Universal property
Products

A categorical product can be characterized by a universal construction. For concreteness, one may consider the Cartesian product in Set, the direct product in Grp, or the product topology in Top, where products exist.

Let X {\displaystyle X} and Y {\displaystyle Y} be objects of a category C {\displaystyle {\mathcal {C}}} with finite products. The product of X {\displaystyle X} and Y {\displaystyle Y} is an object X {\displaystyle X} × Y {\displaystyle Y} together with two morphisms

such that for any other object Z {\displaystyle Z} of C {\displaystyle {\mathcal {C}}} and morphisms f: Z → X {\displaystyle f:Z\to X} and g: Z → Y {\displaystyle g:Z\to Y} there exists a unique morphism h: Z → X × Y {\displaystyle h:Z\to X\times Y} such that f = π 1 ∘ h {\displaystyle f=\pi _{1}\circ h} and g = π 2 ∘ h {\displaystyle g=\pi _{2}\circ h}.

To understand this characterization as a universal property, take the category D {\displaystyle {\mathcal {D}}} to be the product category C × C {\displaystyle {\mathcal {C}}\times {\mathcal {C}}} and define the diagonal functor

by Δ (X) = (X, X) {\displaystyle \Delta (X)=(X,X)} and Δ (f: X → Y) = (f, f) {\displaystyle \Delta (f:X\to Y)=(f,f)}. Then (X × Y, (π 1, π 2)) {\displaystyle (X\times Y,(\pi _{1},\pi _{2}))} is a universal morphism from Δ {\displaystyle \Delta } to the object (X, Y) {\displaystyle (X,Y)} of C × C {\displaystyle {\mathcal {C}}\times {\mathcal {C}}}: if (f, g) {\displaystyle (f,g)} is any morphism from (Z, Z) {\displaystyle (Z,Z)} to (X, Y) {\displaystyle (X,Y)}, then it must equala morphism Δ (h: Z → X × Y) = (h, h) {\displaystyle \Delta (h:Z\to X\times Y)=(h,h)} from Δ (Z) = (Z, Z) {\displaystyle \Delta (Z)=(Z,Z)} to Δ (X × Y) = (X × Y, X × Y) {\displaystyle \Delta (X\times Y)=(X\times Y,X\times Y)} followed by (π 1, π 2) {\displaystyle (\pi _{1},\pi _{2})}. As a commutative diagram:

For the example of the Cartesian product in Set, the morphism (π 1, π 2) {\displaystyle (\pi _{1},\pi _{2})} comprises the two projections π 1 (x, y) = x {\displaystyle \pi _{1}(x,y)=x} and π 2 (x, y) = y {\displaystyle \pi _{2}(x,y)=y}. Given any set Z {\displaystyle Z} and functions f, g {\displaystyle f,g} the unique map such that the required diagram commutes is given by h = ⟨ x, y ⟩ (z) = (f (z), g (z)) {\displaystyle h=\langle x,y\rangle (z)=(f(z),g(z))}.
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