# Category Theory
## Core Concepts
### Categories
1. **Definition**
```math
\mathcal{C} = (Ob(\mathcal{C}), Hom(\mathcal{C}), \circ)
```
where:
- Ob is objects
- Hom is morphisms
- ∘ is composition
2. **Functors**
```math
F: \mathcal{C} \to \mathcal{D}
```
Properties:
- F(f ∘ g) = F(f) ∘ F(g)
- F(id_A) = id_{F(A)}
### Natural Transformations
1. **Definition**
```math
η: F \Rightarrow G
```
where:
- F,G are functors
- η is family of morphisms
2. **Naturality Square**
```math
η_B \circ F(f) = G(f) \circ η_A
```
where:
- f: A → B
- η_A, η_B are components
## Advanced Concepts
### Adjoint Functors
1. **Definition**
```math
F \dashv G \iff Hom_\mathcal{D}(F(A),B) \cong Hom_\mathcal{C}(A,G(B))
```
where:
- F: C → D is left adjoint
- G: D → C is right adjoint
2. **Unit and Counit**
```math
η: 1_\mathcal{C} \Rightarrow GF, \quad ε: FG \Rightarrow 1_\mathcal{D}
```
where:
- η is unit
- ε is counit
### Limits and Colimits
1. **Universal Property**
```math
\lim F = \{(x_i)_{i \in I} : F(f)(x_i) = x_j \text{ for all } f: i \to j\}
```
where:
- F is functor
- I is index category
2. **Kan Extensions**
```math
(Lan_K F)(d) = \colim_{(k,α:K(k)\to d)} F(k)
```
where:
- K is functor
- F is functor to extend
## Applications
### Algebraic Structures
1. **Monads**
```math
T: \mathcal{C} \to \mathcal{C}, \quad μ: T^2 \Rightarrow T, \quad η: 1 \Rightarrow T
```
where:
- T is endofunctor
- μ is multiplication
- η is unit
2. **Operads**
```math
P(n) \times P(k_1) \times ... \times P(k_n) \to P(k_1 + ... + k_n)
```
where:
- P(n) is n-ary operations
- × is product
### Homological Algebra
1. **Derived Functors**
```math
LF: D^-(\mathcal{A}) \to D^-(\mathcal{B})
```
where:
- D^- is derived category
- A,B are abelian categories
2. **Spectral Sequences**
```math
E^{p,q}_r \Rightarrow H^{p+q}
```
where:
- E^{p,q}_r are pages
- H^{p+q} is limit
## Implementation
### Category Theory in Code
```python
class Category:
def __init__(self,
objects: Set[Object],
morphisms: Dict[Tuple[Object, Object], Set[Morphism]]):
"""Initialize category.
Args:
objects: Objects of category
morphisms: Morphisms between objects
"""
self.objects = objects
self.morphisms = morphisms
self._validate_category()
def compose(self,
f: Morphism,
g: Morphism) -> Morphism:
"""Compose morphisms.
Args:
f: First morphism
g: Second morphism
Returns:
composition: Composed morphism
"""
if not self._composable(f, g):
raise ValueError("Morphisms not composable")
return self._compose_morphisms(f, g)
def identity(self,
obj: Object) -> Morphism:
"""Get identity morphism.
Args:
obj: Object
Returns:
id: Identity morphism
"""
return self._get_identity(obj)
```
### Functorial Computations
```python
class Functor:
def __init__(self,
source: Category,
target: Category,
object_map: Callable,
morphism_map: Callable):
"""Initialize functor.
Args:
source: Source category
target: Target category
object_map: Object mapping
morphism_map: Morphism mapping
"""
self.source = source
self.target = target
self.object_map = object_map
self.morphism_map = morphism_map
def apply(self,
x: Union[Object, Morphism]) -> Union[Object, Morphism]:
"""Apply functor.
Args:
x: Object or morphism
Returns:
Fx: Mapped object or morphism
"""
if isinstance(x, Object):
return self.object_map(x)
return self.morphism_map(x)
def compose(self,
G: 'Functor') -> 'Functor':
"""Compose with another functor.
Args:
G: Second functor
Returns:
composition: Composed functor
"""
return self._compose_functors(self, G)
```
## Advanced Topics
### Higher Categories
1. **n-Categories**
```math
k\text{-morphisms for } 0 \leq k \leq n
```
where:
- k=0 are objects
- k>0 are higher morphisms
2. **∞-Categories**
```math
\text{Hom}(x,y) \text{ is topological space}
```
where:
- x,y are objects
- Hom is mapping space
### Topos Theory
1. **Elementary Topos**
```math
\Omega: 1 \to \Omega \text{ classifies subobjects}
```
where:
- Ω is subobject classifier
- 1 is terminal object
2. **Geometric Morphisms**
```math
f^* \dashv f_* : \mathcal{F} \to \mathcal{E}
```
where:
- f* is inverse image
- f_* is direct image
## Future Directions
### Emerging Areas
1. **Higher Category Theory**
- (∞,1)-categories
- Model Categories
- Derived Geometry
2. **Homotopy Type Theory**
- Univalent Foundations
- Higher Inductive Types
- Synthetic Homotopy Theory
### Open Problems
1. **Theoretical Challenges**
- Higher Topos Theory
- Derived Categories
- Motivic Theory
2. **Practical Challenges**
- Computational Methods
- Type Theory Integration
- Software Implementation
## Related Topics
1. [[algebraic_geometry|Algebraic Geometry]]
2. [[homological_algebra|Homological Algebra]]
3. [[type_theory|Type Theory]]
4. [[logic|Logic]]