We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground fields, and the proof of new results about topological invariants in three dimensions. The central notion is that of a Frobenius algebra in a tensor category A, which reduces to the classical notion if A = F-Vect, where F is a field. An object X ∈ A with two-sided dual X gives rise to a Frobenius algebra in A, and under weak additional conditions we prove a converse: There exists a bicategory E with ObjE = {A, B} such that EndE(A) ⊗ ≃ A and such that there are J, J: B ⇋ A producing the given Frobenius algebra. Many properties (additivity, sphericity, semisimplicity,...) of A carry over to the bicategory E. We define weak monoidal Morita equivalence of tensor categories, denoted A ≈ B, and establish a correspondence between Frobenius algebras in A and tensor categories B ≈ A. While considerably weaker than equivalence of tensor categories, weak monoidal Morita equivalence A ≈ B has remarkable consequences: A and B have equivalent (as braided tensor categories) quantum doubles (‘centers’) and (if A, B are semisimple spherical or ∗-categories) have equal dimensions and give rise the same state sum invariant of closed oriented 3-manifolds as recently defined by Barrett and Westbury. An instructive example is provided by finite dimensional semisimple and cosemisimple Hopf algebras, for which we prove H − mod ≈ ˆH − mod. The present formalism permits a fairly complete analysis of the center of a semisimple spherical category, which is the subject of the companion paper math.CT/0111205. 1

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It is well known that one can build models of full higher-order dependent type theory (also called the calculus of constructions) using partial equivalence relations (PERs) and assemblies over a partial combinatory algebra (PCA). But the idea of categories of PERs and ERs (total equivalence relations) can be applied to other structures as well. In particular, we can easily dene the category of ERs and equivalencepreserving continuous mappings over the standard category Top 0 of topological T 0 -spaces; we call these spaces (a topological space together with an ER) equilogical spaces and the resulting category Equ. We show that this category|in contradistinction to Top 0 |is a cartesian closed category. The direct proof outlined here uses the equivalence of the category Equ to the category PEqu of PERs over algebraic lattices (a full subcategory of Top 0 that is well known to be cartesian closed from domain theory). In another paper with Carboni and Rosolini (cited herein) a more abstract categorical generalization shows why many such categories are cartesian closed. The category Equ obviously contains Top 0 as a full subcategory, and it naturally contains many other well known subcategories. In particular, we show why, as a consequence of work of Ershov, Berger, and others, the Kleene-Kreisel hierarchy of countable functionals of nite types can be naturally constructed in Equ from the natural numbers object N by repeated use in Equ of exponentiation and binary products. We also develop for Equ notions of modest sets (a category equivalent to Equ) and assemblies to explain why a model of dependent type theory is obtained. We make some comparisons of this model to other, known models. 1

Concurrent computation can be given an abstract mathematical treatment very similar to that provided for sequential computation by domain theory and denotational semantics of Scott and Strachey.

We investigate the development of theories of types and computability via realizability.

This paper gives the first complete axiomatization for higher types in the refinement calculus of predicate transformers.

Set theory is the standard foundation for mathematics, but the majority of general purpose mechanised proof assistants support versions of type theory (higher order logic). Examples include Alf, Automath, Coq, EHDM, HOL, IMPS, LAMBDA, LEGO, Nuprl, PVS and Veritas. For many applications type theory works well and provides, for specification, the benefits of type-checking that are well-known in programming. However, there are areas where types get in the way or seem unmotivated. Furthermore, most people with a scientific or engineering background already know set theory, whereas type theory may appear inaccessable and so be an obstacle to the uptake of proof assistants based on it. This paper describes some experiments (using HOL) in combining set theory and type theory; the aim is to get the best of both worlds in a single system. Three approaches have been tried, all based on an axiomatically specified type V of ZF-like sets: (i) HOL is used without any additions besides V; (ii) an emb...

In this paper I compare two well studied approaches to topological semantics| the domain-theoretic approach, exemplied by the category of countably based equilogical spaces, Equ, and Type Two Eectivity, exemplied by the category of Baire space representations, Rep(B ). These two categories are both locally cartesian closed extensions of countably based T 0 -spaces. A natural question to ask is how they are related.

We present a state sum construction of two-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, which generalizes the state sum of Fukuma–Hosono–Kawai from triangulations of conventional two-dimensional cobordisms to those of open-closed cobordisms, i.e. smooth compact oriented 2-manifolds with corners that have a particular global structure. This construction reveals the topological interpretation of the associative algebra on which the state sum is based, as the vector space that the TQFT assigns to the unit interval. Extending the notion of a two-dimensional TQFT from cobordisms to suitable manifolds with corners therefore makes the relationship between the global description of the TQFT in terms of a functor into the category of vector spaces and the local description in terms of a state sum fully transparent. We also illustrate the state sum construction of an open-closed TQFT with a finite set of D-branes using the example of the groupoid algebra of a finite groupoid.

This paper studies the notions of approximation and passage to the limit in an axiomatic setting. Our axiomatisation is subject to the following criteria: the axioms should be natural (so that they are available in as many contexts as possible) and non-order-theoretic (so that Research supported by SERC grant RR30735 and EC project Programming Language Semantics and Program Logics grant SC1000 795 they explain the order-theoretic structure). Our aim is 1. to provide a justification of Scott's original consideration of ordered structures, and 2. to deepen our understanding of the notion of passage to the limit