This paper discusses the use and importance of category theory in system descriptions for model-based diagnostic reasoning. We argue that category theory provides a precise and convenient conceptual language and tool to model complex systems. System descriptions are regarded as categories, in which each object is a system model and each morphism represents changes and transformations of system ...

Definition 1.1. A database schema is a small category C, and a morphism of schemas from C to C′ is a functor F : C → C′. An object c of C is called a type or a table in C, and an arrow f : c→ c′ is called a column of c valued in c′. A database state on schema C is a functor δ : C → Set. Given an object c ∈ Ob(C), an element x ∈ δ(c) is called a c-representative in δ, or a row of δ(c). We refer ...

We prove that the bounded derived category of coherent sheaves on a quasicompact separated quasiexcellent scheme finite dimension has strong generator in sense Bondal-Van den Bergh. This extends recent result Neeman and is new even affine case. The main ingredient includes Gabber's weak local uniformization theorem notions boundedness descendability morphism schemes.

To a B-coring and a (B, A)-bimodule that is finitely generated and projective as a right A-module an A-coring is associated. This new coring is termed a base ring extension of a coring by a module. We study how the properties of a bimodule such as separability and the Frobenius properties are reflected in the induced base ring extension coring. Any bimodule that is finitely generated and projec...

Given a complete nonsingular algebraic variety X and a divisor D with normal crossings, we say that X is log homogeneous with boundary D if the logarithmic tangent bundle T X (− log D) is generated by its global sections. We then show that the Albanese morphism α is a fibration with fibers being spherical (in particular, rational) varieties. It follows that all irreducible components of D are n...

Given a complete nonsingular algebraic variety X and a divisor D with normal crossings, we say that X is log homogeneous with boundary D if the logarithmic tangent bundle TX(− log D) is generated by its global sections. Then the Albanese morphism α turns out to be a fibration with fibers being spherical (in particular, rational) varieties. It follows that all irreducible components of D are non...

In this paper, we introduce a category of graded commutative rings with certain algebraic morphisms, to investigate the cobordism category of plumbed 3-manifolds. In particular, we define a non-associative distributive algebra that gives necessary conditions for an abstract morphism between the homologies of two plumbed 3-manifolds to be realized geometrically by a cobordism. Here we also consi...

Propositions 4.2 and 4.3 of the author’s article (Theory Appl. Categ. 14 (2005), 451-479) are not correct. We show that their use can be avoided and all remaining results remain correct. Propositions 4.2 and 4.3 of the author’s [3] are not correct and I am grateful to J. F. Jardine for pointing it out. In fact, consider the diagram D sending the one morphism category to the point ∆0 in the homo...

Recently Flaminio and Montagna, [FlMo], extended the language of MV-algebras by adding a unary operation, called a state-operator. This notion is introduced here also for effect algebras. Having it, we generalize the Loomis–Sikorski Theorem for monotone σ-complete effect algebras with internal state. In addition, we show that the category of divisible state-morphism effect algebras satisfying (...

A concept of equation morphism is introduced for every endofuctor F of a cocomplete category C. Equationally defined classes of F–algebras for which free algebras exist are called varieties. Every variety is proved to be monadic over C, and conversely, every monadic category is equivalent to a variety. And the Birkhoff Variety Theorem is proved for “Set–like” categories. By dualizing, we arrive...