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 andcore-thin implies thin
• binary copowers andleft cancellative implies thin
• binary powers andcore-thin implies thin
• binary powers andright cancellative implies thin
• cartesian closed andstrict terminal object implies thin
• cartesian closed andthin implies locally cartesian closed
• cartesian filtered colimits andthin implies exact filtered colimits
• cocartesian coclosed andstrict initial object implies thin
• cocartesian coclosed andthin implies locally cocartesian coclosed
• cocartesian cofiltered limits andthin implies exact cofiltered limits
• cocomplete andessentially small andinfinitary codistributive andthin implies cocartesian coclosed
• cocomplete andessentially small andthin implies complete
• codistributive andthin implies distributive
• coequalizers andregular-quotient-trivial is equivalent to thin
• cofiltered andessentially finite andthin implies initial object
• complete andessentially small andinfinitary distributive andthin implies cartesian closed
• complete andessentially small andthin implies cocomplete
• copowers andessentially small implies thin
• core-thin andgroupoid implies thin
• cosifted andright cancellative implies thin
• cosifted andthin implies cofiltered
• countable copowers andessentially countable implies thin
• countable copowers andlocally finite implies thin
• countable coproducts andessentially countable andthin implies coproducts
• countable powers andessentially countable implies thin
• countable powers andlocally finite 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 andregular-subobject-trivial is equivalent to thin
• essentially countable andthin andℵ₁-cofiltered implies initial object
• essentially countable andthin andℵ₁-filtered implies terminal object
• essentially discrete is equivalent to groupoid andthin
• essentially finite andfiltered andthin implies terminal object
• 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
• 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
• 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
• sifted andthin implies filtered
• strongly connected andthin implies trivial
• terminal object andthin implies powers
• thin implies binary copowers
• thin implies binary powers
• thin implies coequalizers andcogenerating collection andlocally finite andone-way andright cancellative
• thin implies equalizers andgenerating collection andleft cancellative andlocally finite andone-way

Examples

There are 14 categories with this property.

• discrete category of sets
• discrete category on two objects
• empty category
• partially ordered collection of ordinal numbers
• partially ordered set of extended natural numbers
• partially ordered set of natural numbers
• preordered set of integers w.r.t. divisibility
• real interval [0,1]
• trivial category
• walking commutative square
• walking composable pair
• walking isomorphism
• walking morphism
• walking span

Counterexamples

There are 104 categories without this property.

• category of abelian groups
• category of abelian sheaves
• category of algebras
• category of Banach spaces with linear contractions
• category of cochain complexes of abelian groups
• category of combinatorial species
• category of commutative algebras
• category of commutative monoids
• category of commutative rings
• category of compact Hausdorff spaces
• category of coproducts of Euclidean spaces
• category of countable groups
• category of countable sets
• category of countable-dimensional vector spaces
• category of directed graphs
• category of fields
• category of fields of characteristic zero
• 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 bijections
• category of finite sets and injections
• category of finite sets and surjections
• category of finite sets of cardinality a power of 3
• category of finite sets of even cardinality
• category of finite sets of odd cardinality
• category of finite-dimensional vector spaces
• 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 finitely generated free abelian groups
• category of finitely generated free modules over Z x Z
• category of finitely generated projective modules over the ring of dual numbers
• category of free abelian groups
• category of graded abelian groups
• category of graded modules over a graded ring
• category of groups
• category of Hausdorff spaces
• category of Jónsson-Tarski algebras
• category of large families of abelian groups
• category of large families of sets
• category of large families of sets which are mostly empty
• category of large families of sets which are mostly singletons
• category of large families of vector spaces which are mostly zero
• category of large vector spaces over a large field with a small basis
• category of left modules over a division ring
• category of left modules over a non-semisimple ring
• category of left modules over a ring
• category of locally ringed spaces
• category of long transfinite sequences of abelian groups
• 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 partially ordered sets
• category of partially ordered sets without isolated points
• category of pointed sets
• category of pointed topological spaces
• category of preordered sets
• category of pseudo-metric spaces with non-expansive maps
• category of rings
• category of rngs
• category of schemes
• category of semigroups
• category of sequences of abelian groups
• category of set functions and commutative squares
• category of sets
• category of sets and relations
• category of sets with a binary relation
• category of sets with a distinguished subset
• 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 uniform spaces
• category of vector spaces
• category of Z-functors
• cocompletion of a discrete–pair join
• delooping of a group
• delooping of a non-trivial finite group
• delooping of an infinite countable group
• delooping of an infinite uncountable group
• 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
• forked commutative square
• simplex category
• walking coreflexive pair
• walking fork
• walking idempotent
• walking parallel pair
• walking splitting

Unknown

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

—