CatDat

groupoid

A groupoid is a category in which every morphism is an isomorphism.

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

Relevant implications

• essentially discrete is equivalent to   thin and  groupoid

  trivial
• groupoid implies   self-dual and  mono-regular and  pullbacks and  filtered limits and  left cancellative and  well-powered

  easy
• left cancellative and  right cancellative and  balanced implies   groupoid

  trivial
• groupoid and  equalizers implies   thin

  The equalizer of any parallel pair $f,g$ must be an isomorphism, so $f=g$.
• 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 $G=1$.
• groupoid and  initial object implies   trivial

  easy
• groupoid implies   self-dual and  epi-regular and  pushouts and  filtered colimits and  right cancellative and  well-copowered

  [dualized] easy
• groupoid and  coequalizers implies   thin

  [dualized] The equalizer of any parallel pair $f,g$ must be an isomorphism, so $f=g$.
• 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 $G=1$.
• groupoid and  terminal object implies   trivial

  [dualized] easy

Examples

• category of finite sets and bijections
• delooping of a non-trivial finite group
• delooping of an infinite group
• discrete category on two objects
• empty category
• trivial category
• walking isomorphism

Counterexamples

• 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 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 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
• walking morphism
• walking parallel pair of morphisms

Unknown

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

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