Product (Category)

up:: 028 MOC Category Theory

Given a Category $C$, and objects $X_1,X_2\in Ob(C)$, we have that their product $X_1\times X_2$ is defined by the diagram above:

• For every $X_i$, there is a projection morphism $X_1\times X_2\stackrel{{\pi }_i}{⟶}{X}_i$
• If there is some other object $Y\in Ob(C)$ which has morphisms to the individual objects $Y\stackrel{f_i}{⟶}{X}_i$, then there is a unique morphism from it to the product $Y\stackrel{h}{⟶}{X}_1\times X_2$

Examples

• Cartesian products in Set are categorical products

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References

• MAC LANE, Saunders, Categories for the Working Mathematician, New York, NY: Springer New York, 1978.