strong epimorphism in nLab (original) (raw)

Contents

Context

Category theory

category theory

• category
• functor
• natural transformation
• Cat
• universal construction
  • representable functor
  • adjoint functor
  • limit/colimit
  • weighted limit
  • end/coend
  • Kan extension
• Yoneda lemma
• Isbell duality
• Grothendieck construction
• adjoint functor theorem
• monadicity theorem
• adjoint lifting theorem
• Tannaka duality
• Gabriel-Ulmer duality
• small object argument
• Freyd-Mitchell embedding theorem
• relation between type theory and category theory
• sheaf and topos theory
• enriched category theory
• higher category theory
• applications of (higher) category theory

Contents

• Definition
• Properties
• In higher category theory
• References

Definition

A strong epimorphism in a category CC is an epimorphism which is left orthogonal to any monomorphism in CC.

Properties

• The composition of strong epimorphisms is a strong epimorphism. If f∘gf\circ g is a strong epimorphism, then ff is a strong epimorphism.
• If CC has equalizers, then any morphism which is left orthogonal to all monomorphisms must automatically be an epimorphism.
• Every regular epimorphism is strong. The converse is true if CC is regular.
• Every strong epimorphism is extremal. The converse is true if CC has pullbacks.

In higher category theory

A monomorphism in an (∞,1)-category is a (-1)-truncated morphism in an (∞,1)-category CC.

Therefore it makes sense to define an strong epimorphism in an (∞,1)(\infty,1)-category to be a morphism that is part of the left half of an orthogonal factorization system in an (∞,1)-category whose right half is that of (−1)(-1)-truncated morphisms.

If CC is an (∞,1)-topos then it has an n-connected/n-truncated factorization system for all nn. The (−1)(-1)-connected morphisms are also called effective epimorphisms. Therefore in an (∞,1)(\infty,1)-topos strong epimorphisms again coincide with effective epimorphisms.

References

Strong epimorphisms were introduced in:

• Gregory Maxwell Kelly. Monomorphisms, epimorphisms, and pull-backs. Journal of the Australian Mathematical society 9.1-2 (1969): 124-142.

Textbook accounts:

• Francis Borceux, Def. 4.3.5 in: Handbook of Categorical Algebra Vol. 1: Basic Category Theory, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) (doi:10.1017/CBO9780511525858)

Last revised on October 5, 2022 at 11:16:51. See the history of this page for a list of all contributions to it.