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.
A category are directed groupoid, in the same way a poset is a directed set.
Along the same lines, a category is a poset in the next dimension, see (n,r)-category.
A category is the oidification of a monoid — a monoidoid!

In terms of collections

A category $𝖢$ is a mathematical object consisting of: cat

a collection of objects, $O b (𝖢)$ , sometimes referred to as $𝖢$ when its meaning is clear;
for every ordered pair of objects $𝑋, 𝑌 \in O b (𝖢)$ a class¹ 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 $𝑋 \in O b (𝖢)$ , 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. See Objects as identities. Yet another fruitful perspective is Objects as functors. These notions are interchanged as is notationally convenient.

Table of Contents

Category

In terms of collections

See also

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Table of Contents

Category

In terms of collections

See also

Footnotes

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