Equalizers and coequalizers as limits and colimits of a parallel pair

Depends on

• Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties

Used by

• Every equalizer is a monomorphism, and every coequalizer is an epimorphism Corollary
• A reflexive coequalizer of sets not preserved by Set(ℕ,-) Counterexample
• Assuming Choice, nonempty sets have all small products but a parallel pair with no equalizer and hence a diagram with no limit Counterexample
• The group coequalizer of doubling and zero on the integers is not its underlying-set coequalizer Counterexample
• Freyd's axioms A0, A1, A1*, A2, A2*, A3, and A3* for abelian categories Definition
• Kernels and cokernels in a category with zero morphisms as equalizers and coequalizers Definition
• Reflexive parallel pairs and reflexive coequalizers Definition
• Split coequalizer diagrams Definition
• Equalizers in Set are agreement subsets and coequalizers are quotients by the generated equivalence relation Example
• In a poset regarded as a category, products are infima, coproducts are suprema, and equalizers are automatic Example
• The equalizer of two group homomorphisms is their agreement subgroup Example
• FALSE: every category has all small limits False statement
• Supplied created canonical presentations give a quasi-inverse to the comparison functor Lemma
• Under dependent choice, algebras for a finitary monad on a complete cocomplete locally small category have coequalizers Lemma
• Over a cocomplete base, a monadic category is cocomplete exactly when it has coequalizers Proposition
• A complete locally small category with a jointly weakly initial set has an initial object, without class-indexed choice Theorem
• A complete locally small category with a small coseparating set and intersections of all subobject collections has an initial object Theorem
• A set-valued coend is the disjoint union of the diagonal values modulo the dinaturality relation Theorem
• A splitting of an idempotent is simultaneously an equalizer and a coequalizer and is unique up to unique isomorphism Theorem
• An end is the equalizer of two products, and a coend the coequalizer of two coproducts Theorem
• Every algebra is the coequalizer of its canonical pair of free algebras Theorem
• Every small colimit can be constructed as a coequalizer between coproducts over the arrows and objects of the index category Theorem
• Every small limit can be constructed as an equalizer between products over the objects and arrows of the index category Theorem
• Every split coequalizer is a coequalizer and an absolute colimit Theorem
• Finite, nonempty finite, and connected finite (co)limit criteria in terms of products, equalizers, pullbacks, terminal objects, and their duals Theorem
• In a preadditive category, the equalizer of a parallel pair is the kernel of their difference Theorem
• The sheaf axiom is the equalizer condition on a cover Theorem

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• E. Riehl, Category Theory in Context, Definition 3.1.15 (standard reference, not scraped)