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

• balanced and  left cancellative and  right cancellative implies   groupoid
• coequalizers and  left cancellative implies   thin
• groupoid implies   filtered limits and  left cancellative and  mono-regular and  pullbacks and  self-dual and  well-powered
• initial object and  left cancellative implies   strict initial object
• left cancellative and  self-dual implies   right cancellative
• left cancellative and  terminal object implies   strict terminal object
• left cancellative implies   Cauchy complete
• right cancellative and  self-dual implies   left cancellative
• thin implies   equalizers and  left cancellative

Examples

There are 18 categories with this property.

• 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

There are 33 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 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

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

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