Category (mathematics)
Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics.
In general, the objects and arrows may be abstract entities of any kind, and the notion of category provides a fundamental and abstract way to describe mathematical entities and their relationships.
The basic definitions in this article are contained within the first few chapters of any of these books.
A category C consists of such that the following axioms hold: We write f: a → b, and we say "f is a morphism from a to b".
Often the map assigning each object its identity morphism is treated as an extra part of the structure of a category, namely a class function i: ob(C) → mor(C).
Some authors use a slight variant of the definition in which each object is identified with the corresponding identity morphism.
This stems from the idea that the fundamental data of categories are morphisms and not objects.
In fact, categories can be defined without reference to objects at all using a partial binary operation with additional properties.
A category C is called small if both ob(C) and hom(C) are actually sets and not proper classes, and large otherwise.
Since, in small categories, the objects form a set, a small category can be viewed as an algebraic structure similar to a monoid but without requiring closure properties.
The class of all sets (as objects) together with all functions between them (as morphisms), where the composition of morphisms is the usual function composition, forms a large category, Set.
The category Rel consists of all sets (as objects) with binary relations between them (as morphisms).
Abstracting from relations instead of functions yields allegories, a special class of categories.
Any preordered set (P, ≤) forms a small category, where the objects are the members of P, the morphisms are arrows pointing from x to y when x ≤ y.
By the same argument, any partially ordered set and any equivalence relation can be seen as a small category.
Any ordinal number can be seen as a category when viewed as an ordered set.
Consider a topological space X and fix a base point
is the fundamental group of the topological space X and the base point
It is a concrete category, i.e. a category obtained by adding some type of structure onto Set, and requiring that morphisms are functions that respect this added structure.
This is called the dual or opposite category and is denoted Cop.
