Categories & Examples

Objects, arrows, composition, identities — and why a single monoid or ordered set already is a category.

The idea

A category is the common shape of nearly every part of mathematics: a kind of object, and a kind of map between objects. Sets and functions, groups and homomorphisms, vector spaces and linear maps, topological spaces and continuous maps — in each case two maps that meet end to end compose, composition is associative, and every object carries an identity map to itself. The definition keeps exactly those features and forgets what the objects and maps were made of.

Definition (Category).

A category $\mathcal{A}$ consists of a collection of objects; for each pair of objects $A, B$ a collection $\mathcal{A}(A,B)$ of arrows (also called maps or morphisms) from $A$ to $B$, written $f \colon A \to B$; a composition rule turning $f \colon A \to B$ and $g \colon B \to C$ into an arrow $g \circ f \colon A \to C$; and for each object $A$ an identity arrow $1_{A} \colon A \to A$. Two axioms are imposed, $h \circ (g \circ f) = (h \circ g) \circ f, \qquad f \circ 1_{A} = f = 1_{B} \circ f,$ for all arrows for which these composites make sense.

Nothing in the definition mentions elements, and the omission is deliberate: structures that are not sets with structure also qualify. Take the divisors of $12$ as objects, and declare exactly one arrow $a \to b$ whenever $a$ divides $b$. Reflexivity supplies the identities, transitivity supplies the composition, and associativity holds because no two objects are ever joined by more than one arrow. The result is a category whose objects are numbers and whose arrows are not functions at all.

Ways to work on it

Not sure where to start? Take the ten-question placement test.