CatDat

left cancellative

A category is left cancellative if for every morphism $f : A \to B$ and every parallel pair of morphisms $g,h : B \to C$ with $f \circ g = f \circ h$ we have $g = h$. Equivalently, every morphism is a monomorphism.

• Dual property: right cancellative
• Related properties: groupoid
• nLab Link

Relevant implications

• 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.
• 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 $f = g$.
• left cancellative and  initial object implies   strict initial object

  It suffices to prove that in general any monomorphism $f : A \to 0$ into an initial object is an isomorphism. If $g : 0 \to A$ is the unique morphism, then $f \circ g = \mathrm{id}_0$ since $0$ is initial. But then $f$ is a split epimorphism and a monomorphism, hence an isomorphism.
• left cancellative implies   Cauchy complete

  Any idempotent monomorphism must be the identity and therefore splits.
• groupoid implies   self-dual and  mono-regular and  pullbacks and  filtered limits and  left cancellative and  well-powered

  easy
• left cancellative and  right cancellative and  balanced implies   groupoid

  trivial
• left cancellative and  terminal object implies   strict terminal object

  [dualized] Let $f : A \to 0$ be a morphism. Let $g : 0 \to A$ be the unique morphism. It is an epimorphism by assumption. Also, $f \circ g = \mathrm{id}_0$ since $0$ is initial. But then $g$ is a split monomorphism and an epimorphism, hence an isomorphism.
• self-dual and  left cancellative implies   right cancellative

  trivial by self-duality
• self-dual and  right cancellative implies   left cancellative

  trivial by self-duality

Examples

• category of fields
• category of finite sets and bijections
• category of finite sets and injections
• 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
• 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
• walking parallel pair of morphisms

Counterexamples

• category of abelian groups
• category of Banach spaces with linear contractions
• category of combinatorial species
• category of commutative rings
• category of finite abelian groups
• category of finite orders
• category of finite sets
• 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

Unknown

For these categories the database has no info if they satisfy this property or not.

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