Practical Foundations of Mathematics

Paul Taylor

VII. Adjunctions

INTRODUCTION

7.1    EXAMPLES OF UNIVERSAL CONSTRUCTIONS

• Colimits
• Free algebras
• Completions
• Co-universal properties
• Classifying categories

7.2    ADJUNCTIONS

• Applications
• Proof of the equivalence
• Reflections and representables

7.3    GENERAL LIMITS AND COLIMITS

• Adjoints preserve (co)limits
• Comma categories
• Equivalent colimits
• The general adjoint functor theorem

7.4    FINDING LIMITS AND FREE ALGEBRAS

• Limits and colimits in topology and order theory
• Generators and relations
• Computing colimits
• Treating relations as another theory

7.5    MONADS

• The Kleisli and Eilenberg-Moore categories
• Beck's theorem
• Applications

7.6    FROM SEMANTICS TO SYNTAX

• Encoding operations
• Canonical language
• The equivalence
• Conservativity by normalisation

7.7    GLUING AND COMPLETENESS

• The gluing construction
• Conservativity
• Existence and disjunction properties
• Exponentials

EXERCISES  VII