Category theory

Languages such as English have nouns and verbs. A noun identifies an object while a verb identifies an action or process. Thus the sentence "Please lift the tray." conjures an image of a tray on a table, a person who can lift it and the tray in its elevated position.

In a pocket calculator, a datum is a number or pair of numbers. The calculator has a selection of operations which can be performed. Given the number 5, pressing the "square" key produces the number 25.

The mathematical abstraction drawn from these and more sophisticated examples is based on two concepts: objects and the things which act on objects. In category theory, the thing which acts upon an object to produce another object is called a map or morphism. When diverse mathematical structures are recognized to be categorial objects and the relationships between objects to be categorial morphisms, the theory expresses features shared by diverse subjects. For example, arithmetic has the product of a pair of numbers, set theory has the Cartesian product of a pair of sets and logic has the conjunction of a pair of assertions. These three products and many others are instances of the categorial product. Furthermore, the product is a specific instance of a limit and a limit is a specific universal property. By such generalizations, category theory unifies mathematics at a high level of abstraction.

Morphisms can be composed. In the first example the tray can be lifted L and then rotated R. Composition simply means that two actions such as L and R can be thought of as combined into a single action R∘L. The symbol ∘ denotes composition.

Morphisms are associative. Think of three motions of the tray.

L: Lifting of the tray 10 cm above the table.

R: Rotation of the tray 180 degrees clockwise.

S: Shifting of the tray 1 m north while maintaining the elevated position.

The lift and rotation can be thought of as combined into a single motion followed by the shift; denoted S∘(R∘L). Alternatively, the rotation and shift can be thought of as a single motion following the lift: (S∘R)∘L. Associativity simply means that S∘(R∘L) = (S∘R)∘L.

An identity motion is any motion which brings the tray back to a starting position. If M denotes lowering the tray 10 cm then M∘L is an identity motion. The identity rule in the formal definition states that any morphism preceded or followed by the identity is equal to the morphism alone.

The formal definition embodies the preceding concepts in concise mathematical notation.

Formal Definition

A category consists of the following data:

1. A class of "objects," denoted $ob(C)$
2. For objects $A,B,C\in ob(C)$, a set ${\text{Mor}}_{C}(A,B)$ such that ${\text{Mor}}_{C}(A,B)\cap {\text{Mor}}_{C}(A',B')$ is empty if $A\neq A'$ and $B\neq B'$

together with a "law of composition": $\circ :{\text{Mor}}_{C}(B,C)\times {\text{Mor}}_{C}(A,B)\to {\text{Mor}}_{C}(A,C)$ (which we denote by $(g,f)\mapsto g\circ f$) having the following properties:

1. Associativity: $(h\circ g)\circ f=h\circ (g\circ f)$ whenever the compositions are defined
2. Identity: for every object $A\in ob(C)$ there is an element $id_{A}$ such that for all $f\in {\text{Mor}}_{C}(A,B)$, $id_{B}\circ f=f$ and $f\circ id_{A}=f$.

Examples

1. The category of sets:

   Objects are all sets, morphisms are all maps between them.

2. The category of topological spaces:

   Objects are all topological spaces, morphisms are all continuous maps between them.

3. The category of functors: if $C$ and $D$ are two categories, then there is a category consisting of all contravarient functors from $C$ to $D$, where morphisms are natural transformations.
4. The category of schemes is one of the principal objects of study