Category/Categories
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Category - Definition
There are many equivalent definitions of a category. One commonly used definition is as follows. A category 𝐶 consists of
- a class 𝑜𝑏(𝐶) of objects
- a class 𝑚𝑜𝑟(𝐶) of morphisms or arrows
- a domain or source class function 𝑑𝑜𝑚: 𝑚𝑜𝑟(𝐶) → 𝑜𝑏(𝐶)
- a codomain or target class function 𝑐𝑜𝑑: 𝑚𝑜𝑟(𝐶) → 𝑜𝑏(𝐶)
- for every three objects 𝑎, 𝑏, and 𝑐, a binary operation ℎ𝑜𝑚(𝑎, 𝑏) × ℎ𝑜𝑚(𝑏, 𝑐) → ℎ𝑜𝑚(𝑎, 𝑐) called composition of morphisms
- Here ℎ𝑜𝑚(𝑎, 𝑏) denotes the subclass of morphisms 𝑓 in 𝑚𝑜𝑟(𝐶) such that 𝑑𝑜𝑚(𝑓)=𝑎 and 𝑐𝑜𝑑(𝑓)=𝑏
- Morphisms in this subclass are written 𝑓:𝑎→𝑏, and the composite of 𝑓:𝑎→𝑏 and 𝑔:𝑏→𝑐 is often written as 𝑔∘𝑓 or 𝑔𝑓
Indent
such that the following axioms hold:
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- the associative property: if 𝑓:𝑎→𝑏, 𝑔:𝑏→𝑐 and ℎ:𝑐→𝑑 then ℎ∘(𝑔∘𝑓) = (ℎ∘𝑔)∘𝑓
- the (left and right unit laws): for every object 𝑥, there exists a morphism 1𝑥: 𝑥 → 𝑥 (some authors write 𝑖𝑑𝑥) called the identity morphism for 𝑥, such that:
- every morphism 𝑓 : 𝑎 → 𝑥 satisfies 1𝑥 ∘ 𝑓 = 𝑓, and
- every morphism 𝑔 : 𝑥 → 𝑏 satisfies 𝑔 ∘ 1𝑥 = 𝑔
Category - Small & Large
A category 𝐶 is called small if both 𝑜𝑏(𝐶) and ℎ𝑜𝑚(𝐶) are sets and not proper classes, and large otherwise. A locally small category is a category such that for all objects 𝑎 and 𝑏, the hom-class ℎ𝑜𝑚(𝑎, 𝑏) is a set, called a homset. Many important categories in mathematics (such as the category of sets), although not small, are at least locally small. 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. Large categories on the other hand can be used to create “structures” of algebraic structures.
Category - Examples
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. It is the most basic and the most commonly used category in mathematics.
The category Rel consists of all sets (as objects) with binary relations between them (as morphisms).
The category Cat consists of all small categories, with functors between them as morphisms.
The class of all preordered sets with monotonic functions as morphisms form a category, Ord. 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.
|
Category |
Objects |
Morphisms |
|---|---|---|
|
p-times continuously differentiable maps | ||
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R-modules, where R is a ring | ||
|
vector spaces over the field K |
Category - Construction of New Categories
Dual category
Any category 𝐶 can itself be considered as a new category differently: the objects are the same as those in the original category but the arrows are those of the original category reversed. This is called the dual or opposite category and is denoted 𝐶𝑜𝑝.
Product categories
If 𝐶 and 𝐷 are categories, one can form the product category 𝐶×𝐷: the objects are pairs consisting of one object from 𝐶 and one from 𝐷, and the morphisms are also pairs, consisting of one morphism in 𝐶 and one in 𝐷. Such pairs can be composed componentwise.