---- > [!definition]+ Definition. ([[terminal object]]) > Let $\mathsf{C}$ be a [[category]]. > We call an object $I$ of $\mathsf{C}$ **initial in $\mathsf{C}$** if for every object $A$ of $\mathsf{C}$ there exists *exactly one* morphism $I \to A$ in $\mathsf{C}$: > $\text{For all } A \in \text{Obj}(\mathsf{C}), \text{Hom}_{\mathsf{C}}(I,A) \text{ is a singleton}.$ > We call an object $F$ of $\mathsf{C}$ **final in $\mathsf{C}$** if for every object $A$ of $\mathsf{C}$ there exists *exactly one* morphism $A \to F$. $\text{For all } A \in \text{Obj}(\mathsf{C}), \text{Hom}_{\mathsf{C}}(A, F) \text{ is a singleton}.$ > We call an object **terminal in $\mathsf{C}$** if it is initial or final in $\mathsf{C}$. > \ > A **zero object** is one which is *both* initial and final. The **zero morphism** $A \xrightarrow{0} B$ is the unique morphism $A \xrightarrow{} \boldsymbol 0 \to B$ from $A$ to $B$ which factors through the zero object $\boldsymbol 0$. ^definition > [!basicnonexample]+ Warning. > A [[category]] need not have any terminal objects. For example, consider the [[category]] $\mathsf{C}$ from [[category#^basic-example-2|this example]] with $\text{Obj}(\mathsf{C})=\mathbb{Z}$ and morphisms determined by the [[relation]] $\leq$. An initial resp. final object in $\mathsf{C}$ would correspond to an integer smaller resp. larger than all others. > \ > Also, terminal objects — when they exist — may or may not be unique. E.g. in $\mathsf{Set}$, $\emptyset$ is uniquely initial, however every singleton is final. What *is* true is that [[terminal objects are unique up to a unique isomorphism]]. ^basicnonexample > [!basicexample]+ > A [[terminal object|final object]] $A$ in a [[category]] $\mathsf{C}$ is an [[terminal object|initial object]] in the [[opposite category]] $\mathsf{C}^{\text{op}}$. For if $\text{Hom}_{\mathsf{C}}(A, Z)$ is a singleton for all objects $Z$ in $\mathsf{C}$, then $\text{Hom}_{\mathsf{C}^{\text{op}}}(Z,A)$ is a singleton for all objects $Z$ in $\mathsf{C}^{\text{op}}$. ^basicexample ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` #reformatrevisebatch02