Zero objects

category theory

# Zero objects

## Definition

###### Definition

In a category, an object is called a zero object, null object, or biterminator if it is both an initial object and a terminal object.

A category with a zero object is sometimes called a pointed category.

###### Remark

This means that $0 \in \mathcal{C}$ is a zero object precisely if for every other object $A$ there is a unique morphism $A \to 0$ to the zero object as well as a unique morphism $0 \to A$ from the zero object.

###### Remark

If $\mathcal{C}$ is a pointed category, then an object $A$ of $\mathcal{C}$ is a zero object precisely when the only endomorphism of $A$ is the identity morphism.

###### Remark

There is also a notion of zero object in algebra which does not always coincide with the category-theoretic terminology. For example the zero ring $\{0\}$ is not an initial object in the category of unital rings (this is instead the integers $\mathbb{Z}$); but it is the terminal object. However, the zero ring is the zero object in the category of nonunital rings (although it happens to be unital).

## Examples

###### Proposition
• The category of pointed sets has a zero object, namely any one-element set.

• The trivial group is a zero object in the category Grp of groups and in the category Ab of abelian groups.

• For $R$ a ring, the trivial $R$-module (that whose underlying abelian group is the trivial group) is the zero-object in $R$Mod.

In particular for $R = k$ a field, the $k$-vector space of dimension 0 is the zero object in Vect.

• For $R$ and $S$ rings, the trivial $R$-$S$-bimodule (that whose underlying abelian group is the trivial group) is the zero-object in $R$-$S$-Bimod.

• However, the zero ring is not a zero object in the category of rings, at least as long as rings are required to have units (and ring homomorphisms to preserve them).

• For every category $C$ with a terminal object $*$ the under category $pt \downarrow C$ of pointed objects in $C$ has a zero object: the morphism $Id_{pt}$.

###### Proposition

In any category $C$ enriched over the category of pointed sets $(Set_*, \wedge)$ with tensor product the smash product, any object that is either initial or terminal is automatically both and hence a zero object.

###### Proof

Write $* \in Set_*$ for the singleton pointed set. Suppose $t$ is terminal. Then $C(x,t) = *$ for all $x$ and so in particular $C(t,t) = *$ and hence the identity morphism on $t$ is the basepoint in the pointed hom-set. By the axioms of a category, every morphism $f : t \to x$ is equal to the composite

$f : t \stackrel{Id}{\to} t \stackrel{f}{\to} x \,.$

By the axioms of an $(Set_*, \wedge)$-enriched category, since $Id_{t}$ is the basepoint in $C(t,t)$, also this composite is the basepoint in $C(t,x)$ and is hence the zero morphism. So $C(t,x) = *$ for all $x$ and therefore $t$ is also an initial object.

Analogously from assuming $t$ to be initial it follows that it is also terminal.

###### Remark

This is a special case of an absolute limit.

###### Remark

Categories enriched in $(Set_*, \wedge)$ include in particular Ab-enriched categories. So any additive category, hence every abelian category has a zero object.

## Properties

###### Proposition

A category has a zero object precisely if it has an initial object $\emptyset$ and a terminal object $*$ and the unique morphism $\emptyset \to *$ is an isomorphism.

###### Remark

In a category with a zero object 0, there is always a canonical morphism from any object $A$ to any other object $B$ called the zero morphism, given by the composite $A\to 0 \to B$.

Thus, such a category becomes enriched over the category of pointed sets, a partial converse to prop .