Limits and colimits
How can a category express the object that best solves a whole diagram of constraints?
What do products, intersections, quotients, unions, and glued spaces have in common?
Category theory changes the way objects are described. Instead of beginning with internal construction, it asks how an object relates by arrows to other objects.
This becomes especially powerful when several objects and morphisms are already arranged in a pattern.
A ----> C <---- B
or
A <---- C ----> B
Such a pattern is called a diagram. A diagram is a small piece of a category selected for attention. It may show two objects, many objects, parallel arrows, chains, squares, or more complicated shapes.
what object best completes this diagram?
Limits and colimits answer that question.
A limit is the most efficient way to map into a diagram while respecting all of its constraints. A colimit is the most efficient way to map out of a diagram while respecting all of its constraints.
limit:
maps into the diagram
colimit:
maps out of the diagram
The simplest useful limit is a product.
Given two objects A and B, a product is an object A x B equipped with projection morphisms
A x B --p1--> A
A x B --p2--> B
such that any object X with arrows to both A and B
X --f--> A
X --g--> B
factors uniquely through A x B.
X
/ \
f g
/ \
v v
A B
|
| unique
v
X
|
v
A x B
/ \
v v
A B
In Set, this is the ordinary Cartesian product. A function X -> A x B is exactly the same information as a pair of functions X -> A and X -> B.
map into product
=
pair of maps into the factors
But the categorical definition does not mention ordered pairs. It characterizes the product by its mapping behavior.
That is the limit idea in small form.
A pullback is another limit. Suppose we have two morphisms with a common target:
A --f--> C <--g-- B
A pullback is an object that captures pairs from A and B that agree after mapping to C.
In Set, it looks like
{ (a, b) | f(a) = g(b) }
Categorically, the pullback P comes with arrows to A and B:
P ----> B
| |
v v
A ----> C
and the square commutes. Any other object that makes the same agreement happen factors uniquely through P.
any compatible solution
|
| unique map
v
best compatible solution
The pullback therefore solves a compatibility problem in the category.
Limits generalize this. A limit of a diagram is an object with arrows into the diagram, called a cone, that is universal among all such cones.
L
/|\
/ | \
v v v
diagram objects
Every other cone factors uniquely through the limiting cone.
X
|
| unique
v
L
/|\
/ | \
v v v
diagram
So a limit is the best possible coherent way to map into the diagram.
Colimits reverse the direction. The simplest useful colimit is a coproduct. Given two objects A and B, a coproduct is an object A + B equipped with injection morphisms
A --i1--> A + B
B --i2--> A + B
such that any object X receiving arrows from both A and B
A --f--> X
B --g--> X
receives a unique arrow from A + B making everything commute.
A B
\ /
v v
A + B
|
| unique
v
X
In Set, this is disjoint union. To define a function out of a disjoint union, it is enough to define a function out of each part.
map out of coproduct
=
pair of maps out of the summands
Again, the categorical definition does not depend on elements. It depends on mapping behavior.
A pushout is a colimit that expresses gluing.
Given two morphisms out of a common source,
C --f--> A
C --g--> B
a pushout completes the square:
C ----> B
| |
v v
A ----> Q
In many categories, this says: take A and B and identify the parts that come from C. In topology, pushouts describe gluing spaces along a shared subspace. In algebra, they describe amalgamated constructions. The surface content changes, but the categorical pattern is the same.
common source
|
v
glued result
A colimit of a diagram is an object with arrows out of the diagram, called a cocone, that is universal among all such cocones.
diagram
| | |
v v v
K
Every other cocone receives a unique map from the colimit.
diagram
| | |
v v v
K
|
| unique
v
X
This is the dual of the limit pattern. Limits gather compatible information into a universal source for the diagram. Colimits assemble diagram information into a universal target.
limit:
universal source mapping into the diagram
colimit:
universal target receiving the diagram
The language of "best" must be read categorically. A limit is not best because it is largest, smallest, prettiest, or easiest to construct. It is best because every other candidate maps uniquely through it. Its superiority is expressed by a universal mapping property.
That also explains why limits and colimits are unique only up to unique isomorphism. There may be many concrete constructions that satisfy the same role. But if two objects both satisfy the same universal mapping property, then each maps uniquely to the other, and those maps are inverse.
same universal role
|
v
unique isomorphism
So category theory avoids overcommitting to a particular implementation. In Set, a product may be implemented as ordered pairs. In another category, the product may look completely different. The structural role is what matters.
Limits and colimits gather many earlier examples.
products
coproducts
equalizers
coequalizers
pullbacks
pushouts
intersections
quotients
gluing constructions
inverse limits
direct limits
They are not all the same construction. They are different instances of one organizing pattern: solve a diagram by a universal object.
This gives category theory a way to speak about structure without choosing coordinates, elements, or formulas. A product can be defined without saying what its elements are. A quotient-like object can be described by how maps out of it behave. A gluing can be specified by the maps it makes possible.
describe by construction:
what is it made of?
describe by universal behavior:
how do all maps into or out of it factor?
The second description is often more stable. Constructions vary by category. Universal mapping behavior travels across categories and is preserved by suitable functors.
The objects are diagrams in a category and candidate objects that complete them. The morphisms are cones into diagrams for limits and cocones out of diagrams for colimits. What composes are the mediating morphisms that factor one candidate through another. The invariant is universal factorization: every compatible candidate factors uniquely through the universal one. The defining relation is the commuting diagram plus the unique mediating map. Equality of concrete constructions is replaced by unique isomorphism of objects satisfying the same universal role. What remains outside the frame is the general principle behind all such characterizations.
Limits and colimits are universal solutions to diagram-shaped problems.
Products, pullbacks, coproducts, and pushouts show the same deeper pattern. An object can be defined by how every other object maps to or from it. What is that general pattern called?