# Category theory

Category theory attempts to generalize all of mathematics using *categories*.

# Morphisms

A **morphism** is the generalization of functions to mathematical structures beyond sets. It is formally defined as a structure-preserving map from one mathematical structure to another of the same type. For example, in set theory, morphisms are functions, while in group theory they are group homomorphisms, or linear transformations in linear algebra.

The *arrows* in category theory are morphisms between the objects in a category.

# Classes

A **class** is a collection of sets whereby all members share a certain property. Classes are informal structures in standard (ZF) Set theory (as not all collections of things are considered sets under ZF axioms). Classes that are not sets are called *proper classes*, while classes that *are* sets are called *small classes*. Classes are useful because they escape some limitations of formally defined sets; in category theory, they are the containers for objects and morphisms (as seen below).

# Categories

A **category** is a collection of *objects* linked together with *arrows* having two basic properties:

- The ability to compose arrows associatively (i.e. invariance to the order of application)
- The existence of an identity arrow for each object

Common examples include **Sets** (objects) and (set) functions (arrows), rings and ring homomorphisms, and topological spaces and continuous maps. Categories generally serve to represent abstractions of mathematical concepts.

Formally, a category $C$ is defined as consisting of

- A class of objects, denoted $\text{obj}(C)$
- A class of morphisms between objects. Each morphism $f$ has a source object $s$ and target object $t$ written $f: s \rightarrow t$. $\text{hom}(s,t)$ refers to the class of morphisms from $s$ to $t$. Note that $\text{hom}(C) = \cup_{s,t\in\text{obj}(C)} \text{hom}(s,t)$
- Morphisms can be composed such that $\text{hom}(a,b) \times \text{hom}(b,c) \rightarrow \text{hom}(a,c)$ for objects $a$, $b$, and $c$.

# Table of Common Categories

Category | Objects | Morphisms |
---|---|---|

Grp | groups | group homomorphisms |

Mag | magmas | magma homomorphisms |

Manp | smooth manifolds | p-times continuously differentiable maps |

Met | metric spaces | short maps |

R-Mod | R-modules, where R is a ring | R-module homomorphisms |

Mon | monoids | monoid homomorphisms |

Ring | rings | ring homomorphisms |

Set | sets | functions |

Top | topological spaces | ontinuous functions |

Uni | uniform spaces | uniformly continuous functions |

VectK | vector spaces over the field K | K-linear maps |