CatDat

kernel pairs

The kernel pair of a morphism $f : X \to Y$ is the pullback $X \times_Y X \rightrightarrows X$, i.e. the limit of the cospan $X \xrightarrow{f} Y \xleftarrow{f} X$. If the morphism is not clear from the context, we can write $X \times_{f,Y,f} X$ to denote the pullback. We say that a category $\C$ has kernel pairs if every morphism has a kernel pair. Equivalently, each slice category $\C / Y$ has binary powers.

• Dual property: cokernel pairs
• Related properties: binary powers, coequalizers of kernel pairs, kernels, pullbacks, regular
• nLab Link

Relevant implications

• Cauchy complete andkernel pairs andpreadditive implies kernels
• coequalizers of kernel pairs implies kernel pairs
• cokernel pairs andself-dual implies kernel pairs
• kernel pairs andquotients of congruences implies coequalizers of kernel pairs
• kernel pairs andright cancellative implies left cancellative
• kernel pairs andself-dual implies cokernel pairs
• kernel pairs andterminal object implies binary powers
• left cancellative implies kernel pairs
• pullbacks implies kernel pairs

Examples

There are 89 categories with 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 countable groups
• category of countable sets
• category of fields
• category of filtered vector spaces
• category of finite abelian groups
• category of finite groups
• category of finite sets
• category of finite sets and bijections
• category of finite sets and injections
• 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 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 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 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 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 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 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
• 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
• discrete category on two objects
• dual of the category of sets
• dual of the category of topological spaces
• empty category
• forked commutative square
• 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 fork
• walking isomorphism
• walking morphism
• walking parallel pair
• walking span

Counterexamples

There are 8 categories without this property.

• category of finite ordered sets
• category of finite sets and surjections
• category of sets and relations
• category of smooth manifolds
• simplex category
• walking coreflexive pair
• walking idempotent
• walking splitting

Unknown

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

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