Category¶
A category is, in some sense, one of the most general mathematical structures possible. It consists of
- A collection \(\text{ob}(C)\) of objects.
- A collection \(\text{mor}(C)\) of Morphisms, each of which has a source object and a target object.
- A binary operation \(\circ\) on morphisms via which morphisms can be composed.
Additionally, every object \(c \in C\) must have at least one endomorphism called the identity morphism \(id_c\).
Types and Examples of Categories¶
The Size of a Category¶
A category \(C\) for which \(\text{ob}(C)\) and \(\text{mor}(C)\) are sets is called a small category. If instead these collections are proper classes, \(C\) is said to be a large category.
Large categories are often difficult to work with, but not always. A category for which the collection of morphisms between any two objects forms a set (i.e. a Hom-Set) is called a locally small category.
Examples of Categories¶
The numerous and varied examples of categories below serve to demonstrate the ubiquity of category theory and categorical thinking.
Examples of Small Categories¶
There are many useful examples of small categories.
| Category | Objects | Morphisms |
|---|---|---|
| Partially-Ordered Sets | Numbers (perhaps the reals) | \(\leq\) |
| The Path Category of a Graph | Graph Vertices | Paths |
| The Action Groupoid \(G // X\) of a Group \(G\) over a set \(X\) | \(x \in X\) | \(gx\) for \(g \in G\) |
| Programming Languages | n-tuples of Datatypes | Functions |
| The Fundamental Groupoid | Points in a Topological Space | Homotopy Classes of Paths |
Examples of Large Categories¶
There are also many useful examples of large categories, some of which are locally small.
| Category | Objects | Morphisms |
|---|---|---|
| \(\text{Grp}\) | Groups | Homomorphisms |
| \(\text{Top}\) | Topological Spaces | Homeomorphisms |
| \(\text{Vect}\) | Vector Spaces | Linear Maps |
| \(\text{Set}\) | Sets | Functions |
| \(\text{Diff}\) | Smooth Manifolds | Differentiable Functions |
Further Examples of Categories¶
There are many other common examples of categories:
- A category with one element (though perhaps multiple endomorphisms) is called a monoid.
- The category \(\text{Cat}\) is the category of categories, with objects categories and morphisms functors.
- Every category \(C\) admits an opposite category \(C^{\text{op}}\), with the same objects, both with morphisms oriented the other direction.
The Representation of Groups as Categories¶
There are multiple different ways to represent a group \(G\) as a category. The most common is as a monoid, in which case the category is denoted \(\text{B}G\). This category has one element \(G\), and its endomorphisms are the group elements \(g \in G\), with the identity element serving as the identity endomorphism.
Groups themselves collectively form the category \(\text{Grp}\), with morphisms homomorphisms. (This is, in fact, the origin of the name "morphism".) Furthermore, a group's action on a set forms an action groupoid, which itself forms a category. For instance, the Rubik's Cube can be though of as a category in which the objects are the states of the cube and the morphisms are rotations of the faces.