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ABSTRACT. For the 2-monad ((−) 2,I,C)onCAT, with unit I described by identities and multiplication C described by composition, we show that a functor F: K2 � � K satisfying FIK =1K admits a unique, normal, pseudo-algebra structure for (−) 2 if and only if there is a mere natural isomorphism F · F 2 � � � F · CK. We show that when this is the case the set of all natural transformations F · F 2 � � F · CK formsa commutative monoid isomorphic to the centre of K. 1. Preliminaries 1.1. When we speak of ‘the 2-monad (−) 2 on CAT ’ we understand the canonical monad that arises by exponentiation of the cocommutative comonoid structure

This is an expanded, revised and corrected version of the first author's preprint [1]. The discussion of one-dimensional cohomology H1 in a fairly general category E involves passing to the (2-)category Cat(E) of categories in E. In particular, the coe cient object is a category B in E and the torsors that H1 classifies are particular functors in E. We only impose conditions on E that are satisfied also by Cat(E) and argue that H1 for Cat(E) is a kind of H2 for E, and so on recursively. For us, it is too much to ask E to be a topos (or even internally complete) since, even if E is, Cat(E) is not. With this motivation, we are led to examine morphisms in E which act as internal families and to internalize the comprehensive factorization of functors into a nal functor followed by a discrete bration. We de ne B-torsors for a category B in E and prove clutching and classification theorems. The former theorem clutches ƒech cocycles to construct torsors while the latter constructs a coefficient category to classify structures locally isomorphic to members of a given internal family of structures. We conclude with applications to examples.

We study the glueing constructions (comma objects) on general algebraic structures on a 2-category, described in terms of 2-monads and adjunctions. Specifically, lifting theorems for the comma objects and change-of-base results on both algebras of 2-monads and adjunctions in a 2-category are presented. As a leading example, we take the 2-monad on Cat whose algebras are symmetric monoidal categories, and show that many of the constructions in our previous work on models of linear type theories can be derived within this axiomatics. 1 Introduction In the previous work [2, 3] we have considered a glueing construction for symmetric monoidal (closed) categories, for studying the logical predicates for models of linear type theories. In that construction the glueing functor is supposed to be lax symmetric monoidal, thus preserves the structure only up to a few coherent morphisms, not up to isomorphisms or identity. From a view of the study of categories with algebraic structures [8] (which...

We give a categorical account of Kripke-Beth-Joyal models of the - calculus. Kripke models. Emphasize semantics of (terms/representatives/realizers for) consequences. 1 Introduction This paper, \Functorial Kripke-Beth-Joyal models of the -calculus I: type theory and internal logic" (henceforth abbreviated to I), is rst of a sequence of three connected works. It is concerned with the basic model theory of the - calculus considered on the one hand as a system of rst-order dependent function types and on the other as presentation of the f8; g-fragment of minimal rstorder predicate logic with proof-objects. From the point of view of type theory, ... MITCHELL/MOGGI From the point of view of logic, the ... At the core of our denition of Kripke(-Beth-Joyal) models of lies our treatment of comprehension, context extension and (rst-order) dependent function spaces. The essential idea is similar that of earlier work [?, ?, ?]; however, our treatment has the following two advant...

For the 2-monad ((-) 2 , I, C) on CAT, with unit I described by identities and multiplication C described by composition, we show that a functor F : K 2 ## K satisfying F I K = 1 K admits a unique, normal, pseudo-algebra structure for (-) 2 if and only if there is a mere natural isomorphism F F 2 # ## F CK . We show that when this is the case the set of all natural transformations F F 2 ## F CK forms a commutative monoid isomorphic to the centre of K.

are closely related to the orthogonal adjunctions introduced in a previous paper. The pointwise case is treated by introducing internal comma objects, which can be defined in an arbitrary double category.

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In this sequel to [HM02], we explore some of the global aspects of the category of paracategories. We establish its (co)completeness and cartesian closure. From the closed structure we derive the relevant notion of transformation for paracategories. We set-up the relevant notion of adjunction between paracategories and apply it to define (co)completeness and cartesian closure, exemplified by the paracategory of bivariant functors and dinatural transformations. We introduce partial multicategories to account for partial tensor products. We also consider fibrations for paracategories and their indexed-paracategory version. Finally, we instantiate all these concepts in the context of probabilistic automata.