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In this paper we introduce sigma limits (which we write σ -limits), a concept that interpolates between lax and pseudolimits: for a fixed family Σ of arrows of a 2-category A , a σ -cone for a 2-functor A ⟶ F B is a lax cone such that the structural 2-cells corresponding to the arrows of Σ are invertible. The conical σ-limit of F is the universal σ -cone. Similarly we define σ -natural transformations and weighted σ -limits. We consider also the case of bilimits. We develop the theory of σ -limits and σ -bilimits, whose importance relies on the following key fact: any weighted σ-limit (or σ-bilimit) can be expressed as a conical one . From this we obtain, in particular, a canonical expression of an arbitrary C a t -valued 2-functor as a conical σ -bicolimit of representable 2-functors, for a suitable choice of Σ, which is equivalent to the well known bicoend formula. As an application, we establish the 2-dimensional theory of flat pseudofunctors. We define a C a t -valued pseudofunctor to be flat when its left bi-Kan extension along the Yoneda 2-functor preserves finite weighted bilimits. We introduce a notion of 2-filteredness of a 2-category with respect to a class Σ, which we call σ-filtered . Our main result is: A pseudofunctor A ⟶ C a t is flat if and only if it is a σ-filtered σ-bicolimit of representable 2-functors. In particular the reader will notice the relevance of this result for the development of a theory of 2-topoi. [ABSTRACT FROM AUTHOR]

We show that the class of Banach algebras that can be isometrically represented on an L p -space, for p ≠ 2 , is not closed under quotients. This answers a question asked by Le Merdy 20 years ago. Our methods are heavily reliant on our earlier study of Banach algebras generated by invertible isometries of L p -spaces. [ABSTRACT FROM AUTHOR]

In this paper we define a cohomology theory for an arbitrary K-linear semistrict semigroupal 2-category (C,⊗) (called for short a Gray semigroup) and show that its first-order (unitary) deformations, up to the suitable notion of equivalence, are in bijection with the elements of the second cohomology group. Fundamental to the construction is a double complex, similar to the Gerstenhaber–Schack double complex for bialgebras, the role of the multiplication and the comultiplication being now played by the composition and the tensor product of 1-morphisms. We also identify the cohomologies describing separately the deformations of the tensor product, the associator and the pentagonator. To obtain the above results, a cohomology theory for an arbitrary K-linear (unitary) pseudofunctor is introduced describing its purely pseudofunctorial deformations, and generalizing Yetter''s cohomology for semigroupal functors (in: M. Kapranov, E. Getzler (Eds.), Higher Category Theory, AMS Contemporary Mathematics, Vol. 230, Amer. Math. Soc., Providence, RI, 1998, pp. 117–134). The corresponding higher order obstructions will be considered in detail in a future paper. [Copyright &y& Elsevier]

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:Kop→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 TopS of toposes bounded over an elementary topos S, and to its full sub 2-category LTopS 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. [Copyright &y& Elsevier]