Integral categories and calculus categories

Integral Categories and Calculus Categories

Differential categories are now an established abstract setting for differentiation. The paper presents the parallel development for integration by axiomatizing an integral transformation, sA : !A→ !A⊗A, in a symmetric monoidal category with a coalgebra modality. When integration is combined with differentiation, the two fundamental theorems of calculus are expected to hold (in a suitable sense...

Calculus on Categories

In some geometrically interesting cases, a structure on a manifold M can be lifted to a related structure on the groupoid M of paths on M . The well-known relation between connections on M and parallel transport operators provides a motivating example. We examine the problem of lifting geometric constructions such as principal bundles with connection to path space. Additionally, we undertake a ...

Differential Calculus in Categories. I

CLUSTER ALGEBRAS AND CLUSTER CATEGORIES

These are notes from introductory survey lectures given at the Institute for Studies in Theoretical Physics and Mathematics (IPM), Teheran, in 2008 and 2010. We present the definition and the fundamental properties of Fomin-Zelevinsky’s cluster algebras. Then, we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generator...

Convergence and quantale-enriched categories

Generalising Nachbin's theory of ``topology and order'', in this paper we   continue the study of quantale-enriched categories equipped with a compact   Hausdorff topology. We compare these $V$-categorical compact Hausdorff spaces   with ultrafilter-quantale-enriched categories, and show that the presence of a   compact Hausdorff topology guarantees Cauchy completeness and (suitably   defined) ...