+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Category theory +-- {: .hide} [[!include category theory - contents]] =-- #### Additive and abelian categories +-- {: .hide} [[!include additive and abelian categories - contents]] =-- =-- =-- # Zero objects * table of contents {: toc} ## Definition +-- {: .num_defn} ###### 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]]_. +-- {: .num_remark} ###### 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. =-- +-- {: .num_remark} ###### 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. =-- +-- {: .num_remark} ###### 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 ring]]s (although it happens to be unital). =-- ## Examples +-- {: .num_prop} ###### Proposition * The one-point set is the zero object of the [[category of pointed sets]] (denoted $\Set_*$) and of the category of [[pointed topological spaces]] (denoted $\Top_*$), 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 $R$ 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 [[ring|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 $Id_{pt}$. =-- +-- {: .num_prop #InCatsEnrichedInPointedSets} ###### Proposition In any category $C$ [[enriched category|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} ###### Proof Write $* \in Set_*$ for the singleton pointed set. Suppose $t$ is [[terminal object|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. =-- +-- {: .num_remark} ###### Remark This is a special case of an [[absolute limit]]. =-- +-- {: .num_remark} ###### Remark Categories enriched in $(Set_*, \wedge)$ include in particular [[Ab]]-enriched categories. So any [[additive category]], hence every [[abelian category]] has a zero object. =-- * In the [[stable homotopy category]]: [[zero spectrum]]. ## Properties +-- {: .num_prop} ###### 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]]. =-- +-- {: .num_remark} ###### 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 category|enriched]] over the [[category of pointed sets]], a partial converse to prop \ref{InCatsEnrichedInPointedSets}. =-- +-- {: .num_prop} ###### Proposition A category with a zero object is [[cartesian closed category|Cartesian closed]] if and only if it is [[equivalence of categories|equivalent]] to the [[trivial category]]. =-- In other words, as expected, a [[model]] of [[intuitionistic logic]] in which [[true]] and [[false]] coincide is necessarily trivial. ## Related concepts * [[pointed category]] [[pointed (∞,1)-category]], [[pointed model category]] * [[zero morphism]] * [[zero object in a derivator]] ## References * [[Saunders MacLane]], §I.5 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)] [[!redirects zero object]] [[!redirects zero objects]] [[!redirects 0 object]] [[!redirects 0 objects]] [[!redirects 0-object]] [[!redirects null object]] [[!redirects null objects]] [[!redirects biterminator]] [[!redirects biterminators]]