Category theory MOC

Category

Categories are motivated from several perspectives

1. If groups are the algebraic structure which abstract symmetry, categories are the algebraic structure which abstract mathematical theories.
2. A category is a directed groupoid, in the same way a poset is a directed set.
3. Along the same lines, a (univalent) category is a poset in the next dimension, see -category.¹
4. A category is the oidification of a monoid — a monoidoid!

In terms of collections

A category is a mathematical object consisting of: #m/def/cat

• a collection of objects, or , sometimes referred to as when its meaning is clear;
• for every ordered pair of objects a class² of morphisms ; and
• a composition operation so that given and we have ;

and satisfying the following properties

• for any , there exists a unique or which is the left and right identity under composition, i.e. and .
• composition is associative, i.e. .

It is common for to be abandoned in favour of juxtaposition, so . Note that since objects are in correspondence with identity morphisms, it is possible to avoid considering a separate class of objects and instead use identity morphisms. See Objects as identities. Yet another fruitful perspective is Objects as functors. These notions are interchanged as is notationally convenient.

See also

• See also Glossary of categories and Opposite category.
• Morphisms come in different varieties — see Morphism.
• There are also different kinds of category — see Types of Category.
• Reasoning about categories is often done through a Commutative diagram
• Things as categories

#state/tidy | #SemBr | #lang/en