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Because parts of the following 1973 article have been suggestive to workers in several areas, the editors of TAC have kindly proposed to make it available in the present form. The idea on which it is based can be developed considerably further, as initiated in the 1986 article . In the second part of this brief introduction I will summarize, for those… (More)
In this article we see how already in 1967 category theory had made explicit a number of conceptual advances that were entering into the everyday practice of mathematics. For example, local Galois connections (in algebraic geometry, model theory, linear algebra, etc.) are globalized into functors, such as Spec, carrying much more information. Also,… (More)
In the last 60 years, the use of the notion of category has led to a remarkable unification and simplification of mathematics. Conceptual Mathematics, He proposed elementary book is presented, in history of mathematics diverse areas. While the notion of 'category' for this book club lawvere and to satisfy. I read over the 2nd edition's first book to satisfy… (More)
Received by the editors 2005-04-01. Transmitted by M. Hyland, A. Kock, R. Rosebrugh. Reprint published on 2005-05-23. 2000 Mathematics Subject Classification: XXX.
In May 1967 I had suggested in my Chicago lectures certain applications of category theory to smooth geometry and dynamics, reviving a direct approach to function spaces and therefore to functionals. Making that suggestion more explicit led later to elementary topos theory as well as to the line of research now known as synthetic differential geometry. The… (More)
The relation between teaching and research is partly embodied in simple general concepts which can guide the elaboration of examples in both. Notions and constructions, such as the spectral analysis of dynamical systems, have important aspects that can be understood and pursued without the complication of limiting the models to specific classical… (More)
Received by the editors 2005-04-01. Transmitted by M. Hyland, A. Kock, R. Rosebrugh. Reprint published on 2005-05-23. 2000 Mathematics Subject Classification: 18B05, 00A30, 03A05.
Axioms are proposed for the distinctive internal connectedness of a topos that models all spaces of a “general” combinatorial, algebraic, or smooth kind. It is shown that sheaves on any particular space, like representations of any particular group, do not satisfy these axioms. For each object B in a general topos, a topos of the “opposite” or “particular”… (More)