Editing Universal property (section)

==Examples==

Below are a few examples, to highlight the general idea. The reader can construct numerous other examples by consulting the articles mentioned in the introduction.

===Tensor algebras===

Let <math>\mathcal{C}</math> be the [[category of vector spaces]] '''<math>K</math>-Vect''' over a [[field (mathematics)|field]] <math>K</math> and let <math>\mathcal{D}</math> be the category of [[algebra over a field|algebras]] '''<math>K</math>-Alg''' over <math>K</math> (assumed to be [[unital algebra|unital]] and [[associative algebra|associative]]). Let
:<math>U</math> : '''<math>K</math>-Alg''' &rarr; '''<math>K</math>-Vect'''
be the [[forgetful functor]] which assigns to each algebra its underlying vector space.

Given any [[vector space]] <math>V</math> over <math>K</math> we can construct the [[tensor algebra]] <math>T(V)</math>. The tensor algebra is characterized by the fact:
:“Any linear map from <math>V</math> to an algebra <math>A</math> can be uniquely extended to an [[algebra homomorphism]] from <math>T(V)</math> to <math>A</math>.”
This statement is an initial property of the tensor algebra since it expresses the fact that the pair <math>(T(V),i)</math>, where <math>i:V \to U(T(V))</math> is the inclusion map, is a universal morphism from the vector space <math>V</math> to the functor <math>U</math>.

Since this construction works for any vector space <math>V</math>, we conclude that <math>T</math> is a functor from  '''<math>K</math>-Vect''' to '''<math>K</math>-Alg'''.  This means that <math>T</math> is ''left adjoint'' to the forgetful functor <math>U</math> (see the section below on [[#Relation to adjoint functors|relation to adjoint functors]]).

===Products===

A [[categorical product]] can be characterized by a universal construction. For concreteness, one may consider the [[Cartesian product]] in '''[[Set (category theory)|Set]]''', the [[direct product]] in '''[[Grp (category theory)|Grp]]''', or the [[product topology]] in '''[[Top (category theory)|Top]]''', where products exist.

Let <math>X</math> and <math>Y</math> be objects of a category <math>\mathcal{C}</math> with finite products. The product of <math>X</math> and <math>Y</math> is an object <math>X</math> &times; <math>Y</math> together with two morphisms
:<math>\pi_1</math> : <math>X \times Y \to X</math>
:<math>\pi_2</math> : <math>X \times Y \to Y</math>
such that for any other object <math>Z</math> of <math>\mathcal{C}</math> and morphisms <math>f: Z \to X</math> and <math>g: Z \to Y</math> there exists a unique morphism <math>h: Z \to X \times Y</math> such that <math>f = \pi_1 \circ h</math> and <math>g = \pi_2 \circ h</math>.

To understand this characterization as a universal property, take the category <math>\mathcal{D}</math> to be the [[product category]] <math>\mathcal{C} \times \mathcal{C}</math> and define the [[diagonal functor]]
: <math>\Delta: \mathcal{C} \to \mathcal{C} \times \mathcal{C}</math>
by <math>\Delta(X) = (X, X)</math> and <math>\Delta(f: X \to Y) = (f, f)</math>. Then <math>(X \times Y, (\pi_1, \pi_2))</math> is a universal morphism from <math>\Delta</math> to the object <math>(X, Y)</math> of <math>\mathcal{C} \times \mathcal{C}</math>: if <math>(f, g)</math> is any morphism from <math>(Z, Z)</math> to <math>(X, Y)</math>, then it must equal
a morphism <math>\Delta(h: Z \to X \times Y) = (h,h)</math> from <math>\Delta(Z) = (Z, Z)</math>
to <math>\Delta(X \times Y) = (X \times Y, X \times Y)</math> followed by <math>(\pi_1, \pi_2)</math>. As a commutative diagram:
[[File:Universal-property-products.svg|center|484x484px|Commutative diagram showing how products have a universal property.]]For the example of the Cartesian product in '''Set''', the morphism <math>(\pi_1, \pi_2)</math> comprises the two projections <math>\pi_1(x,y) = x</math> and <math>\pi_2(x,y) = y</math>. Given any set <math>Z</math> and functions <math>f,g</math> the unique map such that the required diagram commutes is given by <math>h = \langle x,y\rangle(z) = (f(z), g(z))</math>.<ref>{{Cite arXiv |last1=Fong |first1=Brendan |last2=Spivak |first2=David I. |date=2018-10-12 |title=Seven Sketches in Compositionality: An Invitation to Applied Category Theory |class=math.CT |eprint=1803.05316 }}</ref>

===Limits and colimits===

Categorical products are a particular kind of [[limit (category theory)|limit]] in category theory. One can generalize the above example to arbitrary limits and colimits.

Let <math>\mathcal{J}</math> and <math>\mathcal{C}</math> be categories with <math>\mathcal{J}</math> a [[small category|small]] [[index category]] and let <math>\mathcal{C}^\mathcal{J}</math> be the corresponding [[functor category]]. The ''[[diagonal functor]]''
:<math>\Delta: \mathcal{C} \to \mathcal{C}^\mathcal{J}</math>
is the functor that maps each object <math>N</math> in <math>\mathcal{C}</math> to the constant functor <math>\Delta(N): \mathcal{J} \to \mathcal{C}</math> (i.e. <math>\Delta(N)(X) = N</math> for each <math>X</math> in <math>\mathcal{J}</math> and <math>\Delta(N)(f) = 1_N</math> for each <math>f: X \to Y</math> in <math>\mathcal{J}</math>) and each morphism <math>f : N \to M</math> in <math>\mathcal{C}</math> to the natural transformation <math>\Delta(f):\Delta(N)\to\Delta(M)</math> in <math>\mathcal{C}^{\mathcal{J}}</math> defined as, for every object <math>X</math> of <math>\mathcal{J}</math>, the component <math display="block">\Delta(f)(X):\Delta(N)(X)\to\Delta(M)(X) = f:N\to M</math>
at <math>X</math>. In other words, the natural transformation is the one defined by having constant component <math>f:N\to M</math> for every object of <math>\mathcal{J}</math>.

Given a functor <math>F: \mathcal{J} \to \mathcal{C}</math> (thought of as an object in <math>\mathcal{C}^\mathcal{J}</math>), the ''limit'' of <math>F</math>, if it exists, is nothing but a universal morphism from <math>\Delta</math> to <math>F</math>. Dually, the ''colimit'' of <math>F</math> is a universal morphism from <math>F</math> to <math>\Delta</math>.