CatDat

connected

A category is connected if it is inhabited and every two objects can be joined via a zig-zag path of morphisms. Equivalently, $\mathcal{C}$ is connected if $\mathcal{C} \simeq \coprod_{i \in I} \mathcal{C}_i$ implies $\mathcal{C}_i \simeq 0$ for some $i$.

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

Relevant implications

• essentially discrete and  connected implies   trivial

  trivial
• terminal object implies   connected

  If $1$ denotes the terminal object, then for any two objects $A,B$ we have the zig-zag $A \to 1 \leftarrow B$.
• 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
• connected implies   inhabited

  by definition
• initial object implies   connected

  [dualized] If $1$ denotes the terminal object, then for any two objects $A,B$ we have the zig-zag $A \to 1 \leftarrow B$.
• binary coproducts and  inhabited implies   connected

  [dualized] For any two objects $A,B$ we have the zig-zag $A \to A \times B \to B$.

Examples

• category of abelian groups
• category of Banach spaces with linear contractions
• category of combinatorial species
• category of commutative rings
• category of finite abelian groups
• category of finite orders
• category of finite sets
• category of finite sets and injections
• 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
• 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

• category of fields
• category of finite sets and bijections
• category of finite sets and surjections
• discrete category on two objects
• empty category

Unknown

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

—