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Let K be a finite extension over Q2 and OK the ring of integers. We prove the equivalence of categories between the category of Kisin modules of height 1 and the category of Barsotti-Tate groups over OK .

The general view in psychological science is that natural categories obey a coherent, family-resemblance principle. In this investigation, we documented an example of an important exception to this principle: Results of a multidimensional-scaling study of igneous, metamorphic, and sedimentary rocks (Experiment 1) suggested that the structure of these categories is disorganized and dispersed. This finding motivated us to explore what might be the optimal procedures for teaching dispersed categories, a goal that is likely critical to science education in general. Subjects in Experiment 2 learned to classify pictures of rocks into compact or dispersed high-level categories. One group learned the categories through focused high-level training, whereas a second group was required to simultaneously learn classifications at a subtype level. Although high-level training led to enhanced performance when the categories were compact, subtype training was better when the categories were dispersed. We provide an interpretation of the results in terms of an exemplar-memory model of category learning.

String diagrams for traced and compact categories are oriented 1-cobordisms David I. Spivak ∗ Patrick Schultz ∗ Massachusetts Institute of Technology, Cambridge, MA 02139 Dylan Rupel † , ‡ Northeastern University, Boston, MA 02115 Abstract We give an alternate conception of string diagrams as labeled 1-dimensional oriented cobordisms, the operad of which we denote by Cob / O , where O is the set of string labels. The axioms of traced (symmetric monoidal) categories are fully en- coded by Cob / O in the sense that there is an equivalence between Cob / O -algebras, for varying O , and traced categories with varying object set. The same holds for compact (closed) categories, the difference being in terms of variance in O . As a consequence of our main theorem, we give a characterization of the 2-category of traced categories solely in terms of those of monoidal and compact categories, without any reference to the usual structures or axioms of traced categories. In an appendix we offer a complete proof of the well-known relationship between the 2-category of monoidal categories with strong monoidal functors and the 2-category of monoidal categories whose object set is free with strict functors; similarly for traced and compact categories., United States. Air Force. Office of Scientific Research (grant FA9550–14–1–0031), United States. Office of Naval Research ( grant N000141310260), United States. National Aeronautics and Space Administration (grant NNH13ZEA001N)

Simulations between processes can be understood in terms of coalgebra homomorphisms, with homomorphisms to the final coalgebra exactly identifying bisimilar processes. The elements of the final coalgebra are thus natural representatives of bisimilarity classes, and a denotational semantics of processes can be developed in a final-coalgebra-enriched category where arrows are processes, canonically represented. In the present paper, we describe a general framework for building final-coalgebra-enriched categories. Every such category is constructed from a multivariant functor representing a notion of process, much like Moggi's categories of computations arising from monads as notions of computation. The "notion of process" functors are intended to capture different flavors of processes as dynamically extended computations. These functors may involve a computational (co)monad, so that a process category in many cases contains an associated computational category as a retract. We further discuss categories of resumptions and of hyperfunctions, which are the main examples of process categories. Very informally, the resumptions can be understood as computations extended in time, whereas hypercomputations are extended in space.

In this paper, we consider $n$-perforated Yoneda algebras for $n$-angulated categories, and show that, under some conditions, $n$-angles induce derived equivalences between the quotient algebras of $n$-perforated Yoneda algebras. This result generalizes some results of Hu, K\"{o}nig and Xi. And it also establishes a connection between higher cluster theory and derived equivalences. Namely, in a cluster tilting subcategory of a triangulated category, an Auslander-Reiten $n$-angle implies a derived equivalence between two quotient algebras. This result can be compared with the fact that an Auslander-Reiten sequence suggests a derived equivalence between two algebras which was proved by Hu and Xi., Comment: 19 pages. arXiv admin note: text overlap with arXiv:1102.2790

For a category C, D. Bourn’s categories of points (categories of split epimorphisms with fixed codomain) can be defined as categories of the form ((B,1B)↓(C↓B)) for some B in C. A categorical-algebraic concept of exponentiation, namely, right adjoints for the pullback functors between D. Bourn’s categories of points, was introduced and studied in the author’s Ph.D. Thesis. We show for every category of Lie algebras over a fixed commutative unital ring, that all exponents exist.

By a multi-pointed category we mean a category C equipped with an ideal of null morphisms, i.e. a class N of morphisms satisfying f ∈ N ∨ g ∈ N ⇒ f g ∈ N for any composable pair f , g of morphisms. Such categories are precisely the categories enriched in the category of pairs X = ( X , N ) where X is a set and N is a subset of X , whereas a pointed category has the same enrichment, but restricted to those pairs X = ( X , N ) where N is a singleton. We extend the notion of an “ideal” from regular pointed categories to regular multi-pointed categories, and having “a good theory of ideals” will mean that there is a bijection between ideals and kernel pairs, which in the pointed case is the main property of ideal determined categories. The study of general categories with a good theory of ideals allows in fact a simultaneous treatment of ideal determined and Barr exact Goursat categories: we prove that in the case when all morphisms are chosen as null morphisms, the presence of a good theory of ideals becomes precisely the property for a regular category to be a Barr exact Goursat category. Among other things, this allows to obtain a unified proof of the fact that lattices of effective equivalence relations are modular both in the case of Barr exact Goursat categories and in the case of ideal determined categories.

We address four problems regarding representable functors and give answers to them for functors connecting the category of comodules over a coring to the category of modules over a ring. A functor's property of being Frobenius is restated as a particular case of its representability by imposing the predefinition of the object of representability. Let R, S be two rings, C an R-coring and the category of left C-comodules. The category of all representable functors is shown to be equivalent to the opposite of the category . For U an (S, R)-bimodule we give necessary and sufficient conditions for the induction functor to be: a representable functor, an equivalence of categories, a separable or a Frobenius functor. The latter results generalize and unify the classical theorems of Morita for categories of modules over rings and the more recent theorems obtained by Brezinski, Caenepeel et al. for categories of comodules over corings.

The Lazard correspondence establishes an equivalence of categories between p-groups of nilpotency class less than p and nilpotent Lie rings of the same class and order. The main tools used to achieve this are the Baker–Campbell–Hausdorff formula and its inverse formulae. Here we describe methods to compute the inverse Baker–Campbell–Hausdorff formulae. Using these we get an algorithm to compute the Lie ring structure of a p-group of class

We continue the study of the fundamental series of generalized Harish-Chandra modules initiated in Generalized Harish-Chandra modules with generic minimal $\mathfrak{k}$-type. Generalized Harish-Chandra modules are $(\mathfrak{g}, \mathfrak{k})$-modules of finite type where $\mathfrak{g}$ is a semisimple Lie algebra and $\mathfrak{k} \subset \mathfrak{g}$ is a reductive in $\mathfrak{g}$ subalgebra. A first result of the present paper is that a fundamental series module is a $\mathfrak{g}$-module of finite length. We then define the notions of strongly and weakly reconstructible simple $(\mathfrak{g}, \mathfrak{k})$-modules $M$ which reflect to what extent $M$ can be determined via its appearance in the socle of a fundamental series module. In the second part of the paper we concentrate on the case $\mathfrak{k} \simeq sl(2)$ and prove a sufficient condition for strong reconstructibility. This strengthens our main result from Generalized Harish-Chandra modules with generic minimal $\mathfrak{k}$-type for the case $\mathfrak{k} = sl(2)$. We also compute the $sl(2)$-characters of all simple strongly reconstructible (and some weakly reconstructible) $(\mathfrak{g}, sl(2))$-modules. We conclude the paper by discussing a functor between a generalization of the category $\mathcal{O}$ and a category of $(\mathfrak{g}, sl(2))$-modules, and we conjecture that this functor is an equivalence of categories.

In the paper, the quantum integrals associated to quantum Yetter-Drinfeld ?-modules are defined. We shall prove the following affineness criterion: if there exists ? = {?? : H? ? Hom(H?-1, A)} ? ? ? a total quantum integral and the canonical map ? : A?B A ? ???? H? ? A, ?(a?B b)= ???? S?-1 ??(b[1,?-1?-1?])b[0,0] ? ab[0,0] is subjective. Then the induction functor -?B A : UB ? H YD?A is an equivalence of categories. The affineness criterion proven by Menini and Militaru is recovered as special cases.

It is well known that inverse semigroups are closely related to \'etale groupoids. In particular, it has recently been shown that there is a (non-functorial) equivalence between localic \'etale groupoids, on one hand, and complete and infinitely distributive inverse semigroups (abstract complete pseudogroups), on the other. This correspondence is mediated by a class of quantales, known as inverse quantal frames, that are obtained from the inverse semigroups by a simple join completion that yields an equivalence of categories. Hence, we can regard abstract complete pseudogroups as being essentially ``the same'' as inverse quantal frames, and in this paper we exploit this fact in order to find a suitable replacement for inverse semigroups in the context of open groupoids that are not necessarily \'etale. The interest of such a generalization lies in the importance and ubiquity of open groupoids in areas such as operator algebras, differential geometry and topos theory, and we achieve it by means of a class of quantales, called open quantal frames, which generalize inverse quantal frames and whose properties we study in detail. The resulting correspondence between quantales and open groupoids is not a straightforward generalization of the previous results concerning \'etale groupoids, and it depends heavily on the existence of inverse semigroups of local bisections of the quantales involved., Comment: 55 pages

Previously, employing the apparatus of category theory for abstract description of knowledge in production-type systems was shown to be reasonable. Interest towards applying the obtained techniques to so-called dynamic expert systems, which allow for variation of data and knowledge in the course of system functioning, required a certain generalization of previously developed category theory apparatus related to consideration of so-called folded categories. The paper is devoted to the study of some properties of such categories.

Let $X \hookrightarrow \overline{X}$ be an open immersion of smooth varieties over a field of characteristic $p>0$ such that the complement is a simple normal crossing divisor and let $\overline{Z} \subseteq Z \subseteq \overline{X}$ be closed subschemes of codimension at least $2$. In this paper, we prove that the canonical restriction functor between the category of overconvergent $F$-isocrystals $F\text{-}{\rm Isoc}^{\dagger}(X,\overline{X}) \longrightarrow F\text{-}{\rm Isoc}^{\dagger}(X \setminus Z, \overline{X} \setminus \overline{Z})$ is an equivalence of categories. We also prove an application to the category of $p$-adic representations of the fundamental group of $X$, which is a higher-dimensional version of a result of Tsuzuki., 22 pages

Most often, in a categorical semantics for a programming language, the substitution of terms is expressed by composition and finite products. However this does not deal with the order of evaluation of arguments, which may have major consequences when there are side-effects. In this paper Cartesian effect categories are introduced for solving this issue, and they are compared with strong monads, Freyd-categories and Haskell's Arrows. It is proved that a Cartesian effect category is a Freyd-category where the premonoidal structure is provided by a kind of binary product, called the sequential product. The universal property of the sequential product provides Cartesian effect categories with a powerful tool for constructions and proofs. To our knowledge, both effect categories and sequential products are new notions., 23 pages

We describe an equivalence of categories between the category of mixed Hodge structures and a category of equivariant vector bundles on a toric model of the complex projective plane which verify some semistability condition. We then apply this correspondence to define an invariant which generalizes the notion of R-split mixed Hodge structure and give calculations for the first group of cohomology of possibly non smooth or non-complete curves of genus 0 and 1. Finally, we describe some extension groups of mixed Hodge structures in terms of equivariant extensions of coherent sheaves. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

http://msp.berkeley.edu/ant/2010/4-7/p06.xhtml, Berger and Colmez (2008) formulated a theory of families of overconvergent étale (φ,Γ)-modules associated to families of p-adic Galois representations over p-adic Banach algebras. In contrast with the classical theory of (φ,Γ)-modules, the functor they obtain is not an equivalence of categories. In this paper, we prove that when the base is an affinoid space, every family of (overconvergent) étale (φ,Γ)-modules can locally be converted into a family of p-adic representations in a unique manner, providing the “local” equivalence. There is a global mod p obstruction related to the moduli of residual representations.

The concept of an M-approximate system is introduced. Basic properties of the category of M-approximate systems and in a natural way defined morphisms between them are studied. It is shown that categories related to fuzzy topology as well as categories related to rough sets can be described as special subcategories of the category of M-approximate systems.

2-equivalences are described between the category of small abelian categories with exact functors, the category of definable additive categories with functors which commute with products and direct limits and the category of locally coherent Grothendieck categories with "coherent" morphisms. There is a comparison, for definable additive categories, between the presheaf of finite-type localisations and the presheaf of localisations of associated functor categories. The image of the free abelian category in Mod-R is described and related to special bases of the Ziegler and rep-Zariski spectra restricted to the set of indecomposable injectives. In the coherent case there is a particularly nice form (which is essentially elimination of imaginaries in the model-theoretic sense).

The purpose of this note is to point out that simplicial methods and the well-known Dold-Kan construction in simplicial homotopy theory can be fruitfully applied to convert link homology theories into homotopy theories. Dold and Kan prove that there is a functor from the category of chain complexes over a commutative ring with unit to the category of simplicial objects over that ring such that chain homotopic maps go to homotopic maps in the simplicial category. Furthermore, this is an equivalence of categories. In this way, given a link homology theory, we construct a mapping taking link diagrams to a category of simplicial objects such that up to looping or delooping, link diagrams related by Reidemeister moves will give rise to homotopy equivalent simplicial objects, and the homotopy groups of these objects will be equal to the link homology groups of the original link homology theory. The construction is independent of the particular link homology theory. A simplifying point in producing a homotopy simplicial object in relation to a chain complex occurs when the chain complex is itself derived (via face maps) from a simplicial object that satisfies the Kan extension condition. Under these circumstances one can use that simplicial object rather than apply the Dold-Kan functor to the chain complex. We will give examples of this situation in regard to Khovanov homology. We will investigate detailed working out of this correspondence in separate papers. The purpose of this note is to announce the basic relationships for using simplicial methods in this domain. Thus we do more than just quote the Dold-Kan Theorem. We give a review of simplicial theory and we point to specific constructions, particularly in relation to Khovanov homology, that can be used to make simplicial homotopy types directly., 32 pages, 10 figures, LaTeX document. arXiv admin note: text overlap with arXiv:1107.1524

In this work we give a method for computing sections of homogeneous vector bundles on any rational homogeneous variety G / P of type ADE. Our main tool is the equivalence of categories between homogeneous vector bundles on G / P and finite dimensional representations of a given quiver with relations. Our result generalizes the work of Ottaviani and Rubei (2006) [OR06] .

We consider limits over categories of extensions and show how certain well-known functors on the category of groups turn out as such limits. We also discuss higher (or derived) limits over categories of extensions.

Let ℰ be an additive category and 𝒞 a full subcategory with split idempotents, and closed under isomorphic images and finite direct sums. We give conditions on ℰ and 𝒞 implying that ℰ embeds into an abelian category, so that the objects of 𝒞 turn into injective objects. This construction generalizes the embedding of exactly definable categories into locally coherent categories, while the dual construction generalizes the embedding of finitely accessible categories into Grothendieck categories with a family of finitely generated projective generators. As applications, we characterize exactly definable categories through intrinsic properties and study those locally coherent categories whose fp-injective objects form a Grothendieck category.

We construct a Quillen model structure on the category of spectral categories, where the weak equivalences are the symmetric spectra analogue of the notion of equivalence of categories., 19 pages

We examine the effective categoricity of equivalence structures via Ershov's difference hierarchy. We explore various kinds of categoricity available by distinguishing three different notions of isomorphism available in this hierarchy. We prove several results relating our notions of categoricity to computable equivalence relations: for example, we show that, for such relations, computable categoricity is equivalent to our notion of weak ω-c.e. categoricity, and that -categoricity is equivalent to our notion of graph-ω-c.e. categoricity.

A long-standing assumption in linguistic analysis is that different languages and constructions can be described in terms of the same grammatical categories and relations. Individual grammatical categories and relations are in fact often assumed to be universal. Grammatical categories and relations display however different properties across different languages and constructions, which challenges the idea that the same categories and relations should actually be posited in each case. These facts have been dealt with in two major ways in the functional-typological literature. In a widespread approach, the same categories and relations are posited for different languages and constructions provided that these all have categories and relations that display some selected properties. In a more recent approach, this idea is abandoned, and grammatical categories and relations are argued to be language-specific and construction-specific. This article provides a critical review of these approaches, and a comparison is made with some generatively oriented approaches. In particular, it is argued that a distinction should be made between two views of grammatical categories and relations. In one view, grammatical categories and relations are classificatory labels indicating that a variety of linguistic elements display some selected property. In another view, grammatical categories and relations are proper components of a speaker's mental grammar. While cross-linguistically valid (or possibly universal) and cross-constructionally valid categories and relations can be posited when classifying linguistic elements based on observed grammatical patterns, there is no obvious evidence that such categories and relations exist at the level of mental representation. This is, however, because of the absence of conclusive evidence about the organization of a speaker's mental grammar, rather than because of the linguistic evidence as such.

The notion of equivalence relation is arguably one of the most fundamental ideas of mathematics. Accordingly, it plays an important role in teaching mathematics at all levels, whether explicitly or implicitly. Our success in introducing this notion for its own sake or as a means to teach other mathematical concepts, however, depends largely on our own conceptions of it. This paper considers various conceptions of equivalence, in history, in mathematics today, and in mathematics education. It reveals critical differences in the notion of equivalence at different points in history and a meaning for equivalence proposed by mathematicians and mathematics educators that is at variance with the ways that learners may think. These differences call into question the most popular view of the subject: that the mathematical notion of equivalence relation is the result of spelling out our experience of equivalence. Moreover, the findings of this study suggest that the standard definition of an equivalence relation is ill-chosen from a pedagogical point of view but well-crafted from a mathematical point of view.

There is a well-known correspondence between infinite trees and ultrametric spaces which can be interpreted as an equivalence of categories and comes from considering the end space of the tree. In this equivalence, uniformly continuous maps between the end spaces are translated to some classes of coarse maps (or even classes of metrically proper Lipschitz maps) between the trees.

We describe the (tetra) category of pseudo-categories, pseudo-functors, natural transformations, pseudo-natural transformations, and modifications, as introduced in Martins-Ferreira (JHRS 1:47–78, 2006), internal to an additive 2-category with kernels, as formalized in Martins-Ferreira (Fields Inst Commun 43:387–410, 2004). In the context of a 2-Ab-category, we introduce the notion of a pseudo-morphism and prove the equivalence of categories: PsCat(A)~PsMor(A) between pseudo-categories and pseudo-morphisms in an additive 2-category, A, with kernels– extending thus the well known equivalence Cat(Ab)~Mor(Ab) between internal categories and morphisms of abelian groups. The leading example of an additive 2-category with kernels is Cat(Ab). In the case A=Cat(Ab) we obtain a description of the (tetra) category of internal pseudo-double categories in Ab, and particularize it to a description of the (tetra) category of internal bicategories in abelian groups. As expected, pseudo-natural transformations coincide with homotopies of 2-chain complexes (as in Bourn, J Pure Appl Algebra 66:229–249, 1990).

We show the equivalence of categories of model-theoretic imaginaries (of various kinds) with categories of “small” (finitely generated, finitely presented, coherent) functors. We do this first for certain locally finitely presented categories and then, by localising, for much more general “definable categories” (categories of models of coherent theories). We also investigate the corresponding notion of interpretation.

Greenlees established an equivalence of categories between the homotopy category of rational SO(3)-spectra and the derived category DA(SO(3)) of a certain abelian category. In this paper we lift this equivalence of homotopy categories to the level of Quillen equivalences of model categories. Methods used in this paper provide the first step towards obtaining an algebraic model for the toral part of rational G-spectra, for any compact Lie group G.

For the p-cyclotomic tower of \(\mathbb {Q}_p\) Fontaine established a description of local Iwasawa cohomology with coefficients in a local Galois representation V in terms of the \(\psi \)-operator acting on the attached etale \((\varphi ,\Gamma )\)-module D(V). In this chapter we generalize Fontaine’s result to the case of arbitrary Lubin–Tate towers \(L_\infty \) over finite extensions L of \(\mathbb {Q}_p\) by using the Kisin–Ren/Fontaine equivalence of categories between Galois representations and \((\varphi _L,\Gamma _L)\)-modules and extending parts of [20, 33]. Moreover, we prove a kind of explicit reciprocity law which calculates the Kummer map over \(L_\infty \) for the multiplicative group twisted with the dual of the Tate module T of the Lubin–Tate formal group in terms of Coleman power series and the attached \((\varphi _L,\Gamma _L)\)-module. The proof is based on a generalized Schmid–Witt residue formula. Finally, we extend the explicit reciprocity law of Bloch and Kato [3] Theorem 2.1 to our situation expressing the Bloch–Kato exponential map for \(L(\chi _{LT}^r)\) in terms of generalized Coates–Wiles homomorphisms, where the Lubin–Tate character \(\chi _{LT}\) describes the Galois action on T.

We generalize linear superalgebra to higher gradings and commutation factors, given by arbitrary abelian groups and bicharacters. Our central tool is an extension, to monoidal categories of modules, of the Nekludova-Scheunert faithful functor between the categories of graded-commutative and supercommutative algebras. As a result we generalize (super-)trace, determinant and Berezinian to graded matrices over graded-commutative algebras. For instance, on homogeneous quaternionic matrices, we obtain a lift of the Dieudonn\'e determinant to the skew-field of quaternions., Comment: 44 pages

A finite tensor category is called pointed if all its simple objects are invertible. We find necessary and sufficient conditions for two pointed semisimple categories to be dual to each other with respect to a module category. Whenever the dual of a pointed semisimple category with respect to a module category is pointed, we give explicit formulas for the Grothendieck ring and for the associator of the dual. This leads to the definition of categorical Morita equivalence on the set of all finite groups and on the set of all pairs (G, w), where G is a finite group and w is a 3-cocycle on G. A group-theoretical and cohomological interpretation of this relation is given. A series of concrete examples of pairs of groups that are categorically Morita equivalent but have non-isomorphic Grothendieck rings are given. In particular, the representation categories of the Drinfeld doubles of the groups in each example are equivalent as braided tensor categories and hence these groups define the same modular data., 20 pages; misprints corrected; to appear in Communications in Algebra

To any left system of diagram categories or to any left pointed derivateur (in the sense of Grothendieck) a K-theory space is associated. This K-theory space is shown to be canonically an infinite loop space and to have a lot of common properties with Waldhausen's K-theory. A weaker version of additivity is shown. Also, Quillen's K-theory of a large class of exact categories including the abelian categories is proved to be a retract of the K-theory of the associated derivateur., 50 pages

This paper is devoted to the comparison of different localized categories of differential complexes. The first result is that the canonical functor from the category of complexes of differential operators of order one (defined by Herrera and Lieberman) to the category of differential complexes (of any order, defined by M. Saito), both localized with respect to a suitable notion of quasi-isomorphism, is an equivalence of categories. Then we prove a similar result for a filtered version of the previous categories (defined respectively by Du Bois and M.Saito), localized with respect to graded-quasi-isomorphisms, thus answering a question posed by M. Saito., Comment: 13 pages

Formal (mixed) Hodge structures FHS are introduced in such a way that the Hodge realization of Deligne's 1-motives extends to a realization from Laumon's 1-motives to formal Hodge structures of level 1, providing an equivalence of categories. For the sake of exposition we here confine our study to level 1 mixed Hodge structures. However, it is conceivable and suitable to consider formal mixed Hodge structures with arbitrary Hodge numbers: generalizing our definition herebelow it's not that difficult and we will treat such a matter nextly., 12 pages

We explore the possibility and some potential payoffs of using the theory of accessible categories in the study of categories of logics. We illustrate this by two case studies focusing on the category of finitary structural logics and its subcategory of algebraizable logics. Mathematics Subject Classification (2000). Primary 03B22; Secondary 18C35.

In this paper we put a cofibrantly generated model category struc- ture on the category of small simplicial categories. The weak equivalences are a simplicial analogue of the notion of equivalence of categories.

We explore connections between elementary equivalence of categories of acts over monoids and second-order equivalence of monoids.

We study relative Fourier–Mukai transforms on genus one fibrations with section, allowing explicitly the total space of the fibration to be singular and non-projective. Grothendieck duality is used to prove a skew–commutativity relation between this equivalence of categories and certain duality functors. We use our results to explicitly construct examples of semi-stable sheaves on degenerating families of elliptic curves.

The category-theoretic nature of general frames for modal logic is explored. A new notion of "modal map" between frames is defined, generalizing the usual notion of bounded morphism/p-morphism. The category Fm of all frames and modal maps has reflective subcategories CHFm of compact Hausdorff frames, DFm of descriptive frames, and UEFm of ultrafilter enlargements of frames. All three subcategories are equivalent, and are dual to the category of modal algebras and their homomorphisms. An important example of a modal map that is typically not a bounded morphism is the natural insertion of a frame A into its ultrafilter enlargement EA. This map is used to show that EA is the free compact Hausdorff frame generated by A relative to Fm. The monad E of the resulting adjunction is examined and its Eilenberg-Moore category is shown to be isomorphic to CHFm. A categorical equivalence between the Kleisli category of E and UEFm is defined from a construction that assigns to each frame A a frame A* that is "image-closed" in the sense that every point-image {b : aRb} in A is topologically closed. A* is the unique image-closed frame having the same ultrafilter enlargement as A. These ideas are connected to a category 2U shown by S. K. Thomason to be dual to the category of complete and atomic modal algebras and their homomorphisms. 2U is the full subcategory of the Kleisli category of E based on the Kripke frames.

We define negative K-groups for exact categories and for ``derived categories'' in the framework of Frobenius pairs, generalizing definitions of Bass, Karoubi, Carter, Pedersen-Weibel and Thomason. We prove localization and vanishing theorems for these groups. Devissage (for noetherian abelian categories), additivity, and resolution hold. We show that the first negative K-group of an abelian category vanishes, and that, in general, negative K-groups of a noetherian abelian category vanish. Our methods yield an explicit non-connective delooping of the K-theory of exact categories and chain complexes, generalizing constructions of Wagoner and Pedersen-Weibel. Extending a theorem of Auslander and Sherman, we discuss the K-theory homotopy fiber of e⊕→ e and its implications for negative K-groups. In the appendix, we replace Waldhausen's cylinder functor by a slightly weaker form of non-functorial factorization which is still sufficient to prove his approximation and fibration theorems.

Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not have a substantial counterpart for symmetric tensor categories. In particular, we exhibit various equivalences involving categories of modules over algebras in ribbon categories. Finally we establish a correspondence of ribbon categories that can be applied to, and is in fact motivated by, the coset construction in conformal quantum field theory., 129 pages; several figures. v2: remark 7.4(ii) corrected, conditions in theorem 7.6 and in corollary 7.7 adapted. v3 (version to appear in Adv.Math.): typos corrected