Boolean circuits are used to represent programs on finite data. Reversible Boolean circuits and quantum Boolean circuits have been introduced to modelize some physical aspects of computation. Those notions are essential in complexity theory, but we claim that a deep mathematical theory is needed to make progress in this area. For that purpose, the recent developments of knot theory is a major source of inspiration. Following the ideas of Burroni, we consider logical gates as generators for some algebraic structure with two compositions, and we are interested in the relations satisfied by those generators. For that purpose, we introduce canonical forms and rewriting systems. Up to now, we have mainly studied the basic case and the linear case, but we hope that our methods can be used to get presentations by generators and relations for the (reversible) classical case and for the (unitary) quantum case.

This paper outlines an application of category theory to provide mathematical foundations for the ERA (Entity, Relationship, Attribute) approach to information systems. An ERA specification yields the dynamic category DC of the information system, which is a syntactic object describing all allowable operations. The dynamic category can be "collapsed" to a static category SC, which has a canonical language associated with it. This canonical language is the query language of the information system. A model of the information system is defined to be a functor from DC to the category Set and "snapshots" of a model are described in terms of functors from SC to Set. 1 Introduction In the last three decades the study of syntax and semantics has benefited greatly from category theory [7], [5]. A formal theory has associated with it a category (generally a "free" category in an appropriate sense). We call this category the syntactic category of the formal theory. A model of the theory...

The significance of the 2-dimensional calculus, which goes back to Penrose, has already been pointed out by Joyal and Street in [JoS91]. Independently, Burroni has introduced a general notion of n-dimensional presentation in [Bur91] and he has shown that the equational logic of terms is a special case of 2-dimensional calculus. Here, we propose a combinatorial definition of 2-dimensional diagrams and a simple method for proving that certain monoidal categories are finitely 2-presentable. We illustrate the translation of terms into diagrams and we explain the change from groups to quantum groups in a purely syntactical way. This paper should serve as a reference for our future work on symbolic computation, including a theory of 2-dimensional rewriting and the design of software for interactive diagrammatic reasoning. New address: CNRS - Laboratoire de math'ematiques discr`etes, 163 avenue de Luminy - Case 930, 13288 Marseille Cedex 9, France. Email: lafont@lmd.univ-mrs.fr 2 1 FROM T...

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Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalence is defined; this concept is related to "strongly separable " Frobenius algebras and "weak monoidal Morita equivalence". Wreath products of Frobenius algebras are discussed.

Exponentiable maps in the category Top of topological spaces are characterized by an easy ultrafilter-interpolation property, in generalization of a recent result by Pisani for spaces. From this characterization we deduce that perfect (= proper and separated) maps are exponentiable, generalizing the classical result for compact Hausdorff spaces. Furthermore, in generalization of the Whitehead-Michael characterization of locally compact Hausdorff spaces, we characterize exponentiable maps of Top between Hausdorff spaces as restrictions of perfect maps to open subspaces.

This paper argues that the core of modularity problems is an understanding of how individual components of a large system interact with each other, and that this interaction can be described by a layer structure. We propose a uniform treatment of layers based upon the concept of a monad. The combination of different systems can be described by the coproduct of monads.

ABSTRACT. Inthis paper, we consider those morphisms p: P − → B of posets for which the induced geometric morphism of presheaf toposes is exponentiable in the category of Grothendieck toposes. In particular, we show that a necessary condition is that the induced map p ↓ : P ↓ − → B ↓ is exponentiable in the category of topological spaces, where P ↓ is the space whose points are elements of P and open sets are downward closed subsets of P. Along the way, we show that p ↓ : P ↓ − → B ↓ is exponentiable if and only if p: P − → B is exponentiable in the category of posets and satisfies an additional compactness condition. The criteria for exponentiability of morphisms of posets is related to (but weaker than) the factorization-lifting property for exponentiability of morphisms in the

Abstract. Given a row-finite k-graph Λ with no sources we investigate the K-theory of the higher rank graph C ∗-algebra, C ∗ (Λ). When k = 2 we are able to give explicit formulae to calculate the K-groups of C ∗ (Λ). The K-groups of C ∗ (Λ) for k> 2 can be calculated under certain circumstances. We state that for all k, the torsion-free rank of K0(C ∗ (Λ)) and K1(C ∗ (Λ)) are equal when C ∗ (Λ) is unital, and we determine the position of the class of the unit of C ∗ (Λ) in K0(C ∗ (Λ)). 1.

For a complete cartesian-closed category V with coproducts, and for any pointed endofunctor T of the category of sets satisfying a suitable Beck-Chevalley-type condition, it is shown that the category of lax reflexive (T , V)-algebras is a quasitopos. This result encompasses many known and new examples of quasitopoi. 1.