Categories — Category Theory — S. D. V. Stephens

Definition 1.3.1 (Category). A category consists of:

a collection of objects;
for each , a collection of maps, arrows or morphisms from to ;
for each , a function , , called composition;
for each , an element (or ) of , called the identity on ,

satisfying the following axioms:

associativity: for all , and we have ;
identity laws: for all we have .

Note (Notation).

means .
, and all mean the same thing; is also written .
means .
Another convention: .

From each string we can construct a map , namely .

Example 1.3.2 (A string of length 4). Let the string have length . Then the two maps are the same map, .

For , the map is the identity map.

Commutative diagrams

This diagram commutes .

Domain and codomain

If , then is the domain and is the codomain of .

Special types of categories

: the category of sets, with morphisms the ordinary maps/functions.
: the category of groups (objects = groups), and morphisms are homomorphisms.
: objects are rings, morphisms are homomorphisms.
: for every field there is a category of vector spaces over and linear maps between them.
: topological spaces with continuous maps.

Isomorphisms

Definition 1.3.3 (Isomorphism). is an isomorphism if there exists such that and .

is called the inverse of and is written .
If there exists an isomorphism from to , we say they are isomorphic: .

Example 1.3.4.

On sets, isomorphisms = bijections.
On groups, isomorphisms = bijective homomorphisms.
Rings: same as groups.
: isomorphisms = homeomorphisms (recall that for maps of topological spaces, bijective and continuous isomorphism).

Remark. A general category is one where there need not be elements of objects, or sets with operations, or maps in the usual sense:

the objects of a category needn’t be remotely like sets, AND
the maps needn’t be remotely like functions.

is a category with no objects or maps.

Drawing small categories

We draw the category with one object and only its identity map (often written ) as a single dot, , and we draw as . We don’t know what an “element” of or would be, or how one could regard as a function of any sort!

We easily have more complex examples:

etc.

Definition 1.3.5 (Discrete category). Discrete categories are a collection of objects with ONLY identity maps as morphisms.

Groups as one-object categories

Proposition 1.3.6. A group is the same thing as a category that has only one object and in which all the maps are isomorphisms.

Proposition 1.3.7. For any object , the set of automorphisms of , equipped with composition, is a group.

So a one-object category consists of a set or class , an associative composition function , and a two-sided unit . This would make into a group, since “every map in is an isomorphism” is exactly to say that every element of has an inverse with respect to . We can represent this as:

One-object category	Group
with its object	the group
maps in	elements of
in	in

and the category looks something like this:

Preorders and orders

Definition 1.3.8 (Preorder). A preorder is a reflexive and transitive binary relation. A preordered set is a set with a preorder on it, denoted .

Example 1.3.9.

: , the standard less than or equal to.
: with meaning ( divides ).
is the set of subsets of a set , and is (inclusion).

Category theoretically, this is to say that a preordered set is a category in which, for each , there is at most one morphism from to (one exactly when ). By identity, for all , and by composition, if and then (transitivity). Associativity then comes for free, since there is at most one morphism between any two objects.

Definition 1.3.10 (Order). An order (or poset, partially ordered set) is a preorder with the property that if and then .

1.3 Categories