Category

Categories are motivated from several perspectives

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

In terms of collections

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

a collection of objects, $𝖢 0$ or $O b (𝖢)$ , sometimes referred to as $𝖢$ when its meaning is clear;
for any $𝑋, 𝑌 \in 𝖢 0$ a set of morphisms $𝖢 (𝑋, 𝑌)$ ; and
a composition operation $(\circ)$ so that given $𝑓 \in 𝖢 (𝑋, 𝑌)$ and $𝑔 \in 𝖢 (𝑌, 𝑍)$ we have $𝑔 \circ 𝑓 \in 𝖢 (𝑋, 𝑍)$ ;

and satisfying the following properties

for any $𝑋 \in 𝖢 0$ , there exists a unique $1$ or $i d 𝑋 : 𝑋 \to 𝑋$ which is the left and right identity under composition, i.e. $𝑓 = i d 𝑋 \circ 𝑓$ and $𝑔 = 𝑔 \circ i d 𝑌$ .
composition is associative, i.e. $𝑓 \circ (𝑔 \circ ℎ) = (𝑓 \circ 𝑔) \circ ℎ$ .

It is common for $\circ$ to be abandoned in favour of juxtaposition, so $𝑓 \circ 𝑔 = 𝑓 𝑔$ . 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, as described in Objects as identities.

Examples

For a larger list, see Glossary of categories.

A motivating example is the category $𝖲 𝖾 𝗍$ , where objects are sets and morphisms are functions. Along these lines one might consider $𝖳 𝗈 𝗉$ , where objects are topological spaces and morphisms are continuous functions; or $𝖦 𝗋 𝗉$ , where objects are groups and morphisms are group homomorphisms.
There are also many familiar objects in mathematics which can be recontextualized as a special kind of category, see Things as categories.

Category

In terms of collections

Examples

See also

Footnotes