CatDat

thin

A category is thin when between any pair of objects there is at most one morphism. Such categories correspond to preordered collections of objects.

• Dual property: thin (self-dual)
• nLab Link

Relevant implications

• essentially small and  products implies   thin

  Mac Lane, V.2, Prop. 3. The proof works for any category with products.
• essentially finite and  finite products implies   thin

  Mac Lane, V.2, Prop. 3. The proof can easily be adapted to this case.
• essentially discrete is equivalent to   thin and  groupoid

  trivial
• thin implies   equalizers and  left cancellative

  Any two parallel morphisms are equal, so their equalizer is the identity, and every morphism is a monomorphism as well.
• thin and  inhabited implies   generator

  Any object will be a generator for trivial reasons.
• left cancellative and  coequalizers implies   thin

  If $f,g$ are two parallel morphisms, then their coequalizer is a regular epimorphism, but also a monomorphism by assumption, so it must be an isomorphism. But this means that $f = g$.
• disjoint finite coproducts and  thin implies   trivial

  For every object $A$ the two inclusions $A \rightrightarrows A + A$ must be equal, so their equalizer is $A$, but also $0$ since the coproduct is disjoint. Hence $A = 0$.
• essentially small and  thin and  complete implies   cocomplete

  The supremum of a subset in a (small) partial order is the infimum of the set of upper bounds.
• essentially small and  thin and  complete and  infinitary distributive implies   cartesian closed

  This is an application of the adjoint functor theorem. Specifically, if $P$ is a complete lattice in which $\sup_i \inf(t,x_i) = \inf(t, \sup_i y_i)$ always holds, then the functor $\int(t,-)$ is a left adjoint because it preserves all suprema.
• locally presentable and  self-dual implies   thin

  This follows from Adamek-Rosicky, Thm. 1.64.
• groupoid and  equalizers implies   thin

  The equalizer of any parallel pair $f,g$ must be an isomorphism, so $f=g$.
• thin and  finitely complete implies   Malcev

  In a thin category, every subobject of $X^2 = X$ containing $X$ is already $X$.
• essentially small and  coproducts implies   thin

  [dualized] Mac Lane, V.2, Prop. 3. The proof works for any category with products.
• essentially finite and  finite coproducts implies   thin

  [dualized] Mac Lane, V.2, Prop. 3. The proof can easily be adapted to this case.
• thin implies   coequalizers and  right cancellative

  [dualized] Any two parallel morphisms are equal, so their equalizer is the identity, and every morphism is a monomorphism as well.
• thin and  inhabited implies   cogenerator

  [dualized] Any object will be a generator for trivial reasons.
• right cancellative and  equalizers implies   thin

  [dualized] If $f,g$ are two parallel morphisms, then their coequalizer is a regular epimorphism, but also a monomorphism by assumption, so it must be an isomorphism. But this means that $f = g$.
• essentially small and  thin and  cocomplete implies   complete

  [dualized] The supremum of a subset in a (small) partial order is the infimum of the set of upper bounds.
• groupoid and  coequalizers implies   thin

  [dualized] The equalizer of any parallel pair $f,g$ must be an isomorphism, so $f=g$.

Examples

• discrete category on two objects
• empty category
• 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
• trivial category
• walking isomorphism
• walking morphism

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 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 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 a non-trivial finite group
• delooping of an infinite group
• delooping of the additive monoid of natural numbers
• delooping of the additive monoid of ordinal numbers
• 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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