Null object of a category

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zero (object) of a category

An object (usually denoted by 0) such that for every object of the category the sets and are singletons. The null object, if it exists in a given category, is uniquely determined up to isomorphism. In the category of sets with a distinguished point the null object is a singleton, in the category of groups it is the trivial group, in the category of modules it is the zero module, etc. Not every category contains a null object, but a null object can always be formally adjoined to any given category. Every category with a null object has null morphisms.


An object of a category is called initial if there is just one morphism for any , and terminal (or final) if there is just one morphism for any . Thus a null object is one which is both initial and terminal. If an initial object exists in a given category, it is unique up to isomorphism, and similarly for terminal objects; but a category may have non-isomorphic initial and terminal objects. For example, in the category of sets, the empty set is an initial object and any singleton is terminal. A terminal object of a category may be regarded as a limit for the empty diagram in that category (cf. the editorial comments to Limit for the concept of a limit of a diagram in a category). Conversely, a limit of an arbitrary diagram may be defined as a terminal object in an appropriate category of cones.

How to Cite This Entry:
Null object of a category. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article