Editing Universal property (section)

===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>