Final object
2020 Mathematics Subject Classification: Primary: 18A05 [MSN][ZBL]
terminal object, of a category
A concept formalizing the categorical property of a one-point set. An object 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 |
Final object. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Final_object&oldid=42579