Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Equalizers and coequalizers as limits and colimits of a parallel pair
Definition
For parallel morphisms , an equalizer is a limit of that parallel-pair diagram (Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties). It is a morphism satisfying such that, whenever satisfies , there is a unique with .
A coequalizer is a colimit of the parallel pair. It is a morphism satisfying such that, whenever satisfies , there is a unique with .
Depends on
Used by
- Every equalizer is a monomorphism, and every coequalizer is an epimorphism Corollary
- Assuming Choice, nonempty sets have all small products but a parallel pair with no equalizer and hence a diagram with no limit Counterexample
- Kernels and cokernels in a category with zero morphisms as equalizers and coequalizers 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
- 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
- Finite, nonempty finite, and connected finite (co)limit criteria in terms of products, equalizers, pullbacks, terminal objects, and their duals Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 8 results over 5 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- E. Riehl, Category Theory in Context, Definition 3.1.15 (standard reference, not scraped)