View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 212 (2008) 2069–2091 www.elsevier.com/locate/jpaa Generalized 2-vector spaces and general linear 2-groups Josep Elgueta∗ Dept. Matematica` Aplicada II, Universitat Politecnica` de Catalunya, Spain Received 28 February 2007; received in revised form 28 October 2007 Available online 29 February 2008 Communicated by I. Moerdijk Abstract A notion of generalized 2-vector space is introduced which includes Kapranov and Voevodsky 2-vector spaces. Various kinds of such objects are considered and examples are given. The corresponding general linear 2-groups are explicitly computed in a special case which includes Kapranov and Voevodsky 2-vector spaces. c 2008 Elsevier B.V. All rights reserved. MSC: 18E05; 18D35; 18D20 1. Introduction In the last decade and a half, the need to introduce a categorical analog of the notion of vector space over a field K , usually called a 2-vector space over K , has become more and more obvious. This need was made explicit, for example, when looking for the right framework to study the generalizations of the Yang–Baxter equation to higher dimensions. It was indeed in this setting that the notion of (finite-dimensional) 2-vector space over a field K was first introduced by Kapranov and Voevodsky [14]. The main point in Kapranov and Voevodsky’s definition was to take the category VectK of finite-dimensional vector spaces over K as the analog of the field K and to define a (finite-dimensional) n ≥ 2-vector space over K as a “VectK -module category” equivalent in the appropriate sense to VectK for some n 0 (see [14] for details). Inspired by previous drafts of this work and more or less at the same time, Yetter [22] also studied the theory of VectK -module categories with the purpose of providing “an algebraic footing for the extension to higher dimensions of the successful interaction between 3-manifold topology, quantum field theory and monoidal category theory”. Shortly after, motivated by the study of the representation theory of Hopf categories (the higher-dimensional analog of Hopf algebras), Neuchl [18] introduced a notion of 2-vector space over K that is simpler but basically equivalent to Kapranov and Voevodsky’s: a (finite-dimensional) 2-vector space over K is a K -linear additive category V that admits a (finite) “basis of absolutely simple objects”, i.e., a (finite) family of simple objects whose vector spaces of endomorphisms are 1-dimensional and such that any object is a finite biproduct of them in an essentially unique way. n When such a basis B exists, it can be shown that V is indeed K -linearly equivalent to the category VectK for some ∗ Corresponding address: Universitat Politecnica de Catalunya, Department of Mathematica Aplicada II (UPC), Edifici Omega-Campus NordC/ Jordi Girona, 1-3, 08034 Barcelona, Spain. E-mail address: [email protected]. 0022-4049/$ - see front matter c 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jpaa.2007.12.010 2070 J. Elgueta / Journal of Pure and Applied Algebra 212 (2008) 2069–2091 n ≥ 0 (called its rank). Later on, in their attempt to define a higher-dimensional analog of a Lie algebra, Baez and Crans [2] gave a completely different notion of 2-vector space over K , namely, as an (internal) category in VectK . They proved that the appropriately defined 2-category of such 2-vector spaces is in fact equivalent to the familiar 2-category of length one complexes of vector spaces over K . Recently, we addressed ourselves to a “linear” representation theory of 2-groups, the categorical analog of the usual groups. This theory also requires some notion of 2-vector space. In particular, in [10] we have studied the 2- category of representations of an arbitrary 2-group G as self-equivalences of a Kapranov and Voevodsky 2-vector space. However, some generalization of this notion of 2-vector space seems to be needed. Thus, on the one hand, the representation theory considered in [10] seems good only for (pro)finite 2-groups. Indeed, a self-equivalence of a Kapranov and Voevodsky 2-vector space of rank n is basically given by an element of the permutation group Σn. Hence, modulo some more data described in [10], such a representation of G is given by a group morphism from the group of isomorphism classes of objects of G to Σn. On the other hand, the setup of Kapranov and Voevodsky allows us to define a natural analog of the group algebra only when the 2-group G is discrete (with only identity morphisms) and hence, given by just a group G. However, a notion of “2-group 2-algebra” VectK [G] for a generic 2-group G seems of interest in order to prove a reconstruction theorem of the Tannaka–Krein type for 2-groups. Hence, one would really like to extend the notion to arbitrary 2-groups, not necessarily discrete. This requires a way to replace VectK [X] for X a set with VectK [C] for a category C. The purpose of this paper is to introduce such a generalization of the notion of 2-vector space that includes Kapranov and Voevodsky definition as a particular case. Instead of categorizing the notion of K -vector space, with K replaced by VectK , we turn our attention to the fact that any vector space is isomorphic to the vector space K [X] of all finite formal linear combinations of elements of some set X with coefficients in K , and categorize this construction. The starting point now is going to be not a set X but a category C.A generalized 2-vector space over K is then defined as a K -linear additive category V which is K -linearly equivalent to the free K -linear additive category generated by C, for some category C. By analogy with K [X], such a freely generated K -linear additive category is denoted by VectK [C]. When C is a finite discrete category, we recover (up to equivalence) Kapranov and Voevodsky 2-vector n ≥ [ ] spaces VectK for all n 0. However, not all K -linear additive categories VectK C are of this type for some category C. As it is shown below with examples, in some cases there exists a basis, but not of absolutely simple objects, and arguments are also given which suggest that there may exist no basis at all. There is an important difference between the category setting and the vector spaces setting. Given a set X, in addition to K [X] there is another vector space naturally associated to X. Namely, the vector space K X of all functions on X with values in K . For finite sets, both vector spaces are isomorphic. In fact, both are functorial and define X naturally isomorphic functors from the category FinSet of finite sets to VectK . The analog of K in the category Cop Cop setting is the functor category VectK for C a (finite) category. But even for finite categories C, VectK [C] and VectK Cop are no longer equivalent. As it is discussed below, VectK [C] is equivalent to just a certain (full) subcategory of VectK (cf. Theorem 20). Besides this, the above notion of generalized 2-vector space has some additional drawbacks with respect to Kapranov and Voevodsky 2-vector spaces. For instance, generalized 2-vector spaces are non-Karoubian (hence, nonabelian) categories in general (see Section 3.1). Moreover, they have no dual object in the usual sense except when they are Kapranov and Voevodsky 2-vector spaces. Indeed, let denote the tensor product of K -linear additive categories (see [9]). If for a given K -linear additive category A there exists a K -linear additive category A∗ and ∗ ∗ K -linear functors ev : A A → VectK and coev : VectK → AA satisfying the usual axioms which define a (left or right) dual object, it may be shown that A is necessarily a Kapranov and Voevodsky 2-vector space (see [18], = n where the result is attributed to P. Schauenburg). When A VectK , such a dual object indeed exists, and it is given by the category of K -linear functors A → VectK and natural transformations between these. By analogy with the situation for vector spaces, such a dual object is again a Kapranov and Voevodsky 2-vector space of rank n. Yet another drawback concerns the categories of morphisms between arbitrary generalized 2-vector spaces. Thus, generalized 2- vector spaces are naturally organized into a 2-category 2GVECTK with the K -linear functors as 1-morphisms and all natural transformations between these as 2-morphisms. It then turns out that the categories of morphisms between arbitrary generalized 2-vector spaces are not always generalized 2-vector spaces. However, as a matter of fact this new notion accomplishes our above-mentioned goal, as it provides a natural analog of the group algebras in the category setting. Thus, the free vector space construction K [X] is not just functorial. It J. Elgueta / Journal of Pure and Applied Algebra 212 (2008) 2069–2091 2071 actually defines a monoidal functor K [−] : Sets → VECTK which is left adjoint to the forgetful functor U : VECTK → Sets. The fact that K [−] is monoidal implies that it indeed induces a functor K [−] : Monoids → AlgK between the category of monoids and that of associative K -algebras with unit and hence, also from the category Grps of groups to AlgK . In a completely analogous way, a monoidal structure on 2GVECTK can be defined such that the construction VectK [C] extends to a monoidal 2-functor VectK [−] : Cat → 2GVECTK which is left 2-adjoint to the forgetful 2-functor U : 2GVECTK → Cat (see [9]).