Cut Elimination in Categories

1. Front Matter

   Pages i-xii

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2. Introduction

   • Kosta Došen

   Pages 1-16

3. Categories

   • Kosta Došen

   Pages 17-52

4. Functors

   • Kosta Došen

   Pages 53-69

5. Natural Transformations

   • Kosta Došen

   Pages 70-80

6. Adjunctions

   • Kosta Došen

   Pages 81-148

7. Comonads

   • Kosta Došen

   Pages 149-194

8. Cartesian Categories

   • Kosta Došen

   Pages 195-218

9. Conclusion

   • Kosta Došen

   Pages 219-220

10. Back Matter

    Pages 221-229

    PDF

Proof theory and category theory were first drawn together by Lambek some 30 years ago but, until now, the most fundamental notions of category theory (as opposed to their embodiments in logic) have not been explained systematically in terms of proof theory. Here it is shown that these notions, in particular the notion of adjunction, can be formulated in such as way as to be characterised by composition elimination. Among the benefits of these composition-free formulations are syntactical and simple model-theoretical, geometrical decision procedures for the commuting of diagrams of arrows. Composition elimination, in the form of Gentzen's cut elimination, takes in categories, and techniques inspired by Gentzen are shown to work even better in a purely categorical context than in logic. An acquaintance with the basic ideas of general proof theory is relied on only for the sake of motivation, however, and the treatment of matters related to categories is also in general self contained. Besides familiar topics, presented in a novel, simple way, the monograph also contains new results. It can be used as an introductory text in categorical proof theory.

• Cut-elimination theorem
• category theory
• logic
• proof
• proof theory

• University of Toulouse III, Toulouse, France

  Kosta Došen

• Mathematical Institute, Belgrade, Yugoslavia

  Kosta Došen

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