Null object of a category

2020 Mathematics Subject Classification: Primary: 18B [MSN][ZBL]

zero (object) of a category

An object (usually denoted by $\mathbf{0}$) such that for every object $X$ of the category the sets $H(X,\mathbf{0})$ and $H(\mathbf{0},X)$ are singletons. The null object, if it exists in a given category, is uniquely determined up to isomorphism. In the category of pointed sets (sets with a distinguished point) a singleton is a null object, 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.

References

• S. MacLane, "Categories for the working mathematician" Graduate Texts in Mathematics 5, Springer (1971) ISBN 0-387-98403-8

Comments

An object $I$ of a category is called initial if there is just one morphism $I\to X$ for any $X$, and terminal (or final) if there is just one morphism $X\to I$ for any $X$. Thus a null object is one which is both an initial object and a terminal object. If an initial object exists in a given category, it is unique up to isomorphism, and similarly for terminal objects; but the initial and terminal objects of a category need not be isomorphic. 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.

If $A$, $B$ are objects of a category with a null object $\mathbf{0}$, then there is a unique composite map $A \rightarrow \mathbf{0} \rightarrow B$, the zero or null morphism from $A$ to $B$.