Monomorphism and epimorphism by left and right cancellation

Depends on

• Category, object, morphism, domain, codomain, identity, composition, and hom-collection

Used by

• Right adjoints preserve monomorphisms and left adjoints preserve epimorphisms Corollary
• A functor need not preserve monomorphisms Counterexample
• A monotone functor between poset categories preserves every monomorphism but need not preserve pullbacks Counterexample
• Comember and the dual calculus Definition
• Equivalence of members Definition
• Freyd's axioms A0, A1, A1*, A2, A2*, A3, and A3* for abelian categories Definition
• Normal monomorphisms and conormal epimorphisms Definition
• Subobject and quotient object as mutual-factorisation classes of monomorphisms and epimorphisms Definition
• Subobject classifier Definition
• Well-powered and co-well-powered categories, and supplied well-powerings Definition
• A pullback of a monomorphism is a monomorphism, and a pushout of an epimorphism is an epimorphism Lemma
• The legs of a limiting cone are jointly monic, and the legs of a colimiting cocone are jointly epic Lemma
• Wide pullbacks compute intersections of supplied set-indexed subobject representatives independently of the representatives Lemma
• Identities and composites of monomorphisms or epimorphisms retain cancellation; split monomorphisms are monic and split epimorphisms are epic Proposition
• The kernel of a monomorphism is zero and the cokernel of an epimorphism is zero 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
• Epimorphy is detected by members Theorem
• For a monad, invertibility of multiplication, monicity of every multiplication component, and equality of the two whiskered units are equivalent Theorem
• Fullness and faithfulness of a right adjoint are detected by its counit Theorem
• In a preadditive category with a zero object, a morphism is monic exactly when its kernel is zero Theorem
• In Set, monomorphisms are exactly injections and epimorphisms are exactly surjections Theorem
• Monicity is detected by members Theorem
• Mutual factorisation is an equivalence relation on monomorphisms into an object and dually on epimorphisms out of it Theorem
• The inclusion ℤ↪ℚ is monic and epic but neither surjective nor an isomorphism in Ring 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

• Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)