Category theory Notes
| Cat | object | morphism |
|---|---|---|
| Vect | vector space | linear maps |
| Groups | unit | group homomorphisms |
| Mfd(manifolds) | smooth manifolds | Continuous Maps |
| Top | Topological Spaces | Continuous Maps |
| Rel | Sets | Binary relations |
| Meas | Measurable Spaces | measurable maps |
| Par | sets | partial functions |
1 Basics
1.1 Isomorphisms
- Set: bijective maps
- Mfd(manifolds): diffeomorphisms
- Top : homeomorphisms which are invertible continuous map with continuous inverses.
1.2 Monomorphisms
- Set: Injective maps
- Top: Injective continuous maps
- FVect(finite-dimension Vect): injective linear map
- Group: injective group homeomorphisms
1.3 Split monomorphisms
Set: Every injective function has a left inverse. Therefore every monomorphism is split.
FVect: Every injective linear map has a left inverse. Therefore every monomorphism is split.
Rings: injective map
Top:
Generally, monomorphisms are not conceptually related to splits even though it looks the case in Set and FVect. General case is Top
1.4 Epimorphism
- Set: Surjective map
- FVect: Surjective continuous linear map
- Algebra: inclusion
- Natural number is Included in Integers
- Rings: NOT a surjective map
- Example: \(Int \rightarrow Rationals\) is a epimorphism but obviously not an surjective map
1.5 Split Epimorphism
2 Functors
2.1 Cocycle
Functors are just 1-cocycles
3 Yoneda Embedding Theorem
3.1 Matrix Example
A matrix transformation through row operation is a natural transformation between presheaves.
Matrix transformation by row operation is isomorphic to a matrix
4 PreSheaf
A functor that takes a dual morphism to set category.
\[f^{OP} \Rightarrow Set\]
5 Applicative
6 Natural
- All functors are endofunctors
functor :: Type* => Type*wrt to codingList :: Type* => Type*Maybe :: Type* => Type*
Below is a natural transformation wrt Object a on functors List and Maybe
Natural Transformation can be considered a family of morphisms between functors wrt an object or morphism.
g :: ∀a b. a -> b
naturaltransformation_a :: ∀ a. List a -> Maybe a6.1 Naturality square
7 Algebras
3 equivalent ways to define an algebra
\[(\mathbb{N},0::\top \rightarrow \mathbb{N},(+)::\mathbb{N}\times\mathbb{N} \rightarrow \mathbb{N})\] \[(\mathbb{N},in::\top + \mathbb{N}\times\mathbb{N} \rightarrow \mathbb{N}) \tag{fuse + and 0}\] \[let\ F X = \top + X \times X \qquad (\mathbb{N}, in::F\ \mathbb{N} \rightarrow \mathbb{N})\tag{package into functor}\]
Other examples of an algebra
- \((\mathbb{B},true::\top \rightarrow \mathbb{B},(\equiv)::\mathbb{B}\times\mathbb{B}\rightarrow\mathbb{B})\)
- \((\mathbb{B},false::\top \rightarrow \mathbb{B},(\lor)::\mathbb{B}\times\mathbb{B}\rightarrow\mathbb{B})\)
F Algebras are just a set with operations that return values in that set
Inductive datatypes correspond to initial algebras.
The homomorphisms are catamorphisms or folds.
