3.2 Initial and terminal objects

Initial and terminal objects are special elements in categories with unique connections to all other objects. They're like the starting and ending points of a journey through the category's structure.

These objects have universal properties that make them stand out. Initial objects have a unique morphism to every other object, while terminal objects receive a unique morphism from every other object. This uniqueness gives them a special role in category theory.

Initial and Terminal Objects

Initial and terminal objects

• An object $I$ in a category $\mathcal{C}$ is an initial object if for every object $X$ in $\mathcal{C}$, there exists a unique morphism $f: I \to X$ ($\emptyset$ in the category of sets, trivial group $\{e\}$ in the category of groups)
• An object $T$ in a category $\mathcal{C}$ is a terminal object if for every object $X$ in $\mathcal{C}$, there exists a unique morphism $g: X \to T$ (singleton set $\{x\}$ in the category of sets, trivial group $\{e\}$ in the category of groups)
• The universal property of an initial object $I$ states that for any other object $X$ in the category, there is a unique morphism from $I$ to $X$
• The universal property of a terminal object $T$ states that for any other object $X$ in the category, there is a unique morphism from $X$ to $T$

Uniqueness of universal objects

• Suppose $I_1$ and $I_2$ are both initial objects in a category $\mathcal{C}$
• By the definition of an initial object, there exist unique morphisms $f: I_1 \to I_2$ and $g: I_2 \to I_1$
• The composition $g \circ f: I_1 \to I_1$ is a morphism from $I_1$ to itself and must be equal to the identity morphism $id_{I_1}: I_1 \to I_1$ since $I_1$ is an initial object
• Similarly, $f \circ g = id_{I_2}$, proving that $f$ and $g$ are inverse isomorphisms and $I_1$ and $I_2$ are uniquely isomorphic
• The proof for the uniqueness of terminal objects is dual to the proof for initial objects if $T_1$ and $T_2$ are both terminal objects in a category $\mathcal{C}$, then they are uniquely isomorphic

Examples in concrete categories

• In the category of sets, the empty set $\emptyset$ is an initial object because there is a unique function from $\emptyset$ to any other set and any singleton set $\{x\}$ is a terminal object because there is a unique function from any set to $\{x\}$
• In the category of groups, the trivial group $\{e\}$ is both an initial and terminal object for any group $G$, there is a unique group homomorphism from $\{e\}$ to $G$ mapping $e$ to the identity element of $G$ and a unique group homomorphism from $G$ to $\{e\}$
• In the category of topological spaces, any space with a single point is a terminal object because there is a unique continuous function from any topological space to a single point space the empty space is an initial object because there is a unique continuous function from the empty space to any other topological space

Duality of initial vs terminal

• In category theory, duality refers to the process of reversing the direction of morphisms in a category the dual of a concept is obtained by reversing the direction of morphisms in the definition
• Initial objects and terminal objects are dual notions the definition of an initial object is dual to the definition of a terminal object
• If $I$ is an initial object in a category $\mathcal{C}$, then $I$ is a terminal object in the opposite category $\mathcal{C}^{op}$
• If $T$ is a terminal object in a category $\mathcal{C}$, then $T$ is an initial object in the opposite category $\mathcal{C}^{op}$
• The opposite category $\mathcal{C}^{op}$ of a category $\mathcal{C}$ has the same objects as $\mathcal{C}$, but the morphisms are reversed if $f: A \to B$ is a morphism in $\mathcal{C}$, then $f: B \to A$ is a morphism in $\mathcal{C}^{op}$