Category Theory
Micheal F Atiyah has described mathematics as the “Science of Analogy”. In this view, Category Theory is mathematical analogy, it provides a cross disciplinary language for mathematics designed to delineate general phenomena, which enable the transfer of ideas from one theory to another.
Notes
Categories
- Category
- Isomorphisms
- Opposite Category
- Duality
- Monomorphisms and Epimorphisms
- Functors
- Natural Transformation
- Equivalence of Categories
- Categorization of Equivalent Categories
- Initial, Terminal and Zero Objects
- Concrete Categories
- Commuting of Rectangles and Squares
- Vertical and Horizontal Composition of Natural Transformations
- 2-Categories
Universal Properties
- Representable Functors
- Examples of Natural Transformation with Representable Functor as Domain
- Yoneda Lemma
- Universal Property (Mac Lane)
- Universal Property (Riehl)
- Equivalence of Definitions of Universal Property
- Element Category
Limits and Colimits
- Diagram
- Cones and Cocones
- Limits and Colimits
- Complete and Cocomplete Categories
- Set is Complete
- Small Limits in Set are Equalizers
- Preservation, Reflection and Creation of Limits
- Strictly Creating Limits
- Slice Category Strictly Creates Limits
- Functor Categories inherit Limits and Colimits object-wise
- Representable Universal Property of Limits
- Representable Universal Property of Colimits
- Any Category with Coproducts and Coequalizers is Cocomplete, with Products and Equalizers is Complete
- Any category with Pullback and a terminal object has all finite limits, with pushouts and an initial object has all finite colimits
- A Categorical notion of Equivalence Relation
Complete and Cocomplete Categories
- Set is Complete and Cocomplete
- Top is Complete and Cocomplete
- If C is Complete and Cocomplete, so are the slice and coslice categories over C
- Cat and CAT are complete and Cocomplete
- A Poset is a complete and cocomplete category iff it is a complete lattice
Functoriality of Limits
- Choosing Limits of diagrams in Functorial
- A natural isomorphism between diagrams induces an isomorphism between limits and colimits
Size Matters
Limits and Colimts Commuting
- Limits and Colimits in multiple variables can be taken in any order
- Colimit of Limits gives Limit fo Colimit of a Bifunctor
- Filtered Colimits commute with finite limits in Set
Adjunctions
Unit and Counit
Contravariant and Multivariable Adjoint Functors
- Dualling Both Categories in an Adjunctions
- Mutual Left and Right Adjunctions
- Unique construction of Adjunction
- Adjunction of Bifunctors
- Two Variable Adjunction
Calculus of Adjunctions
- Left and Right Adjoints are Unique
- Composition of Adjunctions
- Any Equivalence can be promoted to an Adjunction
- Adjunctions in Functor Categories
Adjunctions, limits and colimits
- Limits and Colimits as Adjunctions
- RAPL
- Reflective Subcategory
- Fully Faithful Functors and Adjunctions
Existence of Adjoint functors
- Adjoints of Inclusion functor from Ring to Rng
- A functor admits a left adjoint iff all its comma categories have an initial object
- Continuous and Cocontinuous Functor
- Forgetful functor from comma category strictly creates limits
- Weakly initial objects and joint weakly initial sets
- A complete, locally small category with a jointly weakly initial set of objects has an initial object
- General Adjoint Functor Theorem
- Separating and Coseparating sets
- Subobjects
- A locally small, complete category with a small coseparator and intersections of all collections of subobjects has an initial object
- Special Adjunct Functor Theorem
- Stone Cech Compactification from Special Adjunct Functor Theorem
- Locally Small and Complete Categories with a small coseparating set where all collections of subobjects of a fixed object have an intersection then it is cocomplete
- Locally Small, Complete category with a small coseparator where all collections of subobjects of an object have an intersection has all continuous functors form it be representable
- Freyd’s Representability Theorem
- locally presentable categories
- Adjoint Functor Theorem for locally presentable categories
- Constructing the Left Adjoint of Inclusion Functor from Ring to Rng