CatDat

inhabited

A category is inhabited (or non-empty) if it has at least one object.

• Dual property: inhabited (self-dual)
• nLab Link

Relevant implications

• thin and  inhabited implies   generator

  Any object will be a generator for trivial reasons.
• binary products and  inhabited implies   connected

  For any two objects $A,B$ we have the zig-zag $A \to A \times B \to B$.
• zero morphisms and  inhabited implies   connected

  trivial
• groupoid and  binary products and  inhabited implies   trivial

  Let $\mathcal{C}$ be an inhabited groupoid with binary products. Then it is connected, so we may assume $\mathcal{C}=BG$ for a group $G$ with unique object $*$. But then $* \times * = *$, so there are $p,q \in G$ such that $G \to G \times G$, $x \mapsto (px,qx)$ is bijective. From here it is an easy exercise to deduce $G=1$.
• connected implies   inhabited

  by definition
• generator implies   inhabited

  trivial
• thin and  inhabited implies   cogenerator

  [dualized] Any object will be a generator for trivial reasons.
• binary coproducts and  inhabited implies   connected

  [dualized] For any two objects $A,B$ we have the zig-zag $A \to A \times B \to B$.
• groupoid and  binary coproducts and  inhabited implies   trivial

  [dualized] Let $\mathcal{C}$ be an inhabited groupoid with binary products. Then it is connected, so we may assume $\mathcal{C}=BG$ for a group $G$ with unique object $*$. But then $* \times * = *$, so there are $p,q \in G$ such that $G \to G \times G$, $x \mapsto (px,qx)$ is bijective. From here it is an easy exercise to deduce $G=1$.
• cogenerator implies   inhabited

  [dualized] trivial

Examples

• category of abelian groups
• category of Banach spaces with linear contractions
• category of combinatorial species
• category of commutative rings
• category of fields
• category of finite abelian groups
• category of finite orders
• category of finite sets
• category of finite sets and bijections
• category of finite sets and injections
• category of finite sets and surjections
• category of finitely generated abelian groups
• category of free abelian groups
• category of groups
• category of left R-modules
• category of locally ringed spaces
• category of M-sets
• category of measurable spaces
• category of metric spaces with ∞ allowed
• category of metric spaces with continuous maps
• category of metric spaces with non-expansive maps
• category of monoids
• category of non-empty sets
• category of pointed sets
• category of posets
• category of rings
• category of rngs
• category of schemes
• category of sets
• category of sets and relations
• category of simplicial sets
• category of small categories
• category of smooth manifolds
• category of topological spaces
• category of vector spaces
• category of Z-functors
• delooping of a non-trivial finite group
• delooping of an infinite group
• delooping of the additive monoid of natural numbers
• delooping of the additive monoid of ordinal numbers
• discrete category on two objects
• partial order [0,1]
• partial order of extended natural numbers
• partial order of natural numbers
• partial order of ordinal numbers
• preorder of integers w.r.t. divisiblity
• trivial category
• walking isomorphism
• walking morphism
• walking parallel pair of morphisms

Counterexamples

• empty category

Unknown

For these categories the database has no info if they satisfy this property or not.

—