Implication Details
Assumptions: left cancellative
Conclusions: coreflexive equalizers, effective cocongruences, effective congruences, reflexive coequalizers
Proof: Any parallel pair of morphisms with a common section (or retraction) must be a pair of equal isomorphisms. In particular, they are the kernel pair of the identity morphism on the target, and the cokernel pair of the identity morphism on the source.
Show 41 categories using this implication
- empty category
- trivial category
- discrete category on two objects
- category of algebras
- category of finite sets and bijections
- delooping of an infinite countable group
- delooping of a non-trivial finite group
- delooping of the additive monoid of natural numbers
- delooping of the additive monoid of ordinal numbers
- category of small categories
- simplex category
- category of finite sets and injections
- category of fields
- category of locally ringed spaces
- category of smooth manifolds
- category of measurable spaces
- category of metric spaces with non-expansive maps
- category of metric spaces with continuous maps
- category of metric spaces with ∞ allowed
- category of monoids
- poset of natural numbers
- poset of ordinal numbers
- category of pseudo-metric spaces with non-expansive maps
- category of posets
- category of prosets
- category of sets and relations
- category of rings
- category of rngs
- category of schemes
- category of semigroups
- category of non-empty sets
- category of topological spaces
- category of pointed topological spaces
- proset of integers w.r.t. divisibility
- poset [0,1]
- walking coreflexive pair
- walking fork
- walking idempotent
- walking isomorphism
- walking parallel pair
- walking span