groupoid
A groupoid is a category in which every morphism is an isomorphism.
Relevant implications
essentially discrete is equivalent to thin and groupoid
trivialgroupoid implies self-dual and mono-regular and pullbacks and filtered limits and left cancellative and well-powered
easyleft cancellative and right cancellative and balanced implies groupoid
trivialgroupoid and equalizers implies thin
The equalizer of any parallel pair must be an isomorphism, so .groupoid and binary products and inhabited implies trivial
Let be an inhabited groupoid with binary products. Then it is connected, so we may assume for a group with unique object . But then , so there are such that , is bijective. From here it is an easy exercise to deduce .groupoid and initial object implies trivial
easygroupoid implies self-dual and epi-regular and pushouts and filtered colimits and right cancellative and well-copowered
[dualized] easygroupoid and coequalizers implies thin
[dualized] The equalizer of any parallel pair must be an isomorphism, so .groupoid and binary coproducts and inhabited implies trivial
[dualized] Let be an inhabited groupoid with binary products. Then it is connected, so we may assume for a group with unique object . But then , so there are such that , is bijective. From here it is an easy exercise to deduce .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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