thin
A category is thin when between any pair of objects there is at most one morphism. Such categories correspond to preordered collections.
Relevant implications
- binary copowers andleft cancellative implies thin
- binary coproducts andone-way implies thin
- binary powers andright cancellative implies thin
- binary products andone-way implies thin
- cartesian closed andstrict terminal object implies thin
- cartesian filtered colimits andthin implies exact filtered colimits
- cocartesian coclosed andstrict initial object implies thin
- cocomplete andessentially small andinfinitary codistributive andthin implies cocartesian coclosed
- cocomplete andessentially small andthin implies complete
- codistributive andthin implies distributive
- coequalizers andleft cancellative implies thin
- complete andessentially small andinfinitary distributive andthin implies cartesian closed
- complete andessentially small andthin implies cocomplete
- copowers andessentially small implies thin
- cosifted andright cancellative implies thin
- countable copowers andessentially countable implies thin
- countable coproducts andessentially countable andthin implies coproducts
- countable powers andessentially countable implies thin
- countable products andessentially countable andthin implies products
- disjoint finite coproducts andstrict terminal object implies thin
- disjoint finite coproducts andthin implies trivial
- disjoint finite products andstrict initial object implies thin
- disjoint finite products andthin implies trivial
- distributive andthin implies codistributive
- equalizers andright cancellative implies thin
- essentially countable andthin implies ℵ₁-accessible
- essentially discrete is equivalent to groupoid andthin
- essentially finite andfinite copowers implies thin
- essentially finite andfinite coproducts andthin implies coproducts
- essentially finite andfinite powers implies thin
- essentially finite andfinite products andthin implies products
- essentially small andpowers implies thin
- finite products andthin implies natural numbers object
- finitely cocomplete andthin implies co-Malcev
- finitely cocomplete andthin implies coregular
- finitely complete andthin implies Malcev
- finitely complete andthin implies regular
- groupoid andmulti-initial object implies thin
- groupoid andmulti-terminal object implies thin
- groupoid andone-way implies thin
- inhabited andthin andzero morphisms implies trivial
- inhabited andthin implies cogenerator
- inhabited andthin implies generator
- initial object andthin implies copowers
- left cancellative andsifted implies thin
- left cancellative andzero morphisms implies thin
- one-way andzero morphisms implies thin
- regular quotient object classifier andstrict initial object implies thin
- regular subobject classifier andstrict terminal object implies thin
- right cancellative andzero morphisms implies thin
- semi-strongly connected andthin implies locally cartesian closed
- semi-strongly connected andthin implies locally cocartesian coclosed
- strongly connected andthin implies trivial
- terminal object andthin implies powers
- thin implies binary copowers
- thin implies binary powers
- thin implies coequalizers andright cancellative
- thin implies cogenerating set andlocally essentially small andone-way
- thin implies equalizers andleft cancellative
- thin implies generating set andlocally essentially small andone-way
Examples
There are 13 categories with this property.
- discrete category on two objects
- empty category
- poset [0,1]
- poset of extended natural numbers
- poset of natural numbers
- poset of ordinal numbers
- proset of integers w.r.t. divisibility
- trivial category
- walking commutative square
- walking composable pair
- walking isomorphism
- walking morphism
- walking span
Counterexamples
There are 57 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 fields
- category of finite abelian groups
- category of finite groups
- category of finite ordered sets
- category of finite sets
- category of finite sets and bijections
- 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 Hausdorff spaces
- 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 sets
- category of sets and relations
- category of sheaves
- 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 countable group
- delooping of the additive monoid of natural numbers
- delooping of the additive monoid of ordinal numbers
- dual of the category of sets
- simplex category
- walking fork
- walking idempotent
- walking parallel pair
Unknown
There are 0 categories for which the database has no information on whether they satisfy this property.
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