Badiou Alain. Mathematics of the Transcendental: Onto-logy and being-there

London: Bloomsbury Academic, 2014 — 296 p. — ISBN10: 1441189246; ISBN13: 978-1441189240. Edited, translated and with an introduction by A. J. Bartlett and Alex Ling.In Mathematics of the Transcendental, Alain Badiou painstakingly works through the pertinent aspects of category theory, demonstrating their internal logic and veracity, their derivation and distinction from set theory, and the 'thinking of being'. In doing so he sets out the basic onto-logical requirements of his greater and transcendental logics as articulated in his magnum opus, Logics of Worlds. Previously unpublished in either French or English, Mathematics of the Transcendental provides Badiou's readers with a much-needed complete elaboration of his understanding and use of category theory. The book is vital to understanding the mathematical and logical basis of his theory of appearing as elaborated in Logics of Worlds and other works and is essential reading for his many followers.Translator's IntroductionTopos, or Logics of Onto-logy: An Introduction for PhilosophersGeneral aim
First definitions
The size of a category
Limit and universality
Some fundamental concepts
Duality
Isomorphism
Exponentiation
Universe 1: closed Cartesian categories
Structures of immanence 1: philosophical grounds
Immanence 2: sub-object
Immanence 3: elements of an object
'Elementary' clarification of exponentiation
Logic 1: central object (or sub-object classifier)
True, false, negation and more
Central object as linguistic power
Universe 2: the concept of Topos
Ontology of the void and of difference
Mono., Epi., Iso., Equa., and other arrows
Topoi as logical places
Internal algebra of 1
Ontology of the void and excluded middle
A classical miniature
A non-classical miniatureBeing-ThereTranscendental structuresTranscendental connections
Of transcendental connections and logic in its usual sense (propositional logic and first order logic of predicates)
Transcendental connections and the general theory of localisations: topologyTheory of appearing and of objectivity
Transcendental projections: theory of localisation
Theory of relations. The status of worlds