zero object (changes) in nLab

Context

Category theory

Additive and abelian categories

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Proposition

• The one-point set is the zero object of thecategory of pointed sets has (denoted a zero object, namely any one-element set.Set *\Set_*) and of the category of pointed topological spaces (denoted Top *\Top_*), but only the terminal object of Set and Top.

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

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

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

• For RR and SS rings, the trivial RR-SS-bimodule (that whose underlying abelian group is the trivial group) is the zero-object in RR-SS-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 CC with a terminal object ** the under category pt↓Cpt \downarrow C of pointed objects in CC has a zero object: the morphism Id ptId_{pt}.

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:t→xf : t \to x is equal to the composite

f:t→Idt→fx. 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.