nLab
zero object

Zero objects

Definition

Definition

In a category, an object is called a zero object, or null object 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 0𝒞0 \in \mathcal{C} is a zero object precisely if for every other object AA there is a unique morphism A0A \to 0 to the zero object as well as a unique morphism 0A0 \to A from the zero object.

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}\{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
Proposition

In any category CC enriched over the category of pointed sets (Set *,)(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 *Set ** \in Set_* for the singleton pointed set. Suppose tt is terminal. Then C(x,t)=*C(x,t) = * for all xx and so in particular C(t,t)=*C(t,t) = * and hence the identity morphism on tt is the basepoint in the pointed hom-set. By the axioms of a category, every morphism f:txf : t \to x is equal to the composite

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

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

Analogously from assuming tt 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 *,)(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 AA to any other object BB called the zero morphism, given by the composite A0BA\to 0 \to B.

Thus, such a category becomes enriched over the category of pointed sets, a partial converse to prop 2.

Revised on June 19, 2013 00:15:18 by Urs Schreiber (82.169.65.155)