Zero object
Definition
An object in a category is termed a zero object if it is both an initial object and a final object.
Examples
Examples of categories with a zero object
- The category of groups has a zero object: the trivial group. This is both an initial and a final object.
- The category of pointed sets has a zero object: a one-point set. This is both an initial and a final object.
Examples of categories without a zero object
- The category of sets has no zero object. This is because the initial object (the empty set) is not isomorphic to the final object (the one-point set).
- The category of topological spaces has no zero object. This is because the initial object (the empty topological space) is not isomorphic to the final object (the one-point topological space).
- The category of fields has neither an initial object nor a final object. Hence, it has no zero object.