l(X)C = idX = r(X)C, for all objects X of C, so in particular IC ⊗CX =X =X⊗C IC.
To keep track of the different parenthesizations of tensor products we will use the following typical bookkeeping device. Givenn∈ N0, denote byYn the set of allrooted planar binary treeswith(n+1)leaves, which we we typically label with the elements ofJ1, n+ 1Kfrom left to right (see [17], Appendix C.1). For example,
Y0=
1 ,Y1=
1 2 ,Y2=
1 2 3
t2 1
,1 2 3
t2 2
,
Y3=
1 2 3 4
t3 1
,1 2 3 4
t3 2
,
1 2 3 4
t3 3
,
1 2 3 4
t3 4
,
1 2 3 4
t3 5
,
(2.1)
where we have numbered the elements ofY2andY3for future reference.
We will denote the set of vertices oft ∈Yn byVer(t)and the set of leaves by Lvs(t). We recall that a treet∈Ynforn≥1hasnvertices,n−1edges–i.e.edges connecting two vertices–, andn+ 2half-edges, –i.e. edges incident on at most one vertex–, n+ 1of which are leaves and one extra half-edge, called theroot.
The treet∈Y0is supposed to be formed of just the root, which is simultaneously considered as a leaf. We also recall that given rooted planar binary treest∈Ynand t0∈Yn0,graftingthe root oft0to thei-th leaf oftforms a new rooted planar binary treet00∈Yn+n0. We will say thatt∈Ynisthinif given any vertexv∈Ver(t)there is a leafiadjacent onv.
Given a monoidal categoryC andt ∈ Yn, denote byt(⊗C) : Cn+1 → C the functor defined recursively as follows. Ift ∈Y0,t(⊗C)is the identity functor of C, and forn ≥ 1 if t ∈ Yn is obtained from treestl ∈ Yn0 and tr ∈ Yn00 with n0 +n00+ 1 = nby grafting the root oftl(resp.,tr) to the left (resp., right) leaf of t0 ∈ Y1, then t(⊗C) = ⊗C ◦(tl(⊗C), tr(⊗C)). Givent, t0 ∈ Yn, we denote by at→tC 0 : t(⊗C) → t0(⊗C)the natural isomorphism obtained from using the associativity constraint of C, which is unique by Mac Lane’s coherence theorem (see [14], Thm. XI.5.3).
The previous construction can be extended to the case of a monoidal categoryC with a symmetric braidingτC. Forn∈N, ifσ∈Sn+1, we denote byc(σ) :Cn+1→ Cn+1 the auto-functor sending(X1, . . . , Xn+1)to(Xσ−1(1), . . . , Xσ−1(n+1)), for all (n+1)-tuples of objects or morphisms ofC. SetYsymn = Yn×Sn+1. Then, givenT = (t, σ), T0= (t0, σ0)∈Ysymn , we denote byacCT→T0 :t(⊗C)◦c(σ)→t0(⊗C)◦c(σ0) the natural isomorphism obtained from using the associativity constraintaC ofC and the symmetric braidingτC, which is unique by Mac Lane’s coherence theorem (see [18], Thm. XI.1.1). IfT = (t,id
J1,n+1K), T0 = (t, σ)∈Ysymn , we will denote the morphismacCT→T0simply byτC(σ, t).
We recall that a monoidal categoryC is said to besemicocartesianif its unit IC is the initial object of the underlying category ofC (cf.[11]). In this case, given t ∈ Yn, an(n+ 1)-tuple Xˆ = (X1, . . . , Xn+1)of objects ofC and i ∈ J1, n+ 1K, letXˆi be the(n+ 1)-tuple whosej-th component isXiifj = iandIC else, and fˆibe the(n+ 1)-tuple of morphisms ofC whosej-th component isidXi ifj =i andiXj : IC →Xj else. We defineιi,t( ˆX) :Xi → t(⊗C)( ˆX)as the composition of the unique isomorphismXi →t(⊗C)( ˆXi)given by using the left and right unit constraints ofC andt(⊗C)( ˆfi). It is clear thatιi,t( ˆX)is natural inXˆ, and,
at→tC 0( ˆX)◦ιi,t( ˆX) =ιi,t0( ˆX), (2.2) for any t, t0 ∈ Yn, by Mac Lane’s coherence theorem for monoidal categories, which also tells us the following result. Givent0 ∈ Ymand t ∈ Yn, as well as tuples Yˆ = (Y1, . . . , Ym+1)and Xˆ = (X1, . . . , Xn+1)of objects of F, such that Yj =t(⊗C)( ˆX)for somej ∈J1, m+ 1K, considert00∈Yn+mobtained by grafting the root ofton thej-th leaf oft0, andZˆ = (Z1, . . . , Zn+m+1)given byZi = Yi if i∈J1, j−1K,Zi=Xi−j+1ifi∈Jj, j+nK, andZi=Yi−nifi∈Jj+n+ 1, m+n+ 1K.
Then,
ιi,t00( ˆZ) =
ιi,t0( ˆY), ifi∈J1, j−1K, ιj,t0( ˆY)◦ιi,t( ˆX), ifi∈Jj, j+nK,
ιi−n,t0( ˆY), ifi∈Jj+n+ 1, n+mK.
(2.3)
Ift∈Y1, we will denoteιi,t( ˆX)simply byιi( ˆX).
We remark that the notion ofmonoidal(ortensor)functorin [14], Def. XI.4.1, is nowadays calledstrongmonoidal, specially due to the pervasiveness oflaxand
oplaxmonoidal functors (see [1], Section 3.1.1). We will denote thestructure mor- phisms of a(n) (op)lax monoidal functor F : C → D between monoidal cate- gories byϕF(X, Y) :F(X)⊗D F(Y)→F(X⊗C Y)andυF :ID →F(IC)(resp., ϕF(X, Y) :F(X⊗C Y)→F(X)⊗DF(Y)andυF :F(IC)→ID). The morphisms ϕF(X, Y)can be assembled into a family of natural morphisms
ϕF,t( ˆX) :t(⊗D) F(X1), . . . , F(Xn)
−→F t(⊗C)( ˆX) ,
resp.,ϕF,t( ˆX) :F t(⊗C)( ˆX)
−→t(⊗D) F(X1), . . . , F(Xn)
for alln≥2,t∈Yn−1andXˆ = (X1, . . . , Xn)∈Cn. Then, the definition of (op)lax monoidal functor tells us that
ϕF,t0( ˆX)◦aDt→t0 F(X1), . . . , F(Xn)
=F at→tC 0( ˆX)
◦ϕF,t( ˆX)
resp.,at→tD 0 F(X1), . . . , F(Xn)
◦ϕF,t( ˆX) =ϕF,t0( ˆX)◦F at→tC 0( ˆX)
, (2.4) for alln≥2,t, t0 ∈Yn−1andXˆ = (X1, . . . , Xn)∈Cn.
2.3 Some constructions from bicategory theory
We will also assume that the reader is familiar with the basic notions of bicategories (see the short but very clear introduction [15], whose notation we shall follow, or the more comprehensive [4]). For the reader’s convenience, we provide some basic notions in the rather restricted setting we will be interested in. We also provide some fundamental results we will extensively use. They will allow us to effectively organize the several algebraic structures appearing from Section 3 onward, and they will also do some heavy lifting for us.
We recall first that the categoryCatof all categories has a canonical structure of2-category, and letF be a small category, which is considered as a2-category where the morphism spaceMorF(I, J)is considered as a discrete category for ev- ery pair of objectsIandJofF. In particular,F andCatare bicategories.
We recall that a (unit preserving)homomorphism of bicategoriesf: F → Catconsists of a mapI 7→f(I)sending every objectIofF to a categoryf(I), a mapf 7→f(f)sending every morphismf :I→J inF to a functorf(f) :f(I)→ f(J), and natural isomorphismsφf(g, f) : f(g)◦f(f) →f(g◦f)of functors for every pair of composable morphisms f and g inF, satisfying thatf(idI)is the identity functor off(I)for every objectIofF and
φf(h, g◦f)(−)◦f(h) φf(g, f)(−)
=φf(h◦g, f)(−)◦φf(h, g) f(f)(−)
, (2.5) for every triple of composable morphismsf,gandhinF. It is easy to see that any functorf:F →Catof the underlying categories gives astricthomomorphism of bicategories,i.e.such thatφf(g, f)is the identity.
Letf,g:F →Catbe two homomorphisms of bicategories. Then, there exists a natural categoryTransst(f,g), which we recall in elementary terms. The objects of the categoryTransst(f,g), called (unit preserving)strong transformations, are given by functorsζ(I) : f(I) → g(I)for all objectsI ofF, and natural isomor- phisms of functorsζ(f) :g(f)◦ζ(I)→ζ(J)◦f(f)for all morphismsf :I→Jin F such thatζ(idI)is the identity natural transformation and
ζ(g◦f)(−)◦φg(g, f) ζ(I)(−)
=ζ(K) φf(g, f)(−)
◦ζ(g) f(f)(−)
◦g(g) ζ(f)(−) , (2.6) for all morphismsf : I → J andg :J → KinF. Given two objectsζandζ0of Transst(f,g), a morphismΓ : ζ → ζ0inTransst(f,g), called amodification, is
given by natural transformationsΓ(I) : ζ(I) → ζ0(I)for all objectsIofF such that
ζ0(f)(−)◦g(f) Γ(I)(−)
= Γ(J) f(f)(−)
◦ζ(f)(−) (2.7) for all morphismsf :I →JinF. The composition of modificationsΓ00 :ζ00 →ζ andΓ0 :ζ →ζ0is the modification given by the natural transformationΓ(I)(−) = Γ0(I)(−)◦Γ(I)(−), for every objectIofF.
We also recall that a strong transformationζ : f→f0 of homomorphisms of bicategories is said to bestrict if ζ(f) is the identity for all morphismsf inF. Given a homomorphisms of bicategoriesf:F →Catdefine theidentity strong transformationidf:f→fas the unique strict natural transformation such that idf(I) :f(I)→f(I)is the identity functor for all objectsIofF. Moreover, given homomorphisms of bicategoriesf00,f,f0 : F →Cat and strong transformations ζ00:f00 →fandζ0 :f→f0, define thecomposedstrong transformationζ0◦ζ00to be the strong transformationζ¯such thatζ(I) =¯ ζ(I)◦ζ0(I)for all objectsIofF, andζ(f¯ )(−) = ζ(J)(ζ0(f)(−))◦ζ(f)(ζ0(I)(−))for all morphismsf :I→J inF. It is straightforward to check thatζ0◦ζ00indeed defines a strong transformation.
The categoryTransst(f,g)is just the space of morphisms fromftogin the 2-category of homomorphisms from F to Cat (see the 2-category [F,Cat] in [15], Section 2.0). We will only need a very small fragment of this well-known 2-category structure, which is contained in the next result.
Lemma 2.1. Let f,f0,g : F → Cat be homomorphisms of bicategories. A strong transformationζ0 :f→f0induces a functor
Transst(ζ0,g) :Transst(f0,g)−→Transst(f,g) (2.8) satisfying thatTransst(idf,g)is the identity functor, and
Transst(ζ0◦ζ00,g) =Transst(ζ00,g)◦Transst(ζ0,g),
for all strong transformationsζ0 :f→f0 andζ00 : f00 →f, and all homomorphisms of bicategoriesf00:F →Cat.
Proof. Given an objectζofTransst(f0,g), letTransst(ζ0,g)(ζ)be the objectζ◦ζ0 ofTransst(f,g). Moreover, given a modificationΓ :ζ1 →ζ2for objectsζ1andζ2
ofTransst(f0,g), letTransst(ζ0,g)(Γ)be the modificationΓ¯ given byΓ(I)(−) =¯ Γ(I)(ζ0(I)(−))for all objectsI ofF. This gives a well-defined functor (2.8), due to the strong hypothesis onζ0. The last part of the statement follows immediately
from the definitions.
We will also make use of the previous result where the strong transformation ζ0:g→g0appears in the second argument.
We have not been to able to find the following definition in the literature, but we believe it should be well known. Given a homomorphism of bicategories g: F →Cat, we define ag-moduleM as a collection of objectsMI ∈ g(I)indexed by all objectsI inF and a family of morphismsMf : g(f)(MI) → MJ in g(J) indexed by all morphismsf :I→J inF such thatMidIis the identity ofMI for allIinF, and
Mg◦g(g)(Mf) =Mg◦f◦φg(g, f)(MI), (2.9) for all morphismsf : I → J andg : J → KinF. Amorphism of g-modules fromM0 toM is a collection of morphismsFI : MI0 → MI for all objectsI ofF, such that
FJ◦Mf0 =Mf ◦g(f)(FI), (2.10) for all morphismsf :I→JinF. Then,g-modules and their morphisms, together with the usual composition, form a category that we will denote byg-Mod.
We will make intensive use of the next result.
Proposition 2.2. Letf,f0,g: F →Catbe homomorphisms of bicategories. Suppose further thatg(I)is cocomplete for all objectsIofF, andg(f)preserves colimits for all morphismsf :I→JinF. Taking colimits induces a functor
colimf,g:Transst(f,g)−→g-Mod. (2.11) Moreover, a strong transformationζ0 :f→f0induces a natural transformation
Z0 : colimf,g◦Transst(ζ0,g)−→colimf0,g,
which is an isomorphism ifζ0(I)is an equivalence for all objectsIofF.
Proof. Given an objectζofTransst(f,g), we define theg-modulecolimf,gζsuch that(colimf,gζ)I is the colimit of the functorζ(I)forIinF, and, for every mor- phismf : I → J inF,(colimf,gζ)f : g(f)((colimf,gζ)I) → (colimf,gζ)J is the composition of
g(f) colimζ(I) ∼
−→colim g(f)◦ζ(I) ∼
−→colim ζ(J)◦f(f)
−→colimζ(J), (2.12) where the first map follows from the fact that g(f)preserves colimits, the second map is induced byζ(f)and the last one follows from the general property of colim- its. A straightforward computation using (2.6) and the universal property of colim- its tells us thatcolimf,gζis indeed ag-module. IfΓ :ζ→ζ00is a modification, for ζ andζ00inTransst(f,g),Γ(I)induces a morphismcolimζ(I)→ colimζ00(I)for every objectIofF, defining a morphism ofg-modulescolimf,gζ →colimf,gζ00, by (2.7) and the universal property of colimits. It is straightforward to check that this definition is indeed a functor, since taking colimits is functorial.
Finally, given an objectζ of Transst(f,g) and an objectI of F, define the morphism
Z0(ζ)I : colim ζ(I)◦ζ0(I)
−→colimζ(I) (2.13)
by the general property of colimits. Using this and the definition of ζ ◦ ζ0(f) we see that the previous construction gives a morphism of g-modules Z0(ζ) : colimf,g◦Transst(ζ0,g)(ζ)→ colimf0,g(ζ). The naturality inζfollows from (2.7) and the universal property of colimits, soZ0 is a natural transformation. Finally, it is clear that ifζ0(I)is an equivalence for all objectsIofF, then (2.13) is also an isomorphism, which implies thatZ0(ζ)is an isomorphism ofg-modules.
We will usually omit the subscripts of the functor (2.11), if they are clear from the context, so we will simply writecolim.
2.4 Review of basic facts on (co)algebras and (co)modules
We also assume that the reader is familiar with the basic notions of (co)algebras (also called (co)monoids) and their (co)modulesin monoidal categories, as well asbialgebras(orbimonoids) in braided monoidal categories (see [21], Ch. 1, [19, 20,22,23] or the nice reference [1]). All (co)algebras in this article are assumed to be (co)unitary, their morphisms respect the (co)unit, and all (co)modules over them are assumed to be compatible with the (co)unit, as well as their morphisms.
IfA(resp.,C) is an algebra (resp., a coalgebra), we will denote its (co)product by µA:A⊗CA→A(resp.,∆C:C→C⊗CC) and its (co)unit byηA:IC →A(resp., C : C → IC). IfAis an algebra in a monoidal category(C,⊗C,IC), we denote byAMod(C)the category of leftA-modules inC and morphisms ofA-modules.
Each A-module is typically written as a pair(M, ρM), with structure morphism ρM : A⊗C M →M, or just byM. RightA-modules are defined analogously, but we will consider left modules unless otherwise stated.
An algebraA(resp., a coalgebraC) in a braided monoidal categoryC is said to be (co)commutativeifµA◦τC(A, A) =µA(resp.,τC(C, C)◦∆C= ∆C). We denote byAlg(C)(resp.,coAlg(C)) the subcategory ofC formed by all (co)algebras and morphisms of (co)algebras. Given an algebraA(resp., a coalgebraC) in a monoidal categoryC andt∈Yn, denote byAˆ(resp.,Cˆ) the(n+ 1)-tuple whose each entry isA(resp.,C). We will denote byµtA:t(⊗C)( ˆA)→A(resp.,∆tC:C→t(⊗C)( ˆC)) the morphism defined recursively asµtA= idA(resp.,∆tC= idC) ifn= 0, and, for n ≥1, iftis obtained from treestl∈ Yn0 andtr ∈ Yn00withn0+n00+ 1 =nby grafting the root oftl(resp.,tr) to the left (resp., right) leaf oft0∈Y1, then
µtA=µA◦(µtAl⊗C µtAr)
resp.,∆tC= (∆tCl⊗C ∆tCr)◦∆C
,
We also setµ[0]A =ηA(resp.,∆[0]C =C). By making use of the (co)associativity of µA (resp.,∆C), givenn ∈ N, we also writeµ[n]A (resp.,∆[n]C ) instead ofµtA (resp.,
∆tC) fort∈Yn−1.
We remark that, given two algebrasAandA0inC anA-A0-bimodule is a pair (M, ρ), with structure morphismρ : A⊗C M ⊗C A0 → M satisfying the usual associativity and unit conditions. It can be equivalently defined as the datum of a left A-module structureρl : A⊗C M → M and a right A0-module structure ρr : M ⊗C A0 → M such thatρr◦(ρl⊗C idA0) =ρl◦(idA⊗Cρr). It is a simple verification that, if C is (resp., finitely) cocomplete, i.e. C has all (resp., finite) colimits, and the tensor product⊗C commutes with (resp., finite) colimits on both sides, then AMod(C)is cocomplete and the inclusion functor AMod(C) → C preserves (resp., finite) colimits. The analogous result holds for right modules and bimodules as well.
IfC has finite colimits, given a rightA-module(M, ρM,r)and a leftA-module (N, ρN,l)one further definesM ⊗AN as the coequalizer inC of the pair of maps ρM,r⊗C idN,idM⊗CρN,l:M ⊗C A⊗C N → M ⊗C N. Assume further that the tensor product ofC commutes with finite colimits on each side and suppose that M has anA0-A-bimodule structure for a leftA0-action given byρM,l. Then,M⊗AN has a natural structure of leftA0-module induced by ρM,l⊗C idN. Analogously, given a morphismg : N → N0 of leftA-modules, idM⊗Cg naturally induces a morphism of left A0-modules idM⊗Ag : M ⊗AN → M ⊗AN0. The previous construction gives a functor
M ⊗A(−) :AMod(C)−→A0Mod(C).
Note that, if M = A has the A-A-bimodule structure given by µ[3]A and N is a left A-module, thenρN,l : A⊗C N → N is precisely a coequalizer of the pair ρA,r⊗C idN and idA⊗CρN,l. This allows us to set A⊗AN = N, where the left A-module structures also coincide. Analogously, we can (and will) set the identity N0⊗AA=N0of rightA-modules, for every rightA-moduleN0.
Iff :A→A0is a morphism of algebras inC, one defines a functor
Resf :A0Mod(C)−→AMod(C) (2.14) that sends anA0-module(M, ρ)to theA-module(M, ρ◦(f ⊗C idM))and is the identity on morphisms. IfC has finite colimits and the tensor product commutes with them on both sides, then a standard computation shows thatResfhas the left adjoint
Indf =A0⊗A(−) :AMod(C)−→A0Mod(C), (2.15) whereA0has theA0-A-bimodule structure morphismµ[3]A0◦(idA0⊗CidA0⊗Cf).
For the following result, consider Alg(C) as a2-category where every mor- phism space is a discrete category, andCathas the usual structure of2-category.
The proof is straightforward.
Fact 2.3. Assume thatC is a monoidal category. Consider the map
R :Alg(C)op−→Cat (2.16)
sending an algebraAtoAMod(C)and a morphismf :A0→Aof algebras to the functor Resfgiven in(2.14). Then,(2.16)is a functor, so a strict homomorphism of bicategories.
Assume moreover thatC is finitely cocomplete and the tensor product commutes with colimits on both sides. Consider the map
I :Alg(C)−→Cat (2.17)
sending an algebraAtoAMod(C), a morphismf : A0 → Aof algebras toIndf, and, given a pair of morphismsf :A0 →Aandg :A→A00of algebras, consider the unique natural isomorphismφI(g, f) : Indg◦Indf → Indg◦f of functors coming from the fact that both are left adjoints ofResf◦Resg = Resg◦f. Then,(2.17)is a homomorphism of bicategories.
Assume thatAis a commutative algebra in a symmetric monoidal categoryC. As usual, in this case one identifies leftA-module structuresρl : A⊗C M → M and rightA-modules structuresρr:M⊗C A→M on an objectM ofC by means ofρr = ρl◦τC(M, A), and we write eitherρlorρr simply byρ. IfC has finite colimits and the tensor product commutes with them on both sides, we recall that the category AMod(C)endowed with the tensor product ⊗A, the associativity constraint induced by that of C, the unit given by A with the regular structure given by the product ofAand the symmetric braiding induced by that ofC is also a symmetric monoidal category with a strict unit, denoted byAMods(C).1 Lemma 2.4. AssumeC is a finitely cocomplete symmetric monoidal category and the ten- sor product commutes with finite colimits on both sides. Letf :A→A0be an morphism of commutative algebras in C. IfM andN areA-modules, there is a canonical natural isomorphism
Indf(M⊗AN)−→Indf(M)⊗A0Indf(N) (2.18) of A0-modules. Moreover, iff is an epimorphism of commutative algebras inC, given A0-modulesM0andN0, the canonical morphism
Resf(M0)⊗AResf(N0)−→Resf(M0⊗A0N0) (2.19) ofA-modules is an isomorphism.
Proof. The first isomorphism is an immediate consequence of the associativity con- straint and the symmetric braiding ofAMods(C). Let us prove the second. Since (2.19) factors through
Resf(M0)⊗AResf(N0)'Resf(M0⊗A0A0)⊗AResf(A0⊗A0N0) 'Resf M0⊗A0(A0⊗AA0)⊗A0N0
,
it suffices to show that the productµA0 ofA0induces an isomorphismA0⊗AA0→ A0. The last identity follows from the fact that a morphismf :A→A0of commu- tative algebras is an epimorphism if and only if
A A0
A0 A0
f
f idA0
idA0
1Note that we definedA⊗AM=M=M⊗AAthree paragraphs before, so the unit ofAMods(C) is strict regardless of the unit ofCbeing strict or not.
is a push-out, and that a push-out of a pair of morphismsf :A→A0andg:A→ A00of commutative algebras in C is the commutative algebraA¯ = Resf(A0)⊗A
Resg(A00), together with the maps A0 → A¯andA00 → A¯given byidA0⊗Ag and f ⊗AidA00, respectively, and the usual product given as the composition of the canonical projection
(A0⊗AA00)⊗C (A0⊗AA00)−→(A0⊗AA00)⊗A(A0⊗AA00),
the associativity constraint inA = AMods(C)with idA0⊗AτA(A00, A0)⊗AidA00
and µ¯A0 ⊗A µ¯A00, where µ¯X : X ⊗A X → X is the morphism induced by the
productµX:X⊗C X →XofX∈ {A0, A00}.
We recall that, ifC is a monoidal category with a braidingτC andBis a bialge- bra inC, then the tensor productM⊗CNof twoB-modules(M, ρM)and(N, ρN) has a natural structure ofB-module, calleddiagonal, given by
ρM ⊗C ρN
◦ idB⊗CτC(B, M)⊗C idN
◦ ∆B⊗C idM⊗CN
, (2.20) where∆Bdenotes the coproduct ofB, andIC has the structure ofB-module given by the composition ofrC(B)and the counitBofB. With this structureBMod(C) is a monoidal category and the inclusion inside ofC is strong monoidal (see [19], Lemma 1.1). We denote byBModt(C)the categoryBMod(C)endowed with the previous monoidal structure. Recall that aB-module algebrais an algebra in the monoidal categoryBModt(C).
On the other hand, recall that, ifF :C →C0 is a(n) (op)lax monoidal functor, thenF sendsAlg(C)(resp.,coAlg(C)) toAlg(C0)(resp.,coAlg(C0)), and given an algebraAit sends the subcategoryAMod(C)ofC toF(A)Mod(C0). IfC and C0 are braided,B is a bialgebra inC andF is braided strong monoidal, then it restricts to a braided strong monoidal functorBModt(C)→F(B)Modt(C0).
If the braidingτC ofC is further assumed to be symmetric andBis cocommu- tative, thenτC(M, N) :M ⊗C N →N⊗C M is a morphism ofB-modules for all B-modulesM andN, which implies thatBModt(C)is a symmetric monoidal cat- egory and the inclusion inside ofC is a braided strong monoidal functor (see [21], Section 1.8 for the case of vector spaces, but the general case is analogous). We will always assume thatBModt(C)is endowed with the previous symmetric braiding in this situation. We remark that, iff :B→B0is a morphism of (resp., cocommu- tative) bialgebras inC, then (2.14) is a (resp., braided) lax monoidal functor.
Remark 2.5. Given a cocommutative bialgebraBin a braided monoidal categoryC and B-modules M and N in C, the braiding τC(M, N) is notin general a morphism of B-modules (take for instance the categoryC of modules over the Fomin-Kirillov algebra on3generators). In the same spirit, given two algebrasAandA0 in a braided monoidal categoryC, even though their tensor productA⊗CA0is naturally an algebra (see the proof of Lemma2.4), the braidingsτC(A, A0)andτC(A0, A)are notin general morphisms of algebras. Moreover, the tensor product of bialgebras (resp., commutative algebras) in a braided monoidal categoryis notin general a bialgebra (resp., commutative algebra).
2.5 Duoidal (or 2-monoidal) categories
A2-monoidal category (or duoidal category) is a category Cendowed with two monoidal structures (C,⊗C,I⊗) and (C,C,I), satisfying some assumptions.
We denote the associativity, and the left and right unit constraints of(C,C,I)by a(X, Y, Z),l(X)andr(X), respectively, and those of the monoidal structure (C,⊗C,I⊗)bya⊗(X, Y, Z),l⊗(X)andr⊗(X), respectively.
Before providing the definition, it will be useful to establish some terminology to be able to deal later with the parenthesizations. First, forn∈N0, we define
Ylabn =
t= (t, λ) :t∈Yn, λ: Ver(t)→ {•,‚} .
Given t = (t, λ) ∈ Ylabn , denote byt(⊗C,C) : Cn+1 → Cthe functor defined recursively as follows. Ift∈Ylab0 ,t(⊗C,C)is the identity functor ofC. Letn≥1 andt = (t, λ)∈ Ylabn be constructed fromtl = (tl, λl)∈ Yn0 andtr = (tr, λr)∈ Yn00 withn0+n00+ 1 =nas follows. First,tis obtained by grafting the root oftl (resp.,tr) to the left (resp., right) leaf oft0 ∈Y1, whose unique vertex is denoted byv. Moreover, we assume thatλcoincides withλl(resp.,λr) on the vertices oftl (resp.,tr). Then, we set
t(⊗C,C) =
(⊗C ◦ tl(⊗C,C),tr(⊗C,C)
, ifλ(v) =•, C ◦ tl(⊗C,C),tr(⊗C,C)
, ifλ(v) =‚.
We recall now the not so well-known definition of2-monoidal category (see [1], Def. 6.1.1).
Definition 2.6. A2-monoidal category(also calledduoidal categoryin[9,25]) is a tuple(C,⊗C,I⊗,C,I), where(C,⊗C,I⊗)and(C,C,I)are monoidal categories, together with a natural morphism
sh(A, B, C, D) : (A⊗CB)C(C⊗CD)−→(ACC)⊗C(BCD) (2.21) in Cand three morphisms
µ:I⊗CI⊗−→I⊗, ∆⊗ :I−→I⊗CI, and ν :I−→I⊗, (2.22) in Csuch that the following conditions hold.
(i) The triple(I⊗, µ, ν)is an algebra in the monoidal category(C,C,I), and the triple (I,∆⊗, ν)is a coalgebra in the monoidal category(C,⊗C,I⊗).
(ii) For any pair of objectsXandY in C, the diagrams
IC(X⊗CY) (I⊗CI)C(X⊗CY) (X⊗CY)CI (X⊗CY)C(I⊗CI)
X⊗CY (ICX)⊗C(ICY) X⊗CY (XCI)⊗C(Y CI)
l(X⊗CY)
∼
∆⊗CidX⊗CY
sh(I,I,X,Y) ∼ r(X⊗CY) idX⊗CYC∆⊗
sh(X,Y,I,I)
∼ (l(X)⊗Cl(Y))−1
∼ (r(X)⊗Cr(Y))−1
and
(I⊗⊗CX)C(I⊗⊗CY) XCY (X⊗CI⊗)C(Y ⊗CI⊗) XCY
(I⊗CI⊗)⊗C(XCY) I⊗⊗C(XCY) (XCY)⊗C(I⊗CI⊗) (XCY)⊗CI⊗ sh(I⊗,X,I⊗,Y)
l⊗(X)Cl⊗(Y)
∼
l⊗(XCY)−1
∼ sh(X,I⊗,Y,I⊗)
r⊗(X)Cr⊗(Y)
∼
r⊗(XCY)−1
∼
µ⊗CidX
CY idX
CY⊗Cµ
commute.
(iii) GivenX1, Y1, X2, Y2, X3, Y3objects in C, the following diagrams
(X1⊗CY1 )C(X2⊗CY2 )
C(X3⊗CY3 )
(X1⊗CY1 )C (X2⊗CY2 )C(X3⊗CY3 )
(X1⊗CY1 )C (X2CX3 )⊗C(Y2CY3 )
X1C(X2CX3 )⊗C Y1C(Y2CY3 ) (X1CX2 )CX3
⊗C (Y1CY2 )CY3 (X1CX2 )⊗C(Y1CY2 )
C(X3⊗CY3 )
a(X1⊗CY1, X2⊗CY2, X3⊗CY3 ) sh(X1, Y1, X2, Y2 )CidX3⊗CY3
sh(X1, Y1, X2CX3, Y2CY3 ) a(X1, X2, X3 )⊗C a(Y1, Y2, Y3 )
sh(X1CX2, Y1CY2, X3, Y3 ) idX1⊗CY1Csh(X2, Y2, X3, Y3 )
and
(X1CY1 )⊗C(X2CY2 )
⊗C(X3CY3 )
(X1CY1 )⊗C (X2CY2 )⊗C(X3CY3 )
(X1CY1 )⊗C (X2⊗CX3 )C(Y2⊗CY3 )
X1⊗C(X2⊗CX3 )
C Y1⊗C(Y2⊗CY3 ) (X1⊗CX2 )⊗CX3
C (Y1⊗CY2 )⊗CY3 (X1⊗CX2 )C(Y1⊗CY2 )
⊗C(X3CY3 ))
a⊗(X1CY1, X2CY2, X3CY3 ) sh(X1, X2, Y1, Y2 )⊗CidX3CY3
sh(X1, X2⊗CX3, Y1, Y2⊗CY3 ) a⊗(X1, X2, X3 )C a⊗(Y1, Y2, Y3 )
sh(X1⊗CX2, X3, Y1⊗CY2, Y3 ) idX1CY1⊗Csh(X2, X3, Y2, Y3 )
commute.
Moreover, a2-monoidal category(C,⊗C,I⊗,C,I,sh)is said to be⊗C-braidedif the monoidal category(C,⊗C,I⊗)is provided with a braidingτ⊗such that(I,∆⊗, ν)is a cocommutative coalgebra in(C,⊗C,I⊗, τ⊗), and for any objectsA,B,CandDin Cthe diagram
(A⊗CB)C(C⊗CD) (ACC)⊗C(BCD)
(B⊗CA)C(D⊗CC) (BCD)⊗C(ACC)
sh(A,B,C,D)
τ⊗(A,B)Cτ⊗(C,D) τ⊗(ACC,BCD)
sh(B,A,D,C)
commutes. The analogous definition ofC-braided2-monoidal category is clear. A2- monoidal category(C,⊗C,I⊗,C,I,sh)is calledbraidedif it is⊗C-braided andC- braided. The analogous definitions for symmetric braidings are immediate.
Example 2.7. Any braided (resp., symmetric) monoidal category (C,⊗C,IC, τC)can be regarded as a braided (resp., symmetric)2-monoidal category where both braided (resp., symmetric) monoidal structures coincide, the unitIC has the obvious algebra and coalgebra structures, andsh(A, B, C, D)is given byidA⊗CτC(B, C)⊗CidD(see[1], Prop. 6.10).
Remark 2.8. Given two structures of monoidal categories(C,⊗C,I⊗)and(C,C,I) on Cwith structure morphisms(2.21)and(2.22), they define a structure of2-monoidal category if either of the following equivalent conditions holds:
(a) the functors⊗C : (C,C,I)×(C,C,I) → (C,C,I)and I⊗ : e → (C,C,I)are lax monoidal,
(b) the functors C : (C,⊗C,I⊗)×(C,⊗C,I⊗) → (C,⊗C,I⊗)and I : e → (C,⊗C,I⊗)are oplax monoidal,
where eindicates the monoidal category with one object whose space of endomorphisms is the identity. The proof just follows from writing down the definitions (see [1], Prop.
6.4). We may equivalently rephrase(b)(resp.,(a)) as saying that a2-monoidal category is a pseudomonoid in the monoidal 2-categoryopl(Cat)(resp., l(Cat)) whose0-cells are monoidal categories, whose1-cells are oplax (resp., lax) monoidal functors, and whose 2-cells are monoidal natural transformations (see[1], Prop. 6.73).
We recall the following result, the proof of which is just a direct consequence of the definitions.
Proposition 2.9. Let(C,⊗C,I⊗,C,I,sh)be a2-monoidal category, and letAandA0 (resp.,CandC0) be two (co)algebras in(C,C,I)(resp.,(C,⊗C,I⊗)). Then,A⊗CA0 (resp.,CCC0) is a (co)algebra in(C,C,I)(resp.,(C,⊗C,I⊗)) for the (co)product
(µA⊗CµA0)◦sh(A, A0, A, A0)
resp.,sh(C, C, C0, C0)◦(∆CC∆C0)
,
and the (co)unit(ηA⊗CηA0)◦∆⊗(resp.,µ◦(CCC0)).
If Cis alsoC-symmetric (resp.,⊗C-symmetric) andAandA0 (resp.,CandC0) are (co)commutative, thenA⊗CA0(resp.,CCC0) is (co)commutative.
3 Some preparations: categories of functors
3.1 Induced monoidal structures on categories of functors
For the rest of the article we assume thatF is a small category. Given a category C, denote by Fun(F,C)the category whose objects are functors from F to C and whose morphisms are natural transformations. We recall thatFun(F,C)is (co)complete ifC is so, and (co)limits are computed pointwise.
Moreover, ifC is endowed with a (resp., braided, symmetric) monoidal struc- ture with tensor product⊗C and unitIC, thenFun(F,C)is also a (resp., braided, symmetric) monoidal category with tensor product(F⊗G)(I) =F(I)⊗CG(I)and (F⊗G)(f) =F(f)⊗CG(f)for any objectIand any morphismf inF, unit given by the constant functor of valueIC (so in particular it sends any morphism to the identity ofIC), and the associativity, left unit and right unit constraints (resp., as well as braiding,) induced by that ofC. We will call this (resp., braided) monoidal structure onFun(F,C)inducedand we denote it byFuni(F,C).
3.2 Algebras and coalgebras in categories of functors
For the rest of the article we assume that(C,⊗C,IC)is a monoidal category that is cocomplete and such that the tensor product commutes with colimits on each side. The next result follows directly from the definitions.
Fact 3.1. A functorF ∈Fun(F,C)is a (co)algebra for the induced monoidal structure if and only ifFfactors through the canonical inclusion ofAlg(C)(resp.,coAlg(C)) inside ofC. IfC is also braided, the analogous result holds for (co)commutative (co)algebras and bialgebras.
