An initial object in a category is an object such that for any object of , there is a unique morphism . An initial object, if it exists, is unique up to unique isomorphism, so we speak of the initial object.
An initial object may also be called coterminal, universal initial, co-universal, or simply universal.
Initial objects are the dual concept to terminal objects: an initial object in is the same as a terminal object in . An object that is both initial and terminal is called a zero object.
Likewise, the empty category is an initial object in Cat, the empty space is an initial object in Top, and so on.
The trivial group is the initial object (in fact, the zero object) of Grp and Ab.
The integers are the initial object of Ring.
An initial object is called strict if any morphism must be an isomorphism. The initial objects of a poset, of , , , and of any topos (in fact, any extensive category) are strict. At the other extreme, a zero object is only a strict initial object if the category is trivial (equivalent to the terminal category).