So far, we have considered limits and colimits over diagrams whose underlying category is in FinSet, that is,
diagrams with no non-trivial morphisms. Let us now explore the behavior of limits and colimits for other diagrams.
The additional property of limits and colimits that has been so far unused is that they induce commutativity, that is,
any morphisms from the cone of the limit to the same object must be equal.
For a diagram Di :: [D, C], the simplest D :: Cat for which commutativity is not trivial is when D is composed
of two objects and two parallel morphisms, that is, two morphisms with the same origin and destination. Let us call
P :: Cat, the walking parallel morphism, the category with two objects and two non-trivial morphisms that are
parallel. A limit over this diagram is known as an equalizer, and a colimit is known as a coequalizer.
The limit in C :: Cat over diagram Di :: [P, C], if it exists, selects two parallel morphisms in C,
f, g :: C ^ I, f, g :: x -> y, and yields a new object eq(f, g) :: Cat. In the maximalist case, if the
commutativity condition would impose no restrictions, this would be the product x * y. However, the commutativity
condition says that for any other c :: C, and its induced morphisms i : c -> x, j : c -> y, we have that
i . f = j and i . g = j. In other words, i . f = i . g. Since j is determined by i, we can drop it.
The equalizer eq(f, g) :: Cat then has the property that any morphism i : c -> x with the property that
i . f = i . g is equivalent to a morphism h : c -> eq(f, g). Then consider two morphisms h, h' : c -> eq(f, g)
and a morphism e : eq(f, g) -> x so that h . e = h' . e : c -> x. Then we have that h . e . f = h' . e . g,
so if we denote m = h . e = h' . e, we have m . f = m . g. Since both map to the same morphism, we have
h = h'.
This means that any morphism e : eq(f, g) -> x has the property that h . e = h' . e implies h = h'. A morphism
with this property is known as a monomorphism, and can be described as monic. In Set, a monomorphism is an injective
function. Each h : c -> eq(f, g) is in bijection with h . e : c -> x, but we already have that h is in
bijection with i : c -> x. Then we can define e so that h . e = i. This implies for any h, we have
h . e . f = h . e . g. Thus, e . f = e . g, and any other i so that i . f = i . g has a unique h so that
h . e = i.
Then, for f, g: x -> y, the definition of an equalizer eq(f, g) is an object equ :: C and a morphism
e : equ -> x so that e . f = e . g, and any other object equ' :: C with the same property is derivable from equ
by precomposition. The second condition, that all other objects must be derivable from an object in a unique way, is
known as a universal property, and all limits and colimits have such a condition.
To ground this definition, let us consider if f, g :: Set ^ I. Then the equalizer eq(f, g) is composed of a set
equ and a function e : equ -> X with the property that f(e(equ)) = g(e(equ)), and the set is universal, that is,
it’s the most comprehensive set possible with this property. As e is injective, it maps each element of equ to a
unique value x, so we have that f(x) = g(x), and each value of equ corresponds to such a solution. In other
words, equ is the set of values x :: X so that f(x) = g(x).
Then, for f, g : x -> y, coeq(f, g) is the object so that any morphism i : y -> c with the condition that
f . i = g . i is equivalent to a morphism h : coeq(f, g) -> c. If we have two morphisms h, h' : coeq(f, g) -> c
and e : y -> coeq(f, g) so that e . h = e . h', then h = h'. This is called an epimorphism, and the adjective
form is epic. In Set, it is equivalent to a surjective function. We define e so that e . h = i. Then,
f . e . h = g . e . h', so f . e = g . e.
In other words, coeq(f, g) is an object coe :: C and a morphism e : y -> coe so that f . e = g . e and coe is
the object with the universal property.
Let us consider coequalizers in Set. We have a set coe and a surjective function e : Y -> coe, so that
e(f(x)) = e(g(x)). In other words, e makes it so that, if f(x) =/= g(x), their image over e is equal instead,
and coe is the largest possible set with this property that still has e as surjective. In other words, coe is
the quotient set of Y generated by the relation f(x) ~ g(x).
The walking cospan is the category with three objects, a, b, c :: C and two non-trivial morphisms, namely
f : a -> b and g : c -> b. A cospan is a diagram that starts in a walking cospan. The limit over a cospan is known
as a pullback, and the colimit is known as a pushout.
A pullback, then, has the commutativity condition as i . f = i . g. While the definition is the same regardless,
they are easier to reason about if we assume that we have the product d = a * c. In that case, we can define
f', g' : d -> b as f' = first . f, g' = second . f, and this construction is equivalent due to the properties of
products. This makes a pullback into an instance of an equalizer.
As such, a pullback is an object Pb :: C and two morphisms x : Pb -> a, y : Pb -> c with the property that
x . f = y . g and Pb fulfills the universal property. In particular, x and y are known as projection maps,
as they project from Pb, which can be conceptualized as pairs of elements i, j :: a, c, the first or the second
element and ignore the other. In Set, Pb :: X * Y is the set of values so that x :: X, y :: Y, f(x) = g(y).
This is equivalent to an equation in two values.
Similarly, if coproducts exist, pushouts are instances of coequalizers. As such, a pushout is an object Po :: C and
two morphisms x : a -> Po, y : c -> Po so that f . x = g . y and Po is the universal property. In Set,
Po :: X + Y is the quotient set generated by the relation x ~ y where x = f(a) :: X and y = g(a) :: Y for any
a :: X + Y.
A category is complete if it has all limits, and cocomplete if it has all colimits. If it has all limits and colimits, it is referred to as bicomplete. In fact, for a category to be complete, it is sufficient that it has all products and equalizers, or alternatively, all pullbacks and a terminal object. Similarly, a category is cocomplete if it has all coproducts and coequalizers, or alternatively, all pushouts and an initial object. A category is finitely complete or cocomplete if it has all finite limits or colimits.