Categorical foundations of variety-based bornology

Solomon Feferman Categorical Foundations and Foundations of Category Theory*

This paper is divided'into two parts. Part I deals briefly with the thesis that category theory (or something like it) should provide the proper foundations of mathematics, in preference to current foundational schemes. An opposite view is argued here on the grounds that the notions of operation and collection are prior to all structural notions. However, no position is taken as to whether such...

Reflections on a categorical foundations of mathematics

We examine Gödel’s completeness and incompleteness theorems for higher order arithmetic from a categorical point of view. The former says that a proposition is provable if and only if it is true in all models, which we take to be local toposes, i.e. Lawvere’s elementary toposes in which the terminal object is a nontrivial indecomposable projective. The incompleteness theorem showed that, in the...

Categorical Foundations of Distributed Graph Transformation

A distributed graph (N, D) consists of a network graph N and a commutative diagram D over the scheme N which associates local graphs D(ni) and graph morphisms D(e) : D(n1) → D(n2) to nodes n1, n2 and edges e : n1 → n2 in N . Although there are several interesting applications of distributed graphs and transformations, even the basic pushout constructions for the double pushout approach of distr...

Domain Theory for Concurrency —New Categorical Foundations

Categorical foundations for randomly timed automata

The general theory of randomly timed automata is developed: starting with the practical motivation and presentation of the envisaged notion, the categorical theory of minimization, aggregation, encapsulation, interconnection and realization of such automata is worked out. All these constructions are presented universally: minimization and realization as adjunctions, aggregation as product, inte...