Let Var be the category of complex algebraic varieties, Top that of nice topological spaces, Dab be the derived category of finite complexes of finitely generated abelian groups. One has tensor functors Var → Top → Dab, the first assigns to a variety its space equipped with the classical topology, the second one is the singular chain complex functor (the tensor structure for the first two categ...

The (1+1)-dimensional cobordism category of closed 1-dimensional manifolds and oriented surfaces is a most basic mathematical structure. It has played a fundamental role in string theory and conformal field theory. We give a brief account of its structure and indicate what structure it gives vector spaces and other algebraic objects on which it (or an embellished version of it) acts. We explain...

We use modified traces to renormalize Lyubashenko’s closed 3-manifold invariants coming from twist non-degenerate finite unimodular ribbon categories. Our construction produces new topological which we upgrade 2 + 1-TQFTs under the additional assumption of factorizability. The resulting functors provide monoidal extensions mapping class group representations, as discussed in De Renzi et al. (Co...

This paper establishes two new connections between the familiar function ring functor ${mathfrak R}$ on the category ${bf CRFrm}$ of completely regular frames and the category {bf CR}${mathbf sigma}${bf Frm} of completely regular $sigma$-frames as well as their counterparts for the analogous functor ${mathfrak Z}$ on the category {bf ODFrm} of 0-dimensional frames, given by the integer-valued f...

We find a criterion for a morphism of coalgebras over a Barr-exact category to be effective descent and determine (effective) descent morphisms for coalgebras over toposes in some cases. Also, we study some exactness properties of endofunctors of arbitrary categories in connection with natural transformations between them as well as those of functors that these transformations induce between co...

In Lecture 4, we learned about an algebraic method for describing and classifying structures. In this lecture, we look at using algebra to find combinatorial descriptions of topological spaces. We begin by looking at an equivalence relation called homotopy that gives a classification of spaces that is coarser that homeomorphism, but respects the finer classification. That is, two spaces that ha...

The term Stone-type duality often refers to a dual equivalence between category of lattices or other partially ordered structures on one side and topological the other. This paper is part larger endeavour that aims extend web dualities from metric and, more generally, quantale-enriched categories. In particular, we improve our previous work show how certain results for categories [ 0 , 1 ] -enr...