nLab strong epimorphism




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


  • The composition of strong epimorphisms is a strong epimorphism. If fgf\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.


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:

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