For the product set , given , and projections , , the following diagram commutes:
The unique function is given by for every .
The correspondence is given by the bijection:
We write for the pair of arrows and . The construction involves two additional constructions: and . is said to be left adjoint to the product .
The onepoint object serves as identity for the "cartesian product" operation:
given by , .
Monoid and group
A monoid is a set equipped with and such that the following diagrams commute:
This means that and .
Writing it out in terms of the elements, we get the familiar associativity axioms:
and the identity axioms:
Then a category can be defined as a monoid with "product over ":
We describe a group as a monoid with the additional operation corresponding to :
Equivalently, a group is a category with one object, and in which every arrow has a twosided inverse under composition.
is a category with objects all finite ordinals, and arrows all orderpreserving functions. From algebraic topology, it is called the simplicial category.
Functor
A functor is a morphism of categories. Let be a functor of two categories. Then, the object function assigns to each object , an object ; the arrow function assigns each arrow to the arrow , in such a way that:
the latter whenever is defined in .
They arise naturally in algebraic topology, where we have a topological space lifted to a homology group ; topological spaces map to abelian groups as . In algebra, a forgetful functor would erase the group structure yielding .
A functor is full when it is surjective, faithful when it is injective, fully faithful when it is bijective, in settheoretic terms.
Functors also satisfy the law of associativity of compositions, just like the category does.
Subcategory
A subcategory of category includes some of the arrows in , and:

For each arrow , the objects

For each object , the identity arrow

For each pair of arrows , the composition of arrows
A natural transformation is a morphism of functors. Given two functors , let be an NT. Then, every arrow is mapped to , so the following diagram commutes:
If we think of as giving a picture in of all objects and arrows in , we get the following commutative diagrams:
are called the components of the NT. An NT with all components invertible in is called a natural isomorphism. In this case, the the inverses are components of the NT .
The determinant is an NT . Let be the determinant of matrix with entries in commutative ring . is a morphism of groups.
For group , the transform to the factorcommutator group defines a transform from the identity functor on to the factorcommutator functor . Moreover, is natural, because the following diagram commutes:
The double character group yields an example in . Let denote the character group of so that is the set of all homomorphisms with the familiar group structure, so that is the additive group of real numbers modulo . Each arrow determines an arrow , in the opposite direction, in , with for each ; for composable arrows . For this reason, is a contravariant functor on to . However, the twice iterated character group and identity are both functors . For each group , there is a homomorphism:
obtained in the familar way: To each , assign the function given for any character , . Thus, , and is an NT.
Equivalence between categories is defined to be the pair of functors , along with the natural isomorphisms . This allows us to compare categories which are "alike" but of different "sizes".
Monics, epis, and zeros
Arrow is invertible if there exists with . If is unique, , and the isomorphism holds.
For arrow , is the left cancelation property, and is termed monic. is the right cancelation property, and is termed epi. In , monic arrows are injections and epi arrows are surjections.
For arrow , arrow is right inverse if ; is termed as the section of . Similarly, for , , is termed as the retraction of .
If, for each object , there exists exactly one arrow , is termed as the terminal object. On the other hand, for each object , there exists exactly one arrow , is termed as the initial object. In , the empty set is initial, and any onepoint set is terminal. An object which is both initial and final is called the null object ; then there exists a unique arrow , for any two objects called the zero arrow.
A groupoid is a category in which all morphisms are invertible. An example would be the fundamental groupoid , from algebraic topology. A groupoid is said to be connected if there is an arrow between any two objects in the groupoid.
Examples

is the category of all small rings

is the category of topological spaces, where objects are the set of small topological spaces, and morphisms are continuous maps between spaces

is the category of topological spaces, where objects are the set of small topological spaces, and morphisms are homotopies

is the small category of pointed sets

are the categories of small pointed topological spaces; in the latter, morphisms are homotopies with a fixed basepoint

is the category of small left Rmodules

is the category of vector spaces

is the category of small sets, where morphisms are binary relations

A concrete category is the pair , where is a category, and is a faithful functor
HomSets
is used to denote the set of all morphisms from to . A small category can be defined in terms of homsets, with the following data:

Objects

For every pair of objects , a function associating to the pair

For every triple , the composition axiom:

For every object , ; the identity axiom
An Abcategory or preaddive category is a category in which homsets are additive abelian groups, and composition is bilinear: given , ,
are all Abcategories. is used the denote the category of all small Abcategories.
Duality
We define to be the opposite category, with the same objects as , and directions of arrows reversed. For each , is an . Also . The assignments and define a covariant functor .
The opposite functor is defined in the usual way. A functor can be defined in terms of as , a contravariant functor. Then, .
Homsets provide an example of covariant and contravariant functors. For each object , the object function of the covariant homfunctor sends:
and the arrow function sends each arrow to:
For each object , the object function of the contravariant homfunctor sends:
and the arrow function sends each arrow to:
To simplify the noation, introduce for "composition with k on the left" or "composition induced by k", and for "composition with g" on the right:
For and , we have:
Product structure
along with and defines a product. There exists a unique morphism between and preserving the product structure.
are functors called projections of onto and . Elements of are pairs . are said to be universal among functors to and .
Let be two functors; then is the product of two functors, and we yield the following commutative diagram:
Thus, is a pair of functions, assigning to each pair of categories , the category , and to each pair of functors , the functor . Moreover, when and are defined, we have . This makes into a functor:
Functor is defined to be a bifunctor or a functor in two variable objects. Homsets may be described as bifunctors .
Consider an NT between bifunctors . Let be a function which assigns to every pair of objects , the arrow . is called natural in if, for every :
For bifunctors , is an NT of bifunctors , iff is natural in for every and natural in for every .
Consider the product category , where is the category of the twopoint set :
Here are functors from "bottom" to "top". , . If denotes the unique nonidentity arrow in , then are defined by:
for any object . It maps "bottom" to "top", and is natural. We call the universal natural transform from .
Functor category
Consider functors and NTs . Define , which are components of the NT . To show that is natural, let in the commutative diagram:
The functor category may be written as , with objects the functors , and morphisms the NTs between two such functors. The homset for this category is:
If , the category is isomorphic to the powerset of , . For any category , is isomorphic to , and is the category whose objects are arrows of , and whose arrows are those that, along with pair , make the following diagram commute:
Category of all categories
Given three categories and four NTs, we have:
The vertical and horizontal composities satisfy the interchange law:
The collection of all NTs is the set of arrows of two different categories under two different operations of composition, and , which satisfy the above interchange law. Moreover, any arrow that is identity for the composition is also identity for the composition . To prove this, notice that objects for the horizontal composition are the categories, for vertical composition, the functors. Notice that the objects and arrows of may be written as and . Then, the symbols such as have their accepted meaning in a situation such as:
Comma category
If is an object of category , is a category in which objects are objects of , where , and arrows , such that :
Example: If denotes a onepoint set, denotes the category of pointed sets.
If is an object of category , and a functor, the category of objects under has as objects pairs , where and , and arrows all those arrows such that :
Consider where and are NTs. Then the category , also written as is called the comma category. It has as objects triples , and as arrows all pairs of arrows , such that :
We get the following diagram when we consider projections of the comma category:
Graphs and free categories
A graph may be pictured, like a category, with objects as vertices and arrows as edges, except that there are no identity or composite arrows; for this reason, it may be called a precategory.
A directed graph is a set of objects and a set of arrows such that there exists a pair of arrows:
A morphism between graphs is a pair of functions and such that:
A forgetful functor is defined as follows: for every , there exists .
Let be a fixed set. An Ograph has O as its objects; a morphism of graphs will be one with as identity. If and are two graphs, then product over is defined as:
The definitions make this a graph.
A category with objects may be described as an Ograph equipped with the two morphisms and , composition and identity, such that the following commute:
A free category may be "generated by a graph" having the same set of objects , and morphisms composition on edges of the graph, and this is written as . Given , we get the following diagram in terms of underlying graphs:
There is a bijective correspondence between categories and their underlying graphs:
Quotient category
Let be a category and be a relation defined on the homset. Then, quotient category is defined in terms of a universal functor :
If , and is the homotopy equivalence between and , then quotient category is with objects topological spaces and arrows homotopy classes of continuous maps. If is a free category generated by a graph , is said to have generators and relations .