We show that the quotient of a Hom-finite triangulated category C by the kernel of the functor HomC(T, −), where T is a rigid object, is preabelian. We further show that the class of regular morphisms in the quotient admit a calculus of left and right fractions. It follows that the Gabriel-Zisman localisation of the quotient at the class of regular morphisms is abelian. We show that it is equiv...

From 1986 to 1990 a prospective comparative study was undertaken to compare the relative accuracy of computed tomography, endogastric ultrasonography, and intraoperative surgical assessment in evaluating the depth of invasion (T category) and involvement of lymph nodes (N category) of patients with gastric carcinoma. One hundred and eight consecutive patients, who were treated by total gastrect...

A weak entwining structure in a 2-category K consists of a monad t and a comonad c, together with a 2-cell relating both structures in a way that generalizes a mixed distributive law. A weak entwining structure can be characterized as a compatible pair of a monad and a comonad, in 2-categories generalizing the 2-category of comonads and the 2-category of monads in K , respectively. This observa...

Introduction In this thesis we are investigating possibility of assigning homotopical invariants to toposes in an alternative way. The 2-category of (Grothendieck) toposes, geometric morphisms and their natural transformations (which we will denote by Top) has been used by many authors to model homotopy types particularly efficiently. Thus for example a topological space X can be represented in...

Let X be a compact real analytic manifold, and let T ∗X be its cotangent bundle. Let Sh(X) be the triangulated dg category of bounded, constructible complexes of sheaves on X. In this paper, we develop a Fukaya A∞-category Fuk(T ∗X) whose objects are exact, not necessarily compact Lagrangian branes in the cotangent bundle. We write TwFuk(T ∗X) for the A∞-triangulated envelope of Fuk(T ∗X) consi...

We construct an invariant of t-structures on the derived category of a Noetherian ring. This invariant is complete when restricting to the category of quasi-coherent complexes, and also gives a classification of nullity classes with the same restriction. On the full derived category of Z we show that the class of distinct t-structures do not form a set.

As a practical foundation for a homotopy theory of abstract spacetime, we propose a convenient category S , which we show to extend a category of certain compact partially ordered spaces. In particular, we show that S ′ is Cartesian closed and that the forgetful functor S →T ′ to the category T ′ of compactly generated spaces creates all limits and colimits.