groupoid
A groupoid is a category in which every morphism is an isomorphism.
Relevant implications
- balanced and left cancellative and right cancellative implies groupoid
- binary coproducts and groupoid and inhabited implies trivial
- binary products and groupoid and inhabited implies trivial
- coequalizers and groupoid implies thin
- equalizers and groupoid implies thin
- essentially discrete is equivalent to groupoid and thin
- groupoid and initial object implies trivial
- groupoid and terminal object implies trivial
- groupoid implies epi-regular and filtered colimits and pushouts and right cancellative and self-dual and well-copowered
- groupoid implies filtered limits and left cancellative and mono-regular and pullbacks and self-dual and well-powered
Examples
There are 7 categories with this property.
- 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
There are 44 categories without this property.
- 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
There are 0 categories for which the database has no information on whether they satisfy this property.
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