This paper arises from work carried out at the conference on Geometry and Topology in GG okova, Turkey in May 1993 and we should like to start by thanking the organisers of this excellent conference.

For a complete lattice V which, as a category, is monoidal closed, and for a suitable Setmonad T we consider (T,V)-algebras and introduce (T,V)-proalgebras, in generalization of Lawvere’s presentation of metric spaces and Barr’s presentation of topological spaces. In this lax-algebraic setting, uniform spaces appear as proalgebras. Since the corresponding categories behave functorially both in ...

Dialectica categories are a very versatile categorical model of linear logic. They have been used to model many seemingly different systems, for example Petri nets and the Lambek calculus. In this note, we expand our previous work on fuzzy Petri nets to deal with fuzzy topological systems. One basic idea is to use as the dualizing object in the Dialectica categories construction, the unit real ...

Inspired by a construction of Escardó, Lawson, and Simpson, we give a general construction of C-generated objects in a topological construct. When C consists of exponentiable objects, the resulting category is Cartesian-closed. This generalizes the familiar construction of compactly-generated spaces, and we apply this to Krishnan’s categories of streams and prestreams, as well as to Haucourt st...

This paper concerns open, closed, and open-closed topological conformal field theories. We show that the category of open topological conformal field theories, at all genera, is homotopy equivalent to a category of Calabi-Yau A∞ categories. For each such, we show that there is a universal closed TCFT, which is the initial element in the category of compatible open-closed TCFTs. The homology of ...

5 Goguen’s category of V-sets whose objects are functions with values in a pre-ordered set and morphisms are suitable maps is shown to be a topological construct if and only if the pre-ordered set is a complete lattice; in particular the category Set(L) of L-sets, 7 also considered by Goguen, is topological over Set. The special case when the considered pre-ordered set is L with the “mapping to...

It has been common wisdom among mathematicians that Extended Topological Field Theory in dimensions higher than two is naturally formulated in terms of n-categories with n > 1. Recently the physical meaning of these higher categorical structures has been recognized and concrete examples of Extended TFTs have been constructed. Some of these examples, like the Rozansky-Witten model, are of geomet...

The categorical theory of closure operators is used to characterize the epimorphisms in certain categories of separated fuzzy topological spaces (in the sense of Lowen). These include the 0-T0-spaces of Wuyts and Lowen; the FTS-spaces of Ghanim, Kerre and Mashhour; and the α-T2-spaces of Rodabaugh.