CatDat

groupoid

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

• Dual property: groupoid (self-dual)
• Related properties: left cancellative, quotient-trivial, right cancellative, subobject-trivial
• nLab Link

Relevant implications

• balanced andleft cancellative andright cancellative implies groupoid
• connected andgroupoid implies strongly connected
• core-thin andgroupoid implies thin
• essentially discrete is equivalent to groupoid andthin
• essentially small andgroupoid implies finitely accessible andgeneralized variety
• groupoid andmulti-initial object implies thin
• groupoid andmulti-terminal object implies thin
• groupoid implies directed colimits andepi-regular andpushouts andright cancellative andself-dual andwell-copowered
• groupoid implies directed limits andleft cancellative andmono-regular andpullbacks andself-dual andwell-powered
• groupoid implies locally cartesian closed
• groupoid implies locally cocartesian coclosed
• groupoid implies quotient-trivial
• groupoid implies subobject-trivial

Examples

There are 8 categories with this property.

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

Counterexamples

There are 78 categories without this property.

• category of abelian groups
• category of abelian sheaves
• category of algebras
• category of Banach spaces with linear contractions
• category of combinatorial species
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of compact Hausdorff spaces
• category of countable groups
• category of countable sets
• category of fields
• category of filtered vector spaces
• category of finite abelian groups
• category of finite groups
• category of finite ordered sets
• category of finite sets
• category of finite sets and injections
• category of finite sets and surjections
• category of finite-dimensional vector spaces [countable field]
• category of finite-dimensional vector spaces [finite field]
• category of finite-dimensional vector spaces [uncountable field]
• category of finitely generated abelian groups
• category of free abelian groups
• category of groups
• category of Hausdorff spaces
• category of Jónsson-Tarski algebras
• category of left modules over a division ring
• category of left modules over a ring
• category of locally ringed spaces
• category of M-sets
• category of measurable spaces
• category of metric spaces with continuous maps
• category of metric spaces with non-expansive maps
• category of metric spaces with ∞ allowed
• category of monoids
• category of non-empty sets
• category of pairs of sets
• category of pointed sets
• category of pointed topological spaces
• category of posets
• category of prosets
• category of pseudo-metric spaces with non-expansive maps
• category of rings
• category of rngs
• category of schemes
• category of semigroups
• category of sets
• category of sets and relations
• category of sets with finite-to-one maps
• category of sheaves
• category of simplicial sets
• category of small categories
• category of smooth manifolds
• category of topological spaces
• category of torsion abelian groups
• category of torsion-free abelian groups
• category of vector spaces
• category of Z-functors
• delooping of the additive monoid of natural numbers
• delooping of the additive monoid of ordinal numbers
• dual of the category of sets
• dual of the category of topological spaces
• poset [0,1]
• poset of extended natural numbers
• poset of natural numbers
• poset of ordinal numbers
• proset of integers w.r.t. divisibility
• simplex category
• walking commutative square
• walking composable pair
• walking coreflexive pair
• walking fork
• walking idempotent
• walking morphism
• walking parallel pair
• walking span
• walking splitting

Unknown

There are 0 categories for which the database has no information on whether they satisfy this property.

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