Final object

terminal object, of a category

A concept formalizing the categorical property of a one-point set. An object $T$ in a category $\mathfrak{K}$ is called final if for every object $X$ in $\mathfrak{K}$ the set $H_{\mathfrak{K}}(X,T)$ consists of a single morphism. A final object is also called a right null object of $\mathfrak{K}$. A left null or initial object of a category is defined in the dual way.

In the category of sets the final objects are just the one-point sets, and the initial object is the empty set.. In any category with null objects the final objects are the null objects (cf. Null object of a category). Other examples of final objects arise in various categories of diagrams, where the concept of a final object is essentially equivalent to that of the limit of a diagram. For example, suppose that $\alpha,\beta:A \rightarrow B$ and let $\mathrm{Eq}(\alpha,\beta)$ be the category of left equalizers of the pair $(\alpha,\beta)$; in other words, the objects of $\mathrm{Eq}(\alpha,\beta)$ are morphisms $\mu:X \rightarrow A$ for which $\mu\alpha = \mu\beta$, and morphisms in $\mathrm{Eq}(\alpha,\beta)$ are morphisms $\gamma : (X,\mu)\rightarrow (Y,\nu)$ for which $\gamma\nu=\mu$. A final object in $\mathrm{Eq}(\alpha,\beta)$ is a kernel of the pair of morphisms $(\alpha,\beta)$ (cf. Kernel of a morphism in a category).

Comments

The set $H_{\mathfrak{K}}(X,T)$ is, by definition, the set of morphisms in $\mathfrak{K}$ from $X$ to $T$. Note that any two final objects of a category are (canonically) isomorphic, and so are two initial objects.

References

[a1]	J. Adámek, "Theory of mathematical structures" , Reidel (1983)
[a2]	B. Mitchell, "Theory of categories" , Acad. Press (1965) pp. 4