2-Kac-Moody algebras

Introduction

Over the past ten years, we have advocated the idea that there should exist monoidal categories (or $2$-categories) with an interesting “representation theory”: we propose to call “$2$-representation theory” this higher version of representation theory and to call “$2$-algebras” those “interesting” monoidal additive categories. The difficulty in pinning down what is a $2$-algebra (or a Hopf version) should be compared with the difficulty in defining precisely the meaning of quantum groups (or quantum algebras). The analogy is actually expected to be meaningful: while quantization turns certain algebras into quantum algebras, “categorification” should turn those algebras into $2$-algebras. Dequantization is specialization $q\to 1$, while “decategorification” is the Grothendieck group construction — in the presence of gradings, it leads to a quantum object. A large part of geometric representation theory should, and can, be viewed as a construction of “irreducible” $2$-representations as categories of sheaves.

The starting point of the study of $2$-representation theory of Lie algebras was the definition in 2003 of ${\mathfrak{sl}}$-categorifications with Joseph Chuang and its study for ${\mathfrak{sl}}_{2}$ in [ChRou].

A crucial feature of $2$-representation theory is the construction of a machinery that produces new categories out of some given categories (with extra structure). We believe this should be viewed as an algebraic counterpart of the construction of moduli spaces as families of sheaves or other objects on a variety. The following oversimplified diagram explains how our algebraic constructions would reproduce the various counting invariants based on moduli spaces, bypassing the moduli spaces and the difficulties of their construction and the construction of their invariants

While our focus here is on classical algebraic objects (related in some way to $2$-dimensional geometry), it is our belief that there should be $2$-algebras associated with $3$-dimensional geometry, possibly non-commutative, and that their higher representation theory would provide the proper algebraic framework for the various couting invariants (Gromov-Witten, Donaldson-Thomas,…).

In this paper, we define a $2$-category ${{\gothfamily A}}({\mathfrak{g}})$ associated with a Kac-Moody algebra ${\mathfrak{g}}$. Modulo some Hecke algebra isomorphisms, the generalization from type $A$ (finite or affine) defined in joint work with Joseph Chuang is quite natural.

In [Rou2], we define and study tensor structures on the $2$-category of $2$-representations of ${{\gothfamily A}}({\mathfrak{g}})$ on dg-categories, with aim the construction of $4$-dimensional topological quantum field theories. Our $2$-categories associated with Kac-Moody algebras provide a solution to the question raised by Crane and Frenkel [CrFr] of the construction of “Hopf categories”.

The $2$-category ${{\gothfamily A}}({\mathfrak{g}})$ “categorifies” (a completion of) the ${\mathbf{Z}}$-form $U_{\mathbf{Z}}({\mathfrak{g}})$ of the enveloping algebra of ${\mathfrak{g}}$. Consequently, a $2$-representation of ${{\gothfamily A}}({\mathfrak{g}})$ on an exact or a triangulated category ${\mathcal{V}}$ gives rise to an action of $U_{\mathbf{Z}}({\mathfrak{g}})$ on $K_{0}({\mathcal{V}})$. This gives a hint at the very non-semi-simplicity of the theory of $2$-representations of ${{\gothfamily A}}({\mathfrak{g}})$. The presence of gradings actually gives rise to a “categorification” of the associated quantum group.

The Hecke algebras used in [ChRou] are replaced by nil Hecke algebras associated with Cartan matrices. Some of their specializations occur naturally as endomorphisms of correspondences for quiver varieties [Rou3]. In type $A$, they occur when decomposing representations of (degenerate) affine Hecke algebras according to the spectrum of the polynomial subalgebra, and not just the center. These nil Hecke algebras can be defined by generators and relations and they also have a simple construction as a subalgebra of a wreath product algebra.

We construct more generally a flat family of “Hecke” algebras over the space of matrices over $k[u,v]$ which are hermitian with respect to $u\leftrightarrow v$. They are filtered with associated graded algebra a wreath product of a polynomial algebra by a nil Hecke algebra. They satisfy the PBW property.

Consider a monoidal category or a $2$-category defined by generators and relations. A difficulty in $2$-representation theory is to check the defining relations in examples. The philosophy of [ChRou] was, instead of defining first the monoidal category, to describe directly what a $2$-representation should be, using the action on the Grothendieck group. A key result of this paper is to provide a similar approach for Kac-Moody algebras. We show, under certain finiteness assumptions, that it is enough to check the relations $[e_{i},f_{j}]=\delta_{ij}h_{i}$ on $K_{0}$. This is needed to show that the earlier definition of Chuang and the author of type $A$-categorifications coincides with the more general notion defined here. It is also a crucial ingredient for the construction of algebraic and geometric $2$-representations in [Rou3].

Let us describe in more detail the constructions and results of the paper.

We set up some of the formalism to deal with $2$-categories, presentations by generators and relations and $2$-representations in §2.2. An important role is played by biadjoint pairs and §2.3 develops the theory of symmetric algebras over non-commutative rings. In §3.1, we gather classical results on Hecke algebras of type $A$: affine, degenerate affine and nil affine. We introduce Hecke algebras associated with hermitian matrices in §3.2 and show they satisfy a PBW Theorem. We provide specializations associated with Cartan matrices and further specializations associated with quivers (with automorphisms).

Given a Cartan datum, we construct in §4.1.3 a $2$-category ${{\gothfamily A}}$ with set of objects the weight lattice $X$ and with $1$-arrows generated by $E_{s}:\lambda\to\lambda+\alpha_{s}$ and $F_{s}:\lambda\to\lambda-\alpha_{s}$. The $2$-arrows are generated by units and counits of dual pairs $(E_{s},F_{s})$ and by $x_{s}\in\operatorname{End}\nolimits(E_{s})$ and $\tau_{st}\in\operatorname{Hom}\nolimits(E_{s}E_{t},E_{t}E_{s})$. We impose relations so that there is an action of the nil Hecke algebras associated with the Cartan matrix on products $E_{s_{1}}\cdots E_{s_{n}}$ induced by $x_{s}$ and $\tau_{st}$. Finally, we invert certain maps relating $E_{s}F_{t}$, $F_{t}E_{s}$ and a multiple of $\mathbf{1}$ — this accounts for the decomposition of $[e_{s},f_{t}]$ in the corresponding Kac-Moody algebra ${\mathfrak{g}}$. There is a morphism of algebras from a completion of the ${\mathbf{Z}}$-form of the enveloping algebra of ${\mathfrak{g}}$ to the Grothendieck group of ${{\gothfamily A}}$. The category ${{\gothfamily A}}$ is defined over a base ring with indeterminates and a specialization of these leads to a graded category. There is a morphism from the completed quantized enveloping algebra of ${\mathfrak{g}}$ to the graded Grothendieck group.

We introduce integrable $2$-representations of ${{\gothfamily A}}$ in §5.1.1. We show that for integrable $2$-representations of ${{\gothfamily A}}$, there is a canonical adjunction $(F_{s},E_{s})$, giving rise to an action of a $2$-category ${{\gothfamily A}}^{\prime}$ (§4.1.5 and Theorem 5.25). We provide a construction of a $2$-representation ${\mathcal{V}}(\lambda)$ with lowest weight $\lambda\in-X^{+}$ (§5.1.2) and show that lowest weight integrable $2$-representations admit Jordan-Hölder filtrations (Theorem 5.8). The case of ${\mathfrak{sl}}_{2}$ is crucial for several proofs and §5.2 is a study of its $2$-representations ${\mathcal{V}}(\lambda)$. We also introduce three involutions $I$, $D$ and $\iota$ that allow to swap $E_{s}$ and $F_{s}$ in particular (§4.2.1 and §5.3.4).

In §5.3.3, we show that in the case of abelian categories over a field with finite composition series, the notion of ${\mathfrak{sl}}_{2}$-categorifications of [ChRou] coincides with that of a $2$-representation of ${{\gothfamily A}}({\mathfrak{sl}}_{2})$. We generalize this to type $A$ (finite or affine) in §5.3.8. This builds on the isomorphisms between (degenerate) affine Hecke algebras and Hecke algebras associated with Cartan matrices of type $A$ constructed in §3.2.6. This provides a powerful way to construct $2$-representations. We extend to general Kac-Moody algebras two key facts: the relations of type “$[e_{s},f_{t}]=0$ when $s\not=t$” are a consequence of the other axioms (§5.3.5) and for abelian categories as above, the relations of type “$[e_{s},f_{s}]=h_{s}$” follow from their $K_{0}$ version (§5.3.6).

The main results of this paper have been announced at seminars in Orsay, Paris and Kyoto in the Spring 2007. Certain specializations of the nil Hecke algebras associated with quivers and the resulting monoidal categories associated with “half” Kac-Moody algebras have been introduced independently by Khovanov and Lauda [KhoLau1, KhoLau2]. The relations between Hecke algebras associated with affine type $A$ Cartan matrices and representations of finite Hecke algebras of type $A$ have been studied independently by Brundan and Kleshchev [BrKl].

Preliminaries

Given $n\in{\mathbf{Z}}$, we put $[n]=\frac{v^{n}-v^{-n}}{v-v^{-1}}$, $[n]!=\prod_{i=1}^{n}[i]$ for $n\in{\mathbf{Z}}_{\geq 0}$. We put also

Given $\Omega$ a finite interval of ${\mathbf{Z}}$, we denote by ${\mathfrak{S}}(\Omega)$ the symmetric group on $\Omega$, viewed as a Coxeter group with generating set $\{s_{i}=(i,i+1)\}$ where $i$ runs over the non-maximal elements of $\Omega$. We denote by $w(\Omega)$ the longest element of ${\mathfrak{S}}(\Omega)$. Given $E$ a family of disjoint intervals of $\Omega$, we put ${\mathfrak{S}}(E)=\prod_{\Omega^{\prime}\in E}{\mathfrak{S}}(\Omega^{\prime})$ and we denote by ${\mathfrak{S}}(\Omega)^{E}$ (resp. ${{}^{E}{\mathfrak{S}}}(\Omega)$) the set of minimal length representatives of ${\mathfrak{S}}(\Omega)/{\mathfrak{S}}(E)$ (resp. ${\mathfrak{S}}(E)\setminus{\mathfrak{S}}(\Omega)$). We put ${\mathfrak{S}}_{n}={\mathfrak{S}}[1,n]$. Given $w\in{\mathfrak{S}}_{n}$, we put $\delta_{w}=\delta_{1,w}$.

Let $k$ be a commutative ring. We write $\otimes$ for $\otimes_{k}$. Given $M$ a graded $k$-module and $i$ an integer, we denote by $M(i)$ the graded $k$-module given by $M(i)_{n}=M_{n+i}$.

Given $P=\sum_{i\in{\mathbf{Z}}}p_{i}v^{i}\in{\mathbf{Z}}_{\geq 0}[v^{\pm 1}]$ a Laurent polynomial with non-negative coefficients, we put $Pk=\bigoplus_{i\in{\mathbf{Z}}}k^{p_{i}}(-i)$. Given $k^{\prime}$ a $k$-algebra and $M$ a $k$-module, we put $k^{\prime}M=k^{\prime}\otimes M$. We also put $PM=Pk\otimes M$.

Given $A$ a $k$-algebra, $\gamma$ an automorphism of $A$ and $M$ a right $A$-module, we denote by $M_{\gamma}$ the right $A$-module $\gamma^{*}M$: it is equal to $M$ as a $k$-module and the action of $a\in A$ on $M_{\gamma}$ is given by $M_{\gamma}\ni m\mapsto m\cdot\gamma(a)$. Given $M$ an $(A,A)$-bimodule, we put $M^{A}=\{m\in M~|~am=ma,\ \forall a\in A\}$.

An $A$-algebra is an algebra $B$ endowed with a morphism of algebras $A\to B$. Given $B$ an $A$-algebra, we say that a $B$-module is relatively $A$-projective if it is a direct summand of $B\otimes_{A}M$ for some $A$-module $M$.

Categories are denoted by calligraphic letters ${\mathcal{A}},{\mathcal{B}},{\mathcal{C}}$, etc. and $2$-categories are denoted by gothic letters ${{\gothfamily A}},{{\gothfamily B}},{{\gothfamily C}}$, etc.

We denote by ${\operatorname{Ob}\nolimits}({\mathcal{A}})$ or by ${\mathcal{A}}$ the set of objects of a category (or of a $2$-category) ${\mathcal{A}}$. Given $a$ an object, we will denote by $a$ or $\mathbf{1}_{a}$ or $1_{a}$ the identity of $a$.

Given $F,G:{\mathcal{A}}\to{\mathcal{B}}$ two functors, a morphism $F\to G$ is the data of a compatible collection of arrows $F(a)\to G(a)$ for $a\in{\mathcal{A}}$ and we call these natural morphisms.

We say that an endofunctor $F$ of an additive category ${\mathcal{C}}$ is locally nilpotent if for every $M\in{\mathcal{C}}$, there is $n>0$ such that $F^{n}(M)=0$.

We denote by ${\mathcal{S}}ets$ (resp. ${\mathcal{A}}b$) the category of sets (resp. of abelian groups). We denote by $A\operatorname{\!-Mod}\nolimits$ the category of $A$-modules, by $A\operatorname{\!-mod}\nolimits$ the category of finitely generated $A$-modules and by $A\operatorname{\!-free}\nolimits$ is full subcategory of free $A$-modules of finite rank. Here, module means left module. Given ${\mathcal{A}}$ an additive category, we denote by $\operatorname{Comp}\nolimits^{b}({\mathcal{A}})$ the category of bounded complexes of objects of ${\mathcal{A}}$ and by $\operatorname{Ho}\nolimits^{b}({\mathcal{A}})$ the associated homotopy category.

We denote by ${{\gothfamily C}}at$ (resp. ${{\gothfamily A}}dd$, ${{\gothfamily L}}in_{k}$, ${{\gothfamily A}}b$, ${{\gothfamily T}}ri$) the strict $2$-category of categories (resp. of additive categories, of $k$-linear categories, of abelian categories with exact functors, of triangulated categories). When $k$ is a field, we denote by ${{\gothfamily A}}b_{k}^{f}$ the $2$-category of $k$-linear abelian categories all of whose objects have finite composition series and such that $k=\operatorname{End}\nolimits(V)$ for any simple object $V$ ($1$-arrows are $k$-linear exact functors).

2. 22-Categories

We set up in this section the appropriate formalism for $2$-representation theory. At first, we recall the more classical setting of representation theory as a study of functors.

Let ${\mathcal{A}}$ and ${\mathcal{B}}$ be two categories. We denote by ${{\mathcal{H}}om}({\mathcal{A}},{\mathcal{B}})$ the category of functors ${\mathcal{A}}\to{\mathcal{B}}$: we think of these as representations of ${\mathcal{A}}$ in ${\mathcal{B}}$. For example, if ${\mathcal{A}}$ has a unique object $\ast$ and ${\mathcal{B}}={\mathcal{S}}ets$, the category ${{\mathcal{H}}om}({\mathcal{A}},{\mathcal{B}})$ is equivalent to the category of sets acted on by the monoid $\operatorname{End}\nolimits(\ast)$.

Given $a\in{\mathcal{A}}$, we have a functor $\operatorname{Hom}\nolimits(a,-):\rho_{a}:{\mathcal{A}}\to{\mathcal{S}}ets$ (the regular representation when ${\mathcal{A}}$ has a unique object).

We put ${\mathcal{A}}^{\vee}={{\mathcal{H}}om}({\mathcal{A}}^{\operatorname{opp}\nolimits},{\mathcal{S}}ets^{\operatorname{opp}\nolimits})$. The functor

is fully faithful (Yoneda’s Lemma) and we identify ${\mathcal{A}}$ with a full subcategory of ${\mathcal{A}}^{\vee}$ through this embedding.

Assume ${\mathcal{A}}$ is enriched in abelian groups. The additive closure of ${\mathcal{A}}$ is the full additive subcategory ${\mathcal{A}}^{a}$ of the category of functors ${\mathcal{A}}^{\operatorname{opp}\nolimits}\to{\mathcal{A}}b^{\operatorname{opp}\nolimits}$ generated by objects of ${\mathcal{A}}$. Given ${\mathcal{A}}^{\prime}$ an additive category, the restriction functor gives an equivalence from the category of additive functors ${\mathcal{A}}^{a}\to{\mathcal{A}}^{\prime}$ to the category of functors enriched in abelian groups ${\mathcal{A}}\to{\mathcal{A}}^{\prime}$.

Assume ${\mathcal{A}}$ is an additive category. We denote by ${\mathcal{A}}^{i}$ the idempotent completion of ${\mathcal{A}}$. Given ${\mathcal{A}}^{\prime}$ an idempotent-complete additive category, restriction gives an equivalence from the category of additive functors ${\mathcal{A}}^{i}\to{\mathcal{A}}^{\prime}$ to the category of additive functors ${\mathcal{A}}\to{\mathcal{A}}^{\prime}$.

Let $M\in{\mathcal{A}}$ and let $L$ be a right $\operatorname{End}\nolimits(M)$-module. We denote by $L\otimes_{\operatorname{End}\nolimits(M)}M$ the object of ${\mathcal{A}}^{\vee}$ defined by $\operatorname{Hom}\nolimits_{\operatorname{End}\nolimits(M)^{\operatorname{opp}\nolimits}}(L,\operatorname{Hom}\nolimits(M,-))$.

Given $A$ a ring, the category of $A$-modules in ${\mathcal{A}}$ is the category of additive functors $A\to{\mathcal{A}}$, where $A$ is the category with one object $\ast$ and with $\operatorname{End}\nolimits(\ast)=A$. An object of that category is an object $M$ of ${\mathcal{A}}$ endowed with a morphism of rings $A\to\operatorname{End}\nolimits(M)$.

Given an $A$-module $M$ in ${\mathcal{A}}$ and $L$ a right $A$-module, we put $L\otimes_{A}M=(L\otimes_{A}\operatorname{End}\nolimits(M))\otimes_{\operatorname{End}\nolimits(M)}M$. For example, there is a canonical isomorphism ${\mathbf{Z}}^{n}\otimes_{\mathbf{Z}}M\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}M^{n}$.

Let $B$ be a commutative ring endowed with a morphism $B\to Z({\mathcal{A}})$ and let $A$ be a $B$-algebra. We denote by ${\mathcal{A}}\otimes_{B}A$ the additive category with same objects as ${\mathcal{A}}$ and $\operatorname{Hom}\nolimits_{{\mathcal{A}}\otimes_{B}A}(M,N)=\operatorname{Hom}\nolimits_{\mathcal{A}}(M,N)\otimes_{B}A$, where $B$ acts via $Z({\mathcal{A}})$. Let ${\mathcal{A}}^{\prime}$ be $B$-linear category. We denote by ${\mathcal{A}}\otimes_{B}{\mathcal{A}}^{\prime}$ the additive closure of the category with set of objects ${\operatorname{Ob}\nolimits}({\mathcal{A}})\times{\operatorname{Ob}\nolimits}({\mathcal{A}}^{\prime})$ and with $\operatorname{Hom}\nolimits((M,M^{\prime}),(N,N^{\prime}))=\operatorname{Hom}\nolimits_{{\mathcal{A}}}(M,N)\otimes_{B}\operatorname{Hom}\nolimits_{{\mathcal{A}}^{\prime}}(M^{\prime},N^{\prime})$. Given ${\mathcal{A}}^{\prime\prime}$ a $B$-linear category, there is an equivalence between ${{\mathcal{H}}om}_{{{\gothfamily L}}in_{B}}({\mathcal{A}}\otimes_{B}{\mathcal{A}}^{\prime},{\mathcal{A}}^{\prime\prime})$ and the category of $B$-bilinear functors ${\mathcal{A}}\times{\mathcal{A}}^{\prime}\to{\mathcal{A}}^{\prime\prime}$.

An equivalence relation $\sim$ on a category is a relation on arrows such that $f\sim f^{\prime}$ implies $fg\sim f^{\prime}g$ and $gf\sim gf^{\prime}$ (whenever this makes sense). Given ${\mathcal{A}}$ a category and $\sim$ a relation on arrows of ${\mathcal{A}}$, we have a quotient category ${\mathcal{A}}/\!\!\sim$ with same objects as ${\mathcal{A}}$. The quotient functor ${\mathcal{A}}\to{\mathcal{A}}/\!\!\sim$ induces a fully faithful functor ${{\mathcal{H}}om}({\mathcal{A}}/\!\!\sim,{\mathcal{B}})\to{{\mathcal{H}}om}({\mathcal{A}},{\mathcal{B}})$ for any category ${\mathcal{B}}$. A functor is in the image if and only if two equivalent arrows have the same image under the functor. The construction depends only on the equivalence relation on ${\mathcal{A}}$ generated by $\sim$.

Let $k$ be a commutative ring and ${\mathcal{A}}$ a $k$-linear category. Given $S$ a set of arrows of ${\mathcal{A}}$, let $\sim=\sim_{S}$ be the coarsest equivalence relation on ${\mathcal{A}}$ such that $f\sim 0$ for every $f\in S$ and $\{(f,g)\ |\ f\sim g\}$ is a $k$-submodule of $\operatorname{Hom}\nolimits(a,a^{\prime})\oplus\operatorname{Hom}\nolimits(a,a^{\prime})$. We denote by ${\mathcal{A}}/S={\mathcal{A}}/\!\!\sim$ the quotient $k$-linear category: a $k$-linear functor ${\mathcal{A}}\to{\mathcal{B}}$ factors through ${\mathcal{A}}/S$, and then the factorization is unique up to unique isomorphism, if and only if it sends arrows in $S$ to $0$.

Let $I=(I_{0},I_{1},s,t)$ be a quiver: this is the data of

a set $I_{0}$ (vertices) and a set $I_{1}$ (arrows)

maps $s,t:I_{1}\to I_{0}$ (source and target).

We denote by ${\mathcal{P}}={\mathcal{P}}(I)$ the set of paths in $I$, i.e., sequences $(b_{1},\ldots,b_{n})$ of elements of $I_{1}$ such that $t(b_{i})=s(b_{i-1})$ for $1<i\leq n$. It comes with maps $s:{\mathcal{P}}\to I_{0},\ (b_{1},\ldots,b_{n})\mapsto s(b_{n})$ (source) and $t:{\mathcal{P}}\to I_{0},\ (b_{1},\ldots,b_{n})\mapsto t(b_{1})$ (target). We write $b_{1}\cdots b_{n}$ for the element $(b_{1},\ldots,b_{n})$ of ${\mathcal{P}}$.

We denote by ${\mathcal{C}}(I)$ the category generated by $I$. Its set of objects is $I_{0}$ and $\operatorname{Hom}\nolimits(i,j)=(s,t)^{-1}(i,j)$. Composition is concatenation of paths.

Let ${\mathcal{A}}$ be a category. The category of diagrams of type $I$ in ${\mathcal{A}}$ is canonically isomorphic to the category of functors ${\mathcal{C}}(I)\to{\mathcal{A}}$ (the isomorphism is given by restricting the functor).

A graded category is a category endowed with a self-equivalence $T$. Given $M$ an object with isomorphism class $[M]$, we put $v[M]=[T^{-1}(M)]$.

The $2$-category of graded $k$-linear categories is equivalent to the $2$-category of $k$-linear categories enriched in graded $k$-modules:

Let ${\mathcal{C}}$ be a graded $k$-linear category. We define ${\mathcal{D}}$ as the category with objects those of ${\mathcal{C}}$ and with $\operatorname{Hom}\nolimits_{\mathcal{D}}(V,W)=\bigoplus_{i}\operatorname{Hom}\nolimits_{\mathcal{C}}(V,T^{i}W)$. The composition of the maps of ${\mathcal{D}}$ coming from maps $f:V\to T^{i}W$ and $g:W\to T^{j}X$ of ${\mathcal{C}}$ is the map coming from $T^{i}(g)\circ f:V\to T^{i+j}X$.

Let ${\mathcal{D}}$ be a $k$-linear category enriched in graded $k$-modules. Define ${\mathcal{C}}$ as the category with objects families $\{V_{i}\}_{i\in{\mathbf{Z}}}$ with $V_{i}$ an object of ${\mathcal{D}}$ and $V_{i}=0$ for almost all $i$. We put $\operatorname{Hom}\nolimits_{\mathcal{C}}(\{V_{i}\},\{W_{i}\})=\bigoplus_{i,j}\operatorname{Hom}\nolimits_{\mathcal{D}}(V_{i},W_{j})_{j-i}$. We define $T(\{V_{i}\})_{n}=V_{n+1}$.

2.2. Definitions

Our main reference for basic definitions and results on $2$-categories is [Gra] (cf also [Le] for the basic definitions).

A $2$-category ${{\gothfamily A}}$ is the data of

categories ${{\mathcal{H}}om}(a,a^{\prime})$ for $a,a^{\prime}\in{{\gothfamily A}}_{0}$

functors ${{\mathcal{H}}om}(a_{1},a_{2})\times{{\mathcal{H}}om}(a_{2},a_{3})\to{{\mathcal{H}}om}(a_{1},a_{3}),\ (b_{1},b_{2})\mapsto b_{2}b_{1}$ for $a_{1},a_{2},a_{3}\in{{\gothfamily A}}$

functors $I_{a}\in{\mathcal{E}}nd(a)$ for $a\in{{\gothfamily A}}$

natural isomorphisms $(b_{3}b_{2})b_{1}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}b_{3}(b_{2}b_{1})$ for $b_{i}\in{{\mathcal{H}}om}(a_{i},a_{i+1})$ and $a_{1},\ldots,a_{4}\in{{\gothfamily A}}$.

natural isomorphisms $bI_{a}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}b$ for $b\in{{\mathcal{H}}om}(a,a^{\prime})$ and $a,a^{\prime}\in{{\gothfamily A}}$

natural isomorphisms $I_{a}b\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}b$ for $b\in{{\mathcal{H}}om}(a^{\prime},a)$ and $a,a^{\prime}\in{{\gothfamily A}}$

Note that $2$-categories are called bicategories in [Gra]. A strict $2$-category is a $2$-category where the associativity and unit isomorphisms are identity maps: $(b_{3}b_{2})b_{1}=b_{3}(b_{2}b_{1})$ and $bI_{a}=b$, $I_{a}b=b$ (called $2$-category in [Gra]).

Let ${{\gothfamily A}}$ be a $2$-category. Its $1$-arrows (resp. $2$-arrows) are the objects (resp. arrows) of the categories ${{\mathcal{H}}om}(a,a^{\prime})$

Given $b:a\to a^{\prime}$ and $b^{\prime}:a^{\prime}\to a^{\prime\prime}$ two $1$-arrows, we denote by $b^{\prime}b:a\to a^{\prime\prime}$ their composition. The composition of $2$-arrows $c$ and $c^{\prime}$ (viewed as arrows in a category ${{\mathcal{H}}om}(a,a^{\prime})$) is denoted by $c^{\prime}\circ c$. Given $a$, $a^{\prime}$ and $a^{\prime\prime}$ three objects of ${{\gothfamily A}}$, $b_{1},b_{2}:a\to a^{\prime}$, $c:b_{1}\to b_{2}$ and $b^{\prime}_{1},b^{\prime}_{2}:a^{\prime}\to a^{\prime\prime}$, $c^{\prime}:b^{\prime}_{1}\to b^{\prime}_{2}$, we denote by $c^{\prime}c:b^{\prime}_{1}b_{1}\to b^{\prime}_{2}b_{2}$ the “juxtaposition”.

We say that a $1$-arrow $b:a_{1}\to a_{2}$ is

an equivalence if there is a $1$-arrow $b^{\prime}:a_{2}\to a_{1}$ and isomorphisms $I_{a_{1}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}b^{\prime}b$ and $bb^{\prime}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}I_{a_{2}}$

fully faithful if given any object $a^{\prime\prime}$, the functor ${{\mathcal{H}}om}(a^{\prime\prime},b):{{\mathcal{H}}om}(a^{\prime\prime},a_{1})\to{{\mathcal{H}}om}(a^{\prime\prime},a_{2})$ is fully faithful.

Note that these notions coincide with the usual notions for ${{\gothfamily A}}={{\gothfamily C}}at$, ${{\gothfamily A}}={{\gothfamily A}}dd$, ${{\gothfamily A}}={{\gothfamily A}}b$ or ${{\gothfamily A}}={{\gothfamily T}}ri$.

Given a $2$-category ${{\gothfamily A}}$, we denote by ${{\gothfamily A}}_{\leq 1}$ the category with objects those of ${{\gothfamily A}}$ and with arrows the isomorphism classes of $1$-arrows of ${{\gothfamily A}}$.

The opposite $2$-category ${{\gothfamily A}}^{\operatorname{opp}\nolimits}$ of ${{\gothfamily A}}$ has same set of objects as ${{\gothfamily A}}$ and ${{\mathcal{H}}om}_{{{\gothfamily A}}^{\operatorname{opp}\nolimits}}(a,a^{\prime})={{\mathcal{H}}om}_{{{\gothfamily A}}}(a,a^{\prime})^{\operatorname{opp}\nolimits}$, while the rest of the structure is inherited from that of ${{\gothfamily A}}$.

is given by $(b_{1},b_{2})\mapsto b_{1}b_{2}$ (composition in ${{\gothfamily A}}$). The rest of the structure is inherited from that of ${{\gothfamily A}}$.

A $2$-functor $R:{{\gothfamily A}}\to{{\gothfamily B}}$ between $2$-categories is the data of

a map $R:{\operatorname{Ob}\nolimits}({{\gothfamily A}})\to{\operatorname{Ob}\nolimits}({{\gothfamily B}})$

functors $R:{{\gothfamily H}}om(a,a^{\prime})\to{{\gothfamily H}}om(R(a),R(a^{\prime}))$ for $a,a^{\prime}\in{{\gothfamily A}}$

natural isomorphisms $R(b_{2})R(b_{1})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}R(b_{2}b_{1})$ for $b_{1},b_{2}$ $1$-arrows of ${{\gothfamily A}}$

invertible $2$-arrows $I_{R(a)}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}R(I_{a})$ for $a\in{{\gothfamily A}}$

When the $2$-arrows are identity maps $I_{R(a)}=R(I_{a})$, we say that the $2$-functor is strict (called strict pseudo-functor in [Gra]).

A morphism of $2$-functors $\sigma:R\to R^{\prime}$ is the data of

$1$-arrows $\sigma(a):R(a)\to R^{\prime}(a)$

natural isomorphisms $R^{\prime}(b)\sigma(a_{1})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\sigma(a_{2})R(b)$ for all $1$-arrows $b:a_{1}\to a_{2}$

These are quasi-natural transformations with invertible $2$-arrows in [Gra].

We denote by ${{\gothfamily H}}om({{\gothfamily A}},{{\gothfamily B}})$ denotes the $2$-category of $2$-functors ${{\gothfamily A}}\to{{\gothfamily B}}$. When ${{\gothfamily B}}$ is a strict $2$-category, then ${{\gothfamily H}}om({{\gothfamily A}},{{\gothfamily B}})$ is strict as well.

Given a property of functors, we say that a $2$-functor $F:{{\gothfamily A}}\to{{\gothfamily B}}$ has locally that property if the functors ${{\mathcal{H}}om}(a,a^{\prime})\to{{\mathcal{H}}om}(F(a),F(a^{\prime}))$ have the property for all $a,a^{\prime}$ objects of ${{\gothfamily A}}$.

A $2$-functor $F:{{\gothfamily A}}\to{{\gothfamily B}}$ is a $2$-equivalence if there is a $2$-functor $G:{{\gothfamily B}}\to{{\gothfamily A}}$ and equivalences $\operatorname{id}\nolimits_{{{\gothfamily A}}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}GF$ and $FG\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{id}\nolimits_{{{\gothfamily B}}}$. This is equivalent to the requirement that $F$ is locally an equivalence and every object of ${{\gothfamily B}}$ is equivalent to an object in the image of $F$.

Every $2$-category is $2$-equivalent to a strict $2$-category, but there are $2$-functors between strict $2$-categories that are not equivalent to strict ones.

Given $a$ an object of ${{\gothfamily A}}$, then ${\mathcal{E}}nd(a)$ is a monoidal category. Conversely, a monoidal category gives rise to a $2$-category with a single object $\ast$, and the notion of monoidal functor coincides with that of $2$-functor (i.e., there is a $1,2,3$-fully faithful strict $3$-functor from the $3$-category of monoidal categories to that of $2$-categories).

Let $k$ be a commutative ring. A $k$-linear $2$-category is a $2$-category ${{\gothfamily A}}$ that is locally $k$-linear and such that juxtaposition is $k$-linear. Given ${{\gothfamily A}}$ and ${{\gothfamily B}}$ two $k$-linear $2$-categories, we denote by ${{\gothfamily H}}om({{\gothfamily A}},{{\gothfamily B}})$ the $2$-category of $k$-linear $2$-functors ${{\gothfamily A}}\to{{\gothfamily B}}$: this is the locally full sub-$2$-category of the category of $2$-functors obtained by requiring the functors in the definition of $2$-functors to be $k$-linear.

Given ${{\gothfamily A}}$ a $2$-category, we denote by $k{{\gothfamily A}}$ the $k$-linear closure of ${{\gothfamily A}}$: its objects are those of ${{\gothfamily A}}$ and ${{\mathcal{H}}om}_{k{{\gothfamily A}}}(a,a^{\prime})=k{{\mathcal{H}}om}_{{{\gothfamily A}}}(a,a^{\prime})$.

Let $b:a\to a^{\prime}$ be a $1$-arrow. A right adjoint (or right dual) of $b$ is a triple $(b^{\vee},\varepsilon_{b},\eta_{b})$ where $b^{\vee}:a^{\prime}\to a$ is a $1$-arrow and $\varepsilon_{b}:bb^{\vee}\to I_{a^{\prime}}$ and $\eta_{b}:I_{a}\to b^{\vee}b$ are $2$-arrows such that the compositions

are identities. We also say that $(b,\varepsilon_{b},\eta_{b})$ is a left adjoint (or dual) of $b^{\prime}=b^{\vee}$ (and we write $b={{}^{\vee}b}^{\prime}$) and we say that $(b,b^{\vee},\varepsilon_{b},\eta_{b})$ (or simply $(b,b^{\vee})$) is an adjoint quadruple (resp. an adjoint pair).

Given $b_{1}:a\to a^{\prime}$ a $1$-arrow and $(b_{1},b_{1}^{\vee})$ an adjoint pair, we have a canonical isomorphism

Assume now there are dual pairs $(b,b^{\vee})$ and $(b^{\vee},b)$. We have an automorphism

2.3. Generators and relations

An equivalence relation $\sim$ on ${{\gothfamily A}}$ is the data for every $a,a^{\prime}$ objects, for every $b,b^{\prime}:a\to a^{\prime}$ of an equivalence relation on $\operatorname{Hom}\nolimits(b,b^{\prime})$ compatible with composition and juxtaposition, i.e., if $c_{1}\sim c_{2}$, then given a $2$-arrow $c$, we have $c_{1}\circ c\sim c_{2}\circ c$, $c\circ c_{1}\sim c\circ c_{2}$, $c_{1}c\sim c_{2}c$ and $cc_{1}\sim cc_{2}$, whenever this makes sense. Given a relation $\sim$ on $2$-arrows of ${\mathcal{C}}$, the equivalence relation generated by $\sim$ is the coarsest refinement of $\sim$ that is an equivalence relation.

Let ${{\gothfamily A}}$ be a $2$-category and $\sim$ an equivalence relation. We denote by ${{\gothfamily A}}/\!\!\sim$ the $2$-category with same objects as ${{\gothfamily A}}$ and with ${{\mathcal{H}}om}_{{{\gothfamily A}}/\!\sim}(a,a^{\prime})={{\mathcal{H}}om}_{{{\gothfamily A}}}(a,a^{\prime})/\!\!\sim$ (so, ${{\gothfamily A}}/\!\!\sim$ has the same $1$-arrows as ${{\gothfamily A}}$). The local quotient functors induce a strict quotient $2$-functor ${{\gothfamily A}}\to{{\gothfamily A}}/\!\!\sim$. Given a $2$-category ${{\gothfamily B}}$, the quotient strict $2$-functor ${{\gothfamily A}}\to{{\gothfamily A}}/\!\!\sim$ induces a strict $2$-functor ${{\gothfamily H}}om({{\gothfamily A}}/\!\!\sim,{{\gothfamily B}})\to{{\gothfamily H}}om({{\gothfamily A}},{{\gothfamily B}})$ that is locally an isomorphism. A $2$-functor $R$ is in the image if and only if two equivalent $2$-arrows have the same image under $R$.

The canonical strict $2$-functor ${{\gothfamily A}}\to{{\gothfamily A}}[S^{-1}]$ induces a strict $2$-functor ${{\gothfamily H}}om({{\gothfamily A}}[S^{-1}],{{\gothfamily B}})\to{{\gothfamily H}}om({{\gothfamily A}},{{\gothfamily B}})$ that is locally an isomorphism. A $2$-functor $R$ is in the image if and only if the image under $R$ of any $2$-arrow in $S$ is invertible.

Assume ${{\gothfamily A}}$ is a $k$-linear $2$-category. Let $S$ be a set of $2$-arrows of ${{\gothfamily A}}$. Given $a,a^{\prime}$ objects of ${{\gothfamily A}}$, we consider the equivalence relation $\sim_{S(a,a^{\prime})}$ on ${{\mathcal{H}}om}(a,a^{\prime})$. Let $\sim$ be the coarsest equivalence relation on ${{\gothfamily A}}$ that refines the relations $\sim_{S(a,a^{\prime})}$. We put ${{\gothfamily A}}/S={{\gothfamily A}}/\!\!\sim$.

A $2$-quiver $I=(I_{0},I_{1},I_{2},s,t,s_{2},t_{2})$ is the data of

three sets $I_{0}$ (vertices), $I_{1}$ ($1$-arrows) and $I_{2}$ ($2$-arrows)

maps $s,t:I_{1}\to I_{0}$ (source and target)

maps $s_{2},t_{2}:I_{2}\to{\mathcal{P}}={\mathcal{P}}(I_{0},I_{1},s,t)$ (source and target of $2$-arrows) such that $s(s_{2}(c))=s(t_{2}(c))$ and $t(s_{2}(c))=t(t_{2}(c))$ for all $c\in I_{2}$.

The strict $2$-category ${{\gothfamily C}}(I)$ generated by $I$ is defined as follows. Its set of objects is $I_{0}$. We put ${{\mathcal{H}}om}(a,a^{\prime})={\mathcal{C}}(I(a,a^{\prime}))/\!\!\sim$. Composition of $1$-arrows is concatenation of paths. Juxtaposition is given by

Note that the category ${{\gothfamily C}}(I)_{\leq 1}$ is ${\mathcal{C}}(I_{0},I_{1},s,t)$.

Let ${{\gothfamily B}}$ be a strict $2$-category. An $I$-diagram $D$ in ${{\gothfamily B}}$ is the data of

an object $a_{i}$ of ${{\gothfamily B}}$ for any $i\in I_{0}$

a $1$-arrow $b_{j}:a_{s(j)}\to a_{t(j)}$ for any $j\in I_{1}$

a $2$-arrow $c_{k}:b_{s_{2}(k)}\to b_{t_{2}(k)}$ for any $k\in I_{2}$

where given $p=(p_{1},\ldots,p_{n})\in{\mathcal{P}}$, we put $b_{p}=b_{p_{1}}\cdots b_{p_{n}}$.

The data of $b_{j}$’s and $c_{k}$’s is the same as the data, for $i,i^{\prime}\in I_{0}$, of an $I(i,i^{\prime})$-diagram in ${{\mathcal{H}}om}(a_{i},a_{i^{\prime}})$.

A morphism $\sigma:D\to D^{\prime}$ is the data of

$1$-arrows $\sigma_{i}:a_{i}\to a^{\prime}_{i}$ for $i\in I_{0}$

invertible $2$-arrows $\sigma_{j}:b^{\prime}_{j}\sigma_{s(j)}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\sigma_{t(j)}b_{j}$ for every $j\in I_{1}$