A group–category is an additively semisimple category with a monoidal product structure in which the simple objects are invertible. For example in the category of representations of a group, 1–dimensional representations are the invertible simple objects. This paper gives a detailed exploration of “topological quantum field theories” for group–categories, in hopes of finding clues to a better u...

Schanuel has pointed out that there are mathematically interesting categories whose relationship to the ring of integers is analogous to the relationship between the category of finite sets and the semi-ring of non-negative integers. Such categories are inherently geometrical or topological, in that the mapping to the ring of integers is a variant of Euler characteristic. In these notes, I sket...

The Baire theory of category, which classifies sets into two distinct categories, is an important topic in the study of metric spaces. Many results in topology arise from category theory; in particular, the Baire categories are related to a topological property. Because the Baire Category Theorem involves nowhere dense sets in a complete metric space, this paper first develops the concepts of n...

Weak factorization systems, important in homotopy theory, are related to injective objects in comma{categories. Our main result is that full functors and topological functors form a weak factorization system in the category of small categories, and that this is not coobrantly generated. We also present a weak factorization system on the category of posets which is not coobrantly generated. No s...

Our work is a fundamental study of the notion of approximation inQ-categories and in (U,Q)-categories. Our exposition is categorical but kept as close to language of domain theory as possible. Consequently, we introduce auxiliary, approximating and Scott-continuous distributors, the way-below distributor, and continuity of Qand (U,Q)-categories. We fully characterize J-continuous Q-categories (...

1. Terminology. V and M will be differentiable manifolds of dimension n and m respectively; differentiable meaning always of class C. For simplicity, we assume V compact and without boundary. We shall have to consider several categories of maps: (1) the category of continuous maps, (2) the category of topological imbeddings, (3) the category of topological immersions: a map ƒ: F—>M is a topolog...

Our work is a foundational study of the notion of approximation inQ-categories and in (U,Q)-categories, for a quantale Q and the ultrafilter monad U. We introduce auxiliary, approximating and Scott-continuous distributors, the way-below distributor, and continuity of Qand (U,Q)-categories. We fully characterize continuous Q-categories (resp. (U,Q)-categories) among all cocomplete Q-categories (...

We define a notion of morphism for quotient vector bundles that yields both a category QVBun and a contravariant global sections functor C : QVBun → Vect whose restriction to trivial vector bundles with fiber F coincides with the contravariant functor Top → Vect of F -valued continuous functions. Based on this we obtain a linear extension of the adjunction between the categories of topological ...

We introduce continuous Frobenius categories. These are topological categories which are constructed using representations of the circle over a discrete valuation ring. We show that they are Krull-Schmidt with one indecomposable object for each pair of (not necessarily distinct) points on the circle. By putting restrictions on these points we obtain various Frobenius subcategories. The main pur...

We present a biologically inspired computational framework for language processing and grammar acquisition, called the hierarchical prediction network (HPN). HPN fits in the tradition of connectionist models, but it extends their power by allowing for a substitution operation between the nodes of the network. This, and its hierarchical architecture, enable HPN to function as a full syntactic pa...