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The goal of this paper is to give a category theory based definition and classification of “finite subgroups in Uq( s l 2)” where q=eπi/l is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of Uq( s l 2); we show that this definition is a natural generalization of the notion of a subgroup in a reductive group, and that it is also related with extensions of the chiral (vertex operator) algebra corresponding to s l 2 at level k=l−2. We show that “finite subgroups in Uq( s l 2)” are classified by Dynkin diagrams of types An,D2n,E6,E8 with Coxeter number equal to l, give a description of this correspondence similar to the classical McKay correspondence, and discuss relation with modular invariants in ( s l 2)k conformal field theory. The results we get are parallel to those known in the theory of von Neumann subfactors, but our proofs are independent of this theory.

In this paper we introduce the notion of an extensive 2-category, to be thought of as a “2-category of generalized spaces”. We consider an extensive 2-category K equipped with a binary-product-preserving pseudofunctor C : K op → CAT , which we think of as specifying the “coverings” of our generalized spaces. We prove, in this context, a van Kampen theorem which generalizes and refines one of Brown and Janelidze. The local properties required in this theorem are stated in terms of morphisms of effective descent for the pseudofunctor C . We specialize the general van Kampen theorem to the 2-category Top S of toposes bounded over an elementary topos S , and to its full sub 2-category LTop S determined by the locally connected toposes, after showing both of these 2-categories to be extensive. We then consider three particular notions of coverings on toposes corresponding, respectively, to local homeomorphisms, covering projections, and unramified morphisms; in each case we deduce a suitable version of a van Kampen theorem in terms of coverings and, under further hypotheses, also one in terms of fundamental groupoids. An application is also given to knot groupoids and branched coverings. Along the way we are led to investigate locally constant objects in a topos bounded over an arbitrary base topos S and to establish some new facts about them.

In this paper, we study irreducible representations of regular limit subalgebras of AF-algebras. The main result is twofold: every closed prime ideal of a limit of direct sums of nest algebras (NSAF) is primitive, and every prime regular limit algebra is primitive. A key step is that the quotient of an NSAF algebra by any closed ideal has an AF C*-envelope, and this algebra is exhibited as a quotient of a concretely represented AF-algebra. When the ideal is prime, the C*-envelope is primitive. The GNS construction is used to produce algebraically irreducible (in fact n-transitive for all n⩾1) representations for quotients of NSAF algebras by closed prime ideals. Thus the closed prime ideals of NSAF algebras coincide with the primitive ideals. Moreover, these representations extend to *-representations of the C*-envelope of the quotient, so that a fortiori these algebras are also operator primitive. The same holds true for arbitrary limit algebras and the {0} ideal.