A zero object in a category is an object that is both an initial object and a terminal object. Thus there is exactly one morphism from it to each object and exactly one morphism from each object to it. Any two zero objects are related by a unique isomorphism. For example, the trivial group is a zero object in the category of groups, and the zero module is a zero object in the category of modules over a fixed unit ring.
In a preadditive category, an initial object is automatically a terminal object and conversely. An additive category has a zero object, whereas a preadditive category need not have one (The Stacks Project Authors 2026).
