Category Theory

Category Theory is the study of abstractions of composition, otherwise known as Categories.

Understanding What Categories Are

Set Theory Composition

In order to understand what Categories (abstractions of composition) are, we first need to understand how Composition works in Set Theory. Note that set theory composition is a particular Category.

Composition

The act of combining the application of multiple Functions into one function.

• Age is a function that maps People to Integers
• GreaterThan18 is a function that maps Integers to Booleans
• GreaterThan18 $\circ$ Age is the Composition of the above functions that maps People to Booleans
  • It is equivalent to applying Age, then applying GreaterThan18
  • It can be thought of as: GreaterThan18 follows Age
  • GreaterThan18 $\circ$ Age = Age(GreaterThan18(…))
• id is the Identity Morphism, which is depicted above to map People to People
  • It exists for all sets, but is omitted for the others

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Categories

In Category Theory, the above functionality is generalized through Categories.

Category

A Category consists of a collection of

• Objects
• Morphisms

where there is a definition for

• Composition
• Identity Morphism

• $M_{ab}$ is a Morphism from $A$ to $B$
• $M_{bc}$ is a Morphism from $B$ to $C$
• $M_{bc}$ $\circ$ $M_{ab}$ is the Composition of the above morphisms that maps $A$ to $C$
  • It is equivalent to applying $M_{ab}$, then applying $M_{bc}$
  • It can be thought of as: $M_{bc}$ follows $M_{ab}$
  • Composition is Associative $\to (f\circ g)\circ h=f\circ (g\circ h)$
• id is the Identity Morphism, which is depicted above to map $A$ to $A$
  • It exists for all Objects, but is intentionally omitted for $B$ and $C$
• Think of a Category as a generalized version of Elements, Functions, and Operations in Set Theory.

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Morphisms

Morphism

An abstraction of a process or relationship between Objects, a more generalized version of a Function, that acts as a Structure-preserving Map

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The Identity Morphism

Identity Morphism

The morphism mapping Object a Category to itself

• Every object has a identity morphism

Consider the identity of category $X$

$$\begin{array}{c}{\text{id}}_X:X\to X\\ f\circ {\text{id}}_X=f={\text{id}}_X\circ f\end{array}$$

• Depending on the context, the subscript may not be neccessary

Identity Composition

Composing with the identity has no effect.

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Composition

Composition

• $M_{ab}$ is a Morphism from $A$ to $B$
• $M_{bc}$ is a Morphism from $B$ to $C$
• $M_{bc}$ $\circ$ $M_{ab}$ is the Composition of the above functions that maps $A$ to $C$
  • It is equivalent to applying $M_{ab}$, then applying $M_{bc}$
  • It can be thought of as: $M_{bc}$ follows $M_{ab}$
  • Composition is Associative $\to (f\circ g)\circ h=f\circ (g\circ h)$

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Comparison

• In category theory, we study objects not by looking inside of them, but by analyzing the arrows betweeen them and other objects; by studying their relationships to other objects in the category. In fact, we cannot look inside the objects, because we have not defined the properties of that object. We only know objects by the arrows they emit and receive. If we were to “look inside” an object, we would have to define it as a specific object, like a set or something, which would lose the generality of the category, which doesn’t neccessarily need to have a set there. The whole point of CT is that it is very generalized.
• Imagine a group of people, where arrows between the people indicates A is a friend of B. We cannot look inside the people to see their personality, traits, etc, but we can determine who is most popular just from looking at the arrows.

Isomorphism

Isomorphism

Formal Definition

An Isomorphism is a kind of Morphism between objects in a Category that is Invertible.

Let $f_{AB}$ be a Morphism and $A,B$ be objects

$$\begin{array}{c}\exists f_{AB}:A\to B,f_{AB}^{-1}:B\to A\\ f_{AB}^{-1}\circ f_{AB}={\text{id}}_A\wedge f_{AB}\circ f_{AB}^{-1}={\text{id}}_B\\ ⟺\\ A\cong B\end{array}$$

Isomorphism Facts

• Isomorphisms are unique
• If there exists an isomorphism between objects, then those objects are isomorphic with each other
• If $A$ and $B$ are isomorphic, then $A\cong B$
• When objects are isomorphic, we can say that they have the same internal structure, without having to actually look inside the objects
• For example, as Vector Spaces: $\mathbb{C}\cong {\mathbb{R}}^2$ (not as Algebraic Fields though)

Isomorphism of Categories

Categories contain Objects, and Objects can be anything, including Categories!

• Functors connect Categories
• Morphisms connect Objects
• Our Objects are Categories
• Thus, our morphisms are functors The Objects of a Categories being Categories themselves, is just a special case of Category. Therefore, the same general Formal Definition applies. However, it can also be reexpressed in terms of functors, where $\mathcal{A},\mathcal{B}$ are categories, and $F_{\mathcal{A}\mathcal{B}}$ is a functor.

$$\begin{array}{c}\exists F_{\mathcal{A}\mathcal{B}}:\mathcal{A}\to \mathcal{B},F_{\mathcal{A}\mathcal{B}}^{-1}:\mathcal{B}\to \mathcal{A}\\ F_{\mathcal{A}\mathcal{B}}^{-1}\circ F_{\mathcal{A}\mathcal{B}}={\text{id}}_{\mathcal{A}}\\ F_{\mathcal{A}\mathcal{B}}\circ F_{\mathcal{A}\mathcal{B}}^{-1}={\text{id}}_{\mathcal{B}}\\ ⟺\\ \mathcal{A}\cong \mathcal{B}\end{array}$$

Size Intuition

• When there is a notion of size, this invertibility can be thought of as a Bijective Morphism between a pair of Objects; a relationship where one object can be converted and restored to and from another object.
• A Bijective Homomorphism.

Consider the Arrows to and from category $A$ to $B$, that are Bijective because $A$ and $B$ have the size

$$\begin{array}{c}{f}_{AB}:A\to B\\ f_{BA}:B\to A\\ f_{AB}^{-1}=f_{BA}\\ f_{AB}^{-1}\circ f_{AB}={\text{id}}_A\\ f_{AB}\circ f_{AB}^{-1}={\text{id}}_B\end{array}$$

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Equivalence

Equivalence

Formal Definition

Let $\mathcal{A}$ and $\mathcal{B}$ be Categories, Let $\mathcal{A}\simeq \mathcal{B}\equiv$ ”$\mathcal{A}$ and $\mathcal{B}$ are equivalent”, Let $F_{\mathcal{A}\mathcal{B}}$ be a Functor

$$\begin{array}{c}\exists F_{\mathcal{A}\mathcal{B}}:\mathcal{A}\to \mathcal{B},F_{\mathcal{B}\mathcal{A}}:\mathcal{B}\to \mathcal{A}\\ F_{\mathcal{B}\mathcal{A}}\circ F_{\mathcal{A}\mathcal{B}}\cong {\text{id}}_{\mathcal{A}}\\ F_{\mathcal{A}\mathcal{B}}\circ F_{\mathcal{B}\mathcal{A}}\cong {\text{id}}_{\mathcal{B}}\\ ⟺\\ \mathcal{A}\simeq \mathcal{B}\end{array}$$

• This definition is very similar to that of Isomorphism of Categories

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Adjunction

Adjunction

Formal Definition

Adjunction is a relationship only between Functors. Only functors can be Adjoint to each other.

• Let $\mathcal{A}$ and $\mathcal{B}$ be Categories
• Let $L$ and $R$ be Functors.
  • Left Adjoint (Functor)
    • $L:\mathcal{A}\to \mathcal{B}$
  • Right Adjoint (Functor)
    • $R:\mathcal{B}\to \mathcal{A}$
• Let $A\in \text{obj}(\mathcal{A})$
• Let $B\in \text{obj}(\mathcal{B})$

$${\text{hom}}_{\mathcal{B}}(L(A),B)={\text{hom}}_{\mathcal{A}}(A,R(B))$$Link to original

Universal Construction

Construction

A process for building new Mathematical Objects from existing ones.

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Universal Property

A rigourous way of saying that a specific Construction is the quintessential way of satisfying a a certain condition, that all other Objects satisfying said condition must obey.

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Commutative Diagram

A Category diagram where all paths of arrows lead to the same result.

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Universal Construction

A Construction that satisfies a Universal Property

• Can be applied to expose relationships existing in other Categories

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Universal Objects

Universal Object

An Object that has a Universal Property

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Terminal Object

todo

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Initial Object

todo

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Product Object

todo

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Projection

todo

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Sum Object

todo

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Exponential Object

todo

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