Initial Object

up:: 028 MOC Category Theory

Given a Category $C$, we say an object $I\in Ob(C)$ is an initial object if all objects $x\in C$ have a morphism from $I$ to $x$. That is,

$$\forall x\in C:\exists f\in Mor(I,x):I\stackrel{f}{\to }x$$

Properties

• An initial object is a Terminal Object in the Opposite Category $C^{op}$.
• A terminal object is isomorphic to an initial object if there is a morphism from T to I.
• An initial object which is also terminal is called a Zero Object.

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References

• KASHIWARA, Masaki; SCHAPIRA, Pierre, Categories and Sheaves, Berlin, Heidelberg: Springer Berlin Heidelberg, 2006.
• Summary of Truth about Types, a talk by Bartosz Milewski - Rakhim.org