📝Initial/Terminal object

§ Category Theory

Terminal object

\begin{tikzcd} & x & \\ y_1 \arrow[ru] & y_2 \arrow[u] & y_n \arrow[lu] \end{tikzcd}

xx is a terminal object iff ∀y,∃!f:y→x\forall y, \exists! f : y \to x (∀f:y→x,∀g:y→x  ⟹  f=g\forall f : y \to x, \forall g : y \to x \implies f = g).

There is a unique morphism from any object (including itself).

Terminal object corresponds to a singleton set in Set Theory, True in logic, Unit in types.

If there are multiple terminal objects, they are unique up to isomorphism.

Initial object

Initial object is the reverse of terminal object.

\begin{tikzcd} & x \arrow[ld] \arrow[d] \arrow[rd] & \\ y_1 & y_2 & y_n \end{tikzcd}

xx is an initial object iff ∀y,∃!f:x→y\forall y, \exists! f : x \to y (∀f:x→y,∀g:x→y  ⟹  f=g\forall f : x \to y, \forall g : x \to y \implies f = g).

There exists a unique morphism from initial object to any object (including itself).

Initial object corresponds to an empty set in Set Theory, False in logic, Void in types.