• Corpus ID: 115163041

Categories of categories

  title={Categories of categories},
  author={Claudio Pisani},
  journal={arXiv: Category Theory},
  • C. Pisani
  • Published 6 September 2007
  • Philosophy
  • arXiv: Category Theory
A certain amount of category theory is developed in an arbitrary finitely complete category with a factorization system on it, playing the role of the comprehensive factorization system on Cat. Those aspects related to the concepts of finality (in particular terminal objects), discreteness and components, representability, colimits and universal arrows, seem to be best expressed in this very general setting. Furthermore, at this level we are in fact doing not only (E,M)-category theory but, in… 
Balanced category theory
Some aspects of basic category theory are developed in a finitely complete category $\C$, endowed with two factorization systems which determine the same discrete objects and are linked by a simple


The comprehensive factorization of a functor
In this article we show that every functor has a factorization into an initial functor followed by a discrete O-fibration and that this factorization is functorial. Size considerations will be
Adjointness in Foundations
This article sums up a stage of the development of the relationship between category theory and proof theory and shows how already in 1967 category theory had made explicit a number of conceptual advances that were entering into the everyday practice of mathematics.
The Category of Categories as a Foundation for Mathematics
In the mathematical development of recent decades one sees clearly the rise of the conviction that the relevant properties of mathematical objects are those which can be stated in terms of their
We illustrate the formula (#p)x = !(x/p), which gives the reflection #p of a category p : P ! X over X in discrete fibrations. One of its proofs is based on a "complement operator" which takes a
The nature of the spatial background for classical analysis and for modern theories of continuum physics requires more than the partial invariants of locales and cohomology rings for its description.
Foundations and Applications: Axiomatization and Education
The possibility that other methods may be needed to clarify a contradiction introduced by Cantor, now embedded in mathematical practice, is discussed in section 5.
Local homeomorphisms via ultrafilter convergence
Using the ultrafilter-convergence description of topological spaces, we generalize Janelidze-Sobral characterization of local homeomorphisms between finite topological spaces, showing that local
Connected components and colimits
Components, Complements and Reflection Formulas, preprint, math
  • 2007
Functorial Semantics of Algebraic Theories, Phd Thesis, republished in Reprints in Theory and Appl
  • 1963