Functors
- Find the \(6^{th}\) lecture notes linked: Functors Lecture Notes 6
- Functors are morphisms from one category to another that obey nice composition properties.
- Forgetful functor destroys structure associated with a category.
- There’s difference between imposing structure and imposing conditions. Structure is specified (constructive). Properties are answers to yes/no questions. Objects and morphisms are structure whereas associativity or unitality of a morphism is a property.
- \(\textbf{Linearization Functor}\) is a functor constructed from the Category of Sets to the Category of Vector Spaces, that is, to every set, the linearization functor assigns a Vector Space.
- Adjoint functors, discovered by Daniel Kan are functors in the opposing directions which lead to the notion of weak equivalence of categories.
- Natural Transformations: We defined functors between categories to compare them. Why not define another morphism between two functors? This gives rise to Natural Transformations which are 2-morphisms. This also further leads to study of 2-categories which is a higher categroy. Categories can be thought of as graphs so they are one-dimensional. One can think of higher dimensional analogues. This has been a hard topic to study and there’s no rigorous definitions of higher categories except for 2-categories.
- Limits and co-limits, products and co-products, initial and terminal objects are some of the more specifics that can be studied.
- Please refer to the linked page of ncatLab for more details: Functor.
- Some references that can be seen in addition to those mentioned in previous post, are Tom Leinster’s Basic Category theory, Steve Awodey’s Category Theory.