In the previous lecture, we outlined some approaches to describing the cohomology of the classifying space of G-bundles M on a Riemann surface X. For example, we asserted that the cochain complex C * (M; Q) is quasi-isomorphic to a continuous tensor product ⊗ x∈X C * (BG x ; Q). Here it is vital that we work at the level of cochains, rather than cohomology: there is no corresponding procedure t...

1. The problem. The phenomenology of living systems (i.e., their relation to the physical world) does not yet have a satisfactory theory. This is not just a philosophical issue. In order to emulate the skill and subtlety of biological computation in our own machines and systems, we need to understand what are the means by which such veridicality is obtained. Any person is capable of amazing fea...

In [Fokkinga, 1992a], [Fokkinga, 1992b] and [Fokkinga and Meertens, 1994] a calculational approach to category theory is developed. The scheme has many merits, but sacrifices useful type information in the move to an equational style of reasoning. By contrast, traditional proofs by diagram pasting retain the vital type information, but poorly express the reasoning and development of categorical...

Categories are not invariant. This paper attempts to explore the dynamic nature of semantic category, in particular, that of fashion language, based on the cognitive theory of Dawkins’ memetics, a new theory of cultural evolution. Semantic attributes of linguistic memes decrease or proliferate in replication and spreading, which involves a dynamic development of semantic category. More specific...

We define the notions of Azumaya category and Brauer group in category theory enriched over some very general base category V. We prove the equivalence of various definitions, in particular in terms of separable categories or progenerating bimodules. When V is the category of modules over a commutative ring R with unit, we recapture the classical notions of Azumaya algebra and Brauer group and ...

The present note is a recollection of the most striking and useful applications of co/end calculus. We put a considerable effort in making arguments and constructions rather explicit: after having given a series of preliminary definitions, we characterize co/ends as particular co/limits; then we derive a number of results directly from this characterization. The last sections discuss the most i...

Although numerous contributions from divers authors, over the past fifteen years or so, have brought enriched category theory to a developed state, there is still no connected account of the theory, or even of a substantial part of it. As the applications of the theory continue to expand - some recent examples are given below - the lack of such an account is the more acutely felt. The present b...

in this paper, we discuss the equivalent conditions of pretopological and topological $l$-fuzzy q-convergence structures and define $t_{0},~t_{1},~t_{2}$-separation axioms in $l$-fuzzy q-convergence space. {furthermore, $l$-ordered q-convergence structure is introduced and its relation with $l$-fuzzy q-convergence structure is studied in a categorical sense}.

This essay is an attempt to sketch the evolution of type theory from its beginnings early in the last century to the present day. Central to the development of the type concept has been its close relationship with set theory to begin with and later its even more intimate relationship with category theory. Since it is effectively impossible to describe these relationships (especially in regard t...