## From a Geometrical Point of View: A Study of the History and Philosophy of Category TheoryFrom a Geometrical Point of View explores historical and philosophical aspects of category theory, trying therewith to expose its significance in the mathematical landscape. The main thesis is that Klein’s Erlangen program in geometry is in fact a particular instance of a general and broad phenomenon revealed by category theory. The volume starts with Eilenberg and Mac Lane’s work in the early 1940’s and follows the major developments of the theory from this perspective. Particular attention is paid to the philosophical elements involved in this development. The book ends with a presentation of categorical logic, some of its results and its significance in the foundations of mathematics. From a Geometrical Point of View aims to provide its readers with a conceptual perspective on category theory and categorical logic, in order to gain insight into their role and nature in contemporary mathematics. It should be of interest to mathematicians, logicians, philosophers of mathematics and science in general, historians of contemporary mathematics, physicists and computer scientists. |

### What people are saying - Write a review

This is a brilliant book. It explains the unexpected confluences and unifications of category theory as an extension of the Kleinian Erlangen program. It feels like the application of group transformations that "creates" space is simply setting out specific conditions generally without exception yields space. This to me is not merely an implication of theory to all potential practices, but the implicative order generated by the constraints of the transformations. How this avoids tautology and moves swiftly to show that various "complete" views are interchangeable, making them all party to a type of trivial pursuit makes me wonder whether it might not be better for the adoption of category theory into other fields of knowledge that the ultimate definition of morphism should not be changed to a chimera morphism. See, Efferman's Adjoint Functor paper. A chimera morphism at the very foundation of the theory then would have the conceptual seed to graph many analogies to ordinary and extraordinary phenomena. It would also make it easier, more convenient to accept that an asymmetric structure--the arrow--and its impetuous--is really about both "identity" [A is A] and difference [A is not-A]. Leibniz presumed such as core to his mathesis universalis. This intuition is captured and subsumed by the more general and inclusive f: A-> B.

### Contents

9 | |

12 | |

15 | |

The Irrelevance of the Nature of the Elements of a Space | 25 |

124 Why a Transformation Group is not Quite Enough | 28 |

125 But Then Again why a Group is Enough | 29 |

126 Classifying Geometries | 31 |

13 Logical Remarks | 32 |

Adjoint Functors What They are What They Mean | 147 |

51 Adjointness | 148 |

52 Equivalence of Categories Again | 161 |

53 Back to Klein | 164 |

54 From Groups to Groupoids | 166 |

55 The Foundations of Category Theory Again | 175 |

Invariants in Foundations Algebraic Logic | 191 |

61 Lawveres Thesis | 194 |

14 Main Ontological and Epistemological Consequences of Kleins Program | 34 |

Formal Supervenience and Reduction | 36 |

16 Summing Up | 39 |

Introducing Categories Functors and Natural Transformations | 41 |

21 From a Transformation Group to the Algebra of Mappings | 44 |

22 Foundations of Category Theory | 51 |

An Argument Against the Foundational Status of Category Theory | 54 |

24 At Last Natural Transformations | 60 |

25 Extending Kleins Program in the Wrong Direction | 64 |

The First Phase 19451958 | 67 |

Categories as Spaces Functors as Transformations | 73 |

31 Universal Morphisms | 74 |

Doing Duality without Elements | 77 |

312 Universal Morphisms | 86 |

32 Grothendieck and Abelian Categories | 90 |

321 Abelian Categories | 92 |

322 Representable Functors | 102 |

Discovering Fundamental Categorical Transformations Adjoint Functors | 109 |

Homotopy Theory and Category Theory | 114 |

42 Kans Discovery | 125 |

43 Kans 1958 Papers Adjoint Functors | 132 |

62 The Category of Categories as a Foundational Framework | 197 |

63 The Elementary Theory of the Category of Sets | 208 |

the Program | 210 |

65 An Adjoint Presentation of Propositional Logic | 216 |

66 Quantiﬁers as Adjoint Functors | 220 |

Sketches | 225 |

Conceptual and Generic Structures | 234 |

69 Summing Up | 246 |

Invariants in Foundations Geometric Logic | 247 |

Generalized Spaces | 248 |

72 Elementary Toposes | 261 |

73 Invariants Under Geometric Transformations | 267 |

74 Invariants Under Logical Transformations | 271 |

75 Invariant Foundational Frameworks | 276 |

76 Using Geometric and Logical Invariants | 282 |

77 Summing Up | 283 |

Conclusion | 285 |

291 | |

303 | |

### Other editions - View all

From a Geometrical Point of View: A Study of the History and Philosophy of ... Jean-Pierre Marquis No preview available - 2010 |

From a Geometrical Point of View: A Study of the History and Philosophy of ... Jean-Pierre Marquis No preview available - 2009 |

From a Geometrical Point of View: A Study of the History and Philosophy of ... Jean-Pierre Marquis No preview available - 2010 |