# Maths - Category Theory - Universal Constructions

Here we look at properties of categories that are common across all concrete categories from different branches of mathematics. As discussed earlier we are trying to find the properties of a category from its external interactions and universal properties give us a way to do this. In particular we may be looking for unique arrows (morphisms) that have some particular property.

In universal properties there is a unique isomorphism somewhere, this gives the 'best' of such a property.

Universal constructions happen in dual pairs:

### Limits and Colimits

A limit is a way to take a diagram and encapsulate its properties into one object as simply as possible.

This is a generalisation that includes:

 initial/terminal objects (co)products (co)equalizers pullbacks/pushout. These generate a construction (c) which depends on the diagram such that:

• All triangles commute.
• Any other object c' can be factored through c without loosing information (there is a unique arrow to make the cones commute).

The first condition makes 'c' general enough to capture the essence of the construction.

The second condition makes sure that it is only just general enough (no junk).

### Cone and Cocone

This is also called a wedge. For a given pair there may be many wedges. We look for a 'best possible' wedge .

example: highest common factor.

A cone/cocone can be added on to an existing diagram. Cone Cocone
A cone/cocone is defined by the tuple: (X,f,g).  We can give this a universal property if, for other tuples (Xi,fi,gi), there is an arrow m known as the mediating arrow. This makes (X,f,g), in some way, the 'best' for that construction.  Example in set

union intersection ### Initial and Terminal Objects

These are related to the identity elements of an algebra.

In some cases (group, vectors) an object is both initial and terminal, in this case it is called a zero object or null object.

Terminal objects give a category theory version of the concept of 'element' in set theory. 1 -> A allows us to pick out an arbitrary element of the set.

Terminal Initial  Notation 1 0
generalisation a kind of limit a kind of colimit
universal cone over diagram empty diagram

examples: set:

{1}or {a} ...

set with one element (singleton)

Ø = {}

empty set

group (null object) trivial group (just identity element) trivial group (just identity element)
topological space single point empty space
poset greatest element (if exists) least element (if exists)
monoid trivial monoid (consisting of only the identity element) trivial monoid
semigroup singleton semigroup empty semigroup
Rng trivial ring consisting only of a single element 0=1 ring of integers Z
fields does not have terminal object does not have initial object
Vec zero object zero object
Top one-point space empty space
Grf graph with a single vertex and a single loop the null graph (without vertices and edges)
Ω-Alg
algebra with signature Ω
initial (term) algebra whose carrier consists of all finite trees.
Cat category 1 (with a single object and morphism) empty category

### Equaliser and Coequaliser

Equaliser Coequaliser
universal cone over diagram  An equaliser is an arrow 'h' which makes 'f' and 'g' equal. A Coequaliser is an arrow 'h' which makes 'f' and 'g' equal.

Universal Property:

'k' is a unique arrow known as the mediator.

this is equivalent to the following diagrams commuting that is:
j = f•h = g•h

equaliser is monic coequaliser is epic
Related concepts in set theory.

injective functions

Subset - every subset of a set occurs as an equaliser.

surjective functions

Equivalence - a binary relation which is:

• reflexive
• transitive
• symmetric

## Relationship of Equaliser to Pullback

 If the category A is split into two categories A&B then the equaliser becomes a pullback. See pullback on this page. ### Product and Sum

Product
(pullback)

Sum
(Coproduct)
(pushout)

universal cone over diagram   generalisation a kind of limit a kind of colimit
set example

cartesian product {a,b,c}*{x,y}=
{{a,x},{b,x},{c,x},{a,y},{b,y},{c,y}}

disjoint union {a,b,c}+{x,y}=
{a,b,c,x,y}

group the product is given by the cartesian product with multiplication defined component wise. free product
the free product for groups is generated by the set of all letters from a similar "almost disjoint" union where no two elements from different sets are allowed to commute.
Grp (abelian) direct sum

direct sum
consists of the elements of the direct product which have only finitely many nonzero terms (this therefore coincides exactly with the direct product, in the case of finitely many factors)

the group generated by the "almost" disjoint union (disjoint union of all nonzero elements, together with a common zero)

vector space direct sum direct sum
poset greatest lower bound
meet
least upper bound
join
base topological space   wedge
POS

greatest lower bounds (meets)

least upper bounds (joins)

Rng
Top the space whose underlying set is the cartesian product and which carries the product topology disjoint unions with their disjoint union topologies
Grf
category objects: (a,b)
morphism: (a,b)->(a',b')

tensor products are not categorial products.

In the category of pointed spaces, fundamental in homotopy theory, the coproduct is the wedge sum (which amounts to joining a collection of spaces with base points at a common base point).

#### Sum

When generating a sum for objects with structure then the structure associated with the link can be added to the sum object. #### Product ### Pullbacks and Pushout

Pullback:

• Generalisation of both intersection an inverse.
• If a category has binary products and equalisers, then it has pullbacks.

In programming terms it is related to the concept of indexing.

The pullback is like a type of division for function composition, in other words if multiplication is function composition then what is division? This comes in left and right flavors that is:

if h = g o f

• We know f and h, how do we find g?
• We know g and h, how do we find f?

More about pullbacks on page here.

### Completeness and Cocompleteness

• a category is complete if every diagram from a small (i.e. finite) category to it has a limit
• a category is cocomplete if every such functor has a colimit.
Competeness

Cocompeteness

set example

Set
Grp
Rng
Vec
Top
Grf
Cat

## Exponential

This is a universal structure but not a limit.

## Example of a Limit So, as an example, we take these three subsets: {a,b,c} {a,b,d} {a,b,e} There are various other subsets which have arrows to our original diagram: {a} {b} {a,b} (on this diagram I have shown an internal representation of the arrows rather than just the arrows themselves) But only one of these has arrows from all the others.