nLab strict epimorphism

Strict epimorphisms

Strict epimorphisms


A strict epimorphism in a category is a morphism which is the joint coequalizer of all pairs of parallel morphisms that it coequalizes. In other words, f:BCf \colon B\to C is a strict epimorphism if it is the colimit of the (possibly large) diagram consisting of all parallel pairs g,h:ABg,h \colon A \;\rightrightarrows\; B such that fg=fhf g = f h.

Although this definition does not include this explicitly, it follows that ff is an epimorphism.

A strict monomorphism is a morphism such that its dual is strict epimorphism in the dual category.

Relation to other epimorphism classes

If ff has a kernel pair r,s:ker(f)Br,s \,\colon\,ker(f) \;\rightrightarrows\; B (such as if the ambient category has pullbacks), then any such pair g,hg,h factor uniquely through the kernel pair, which is itself such a pair (that is, fr=fsf r = f s). Thus, for any k:BDk \colon B\to D, we have kg=khk g = k h for all g,hg,h with fg=fhf g = f h if and only if kr=ksk r = k s. Therefore, ff is strict epi if and only if it is the coequalizer of its kernel pair, hence if and only if it is an effective epimorphism and therefore a regular epimorphism.

For this reason, some sources define “regular epimorphism” in a category without pullbacks to mean what we have called a “strict epimorphism.”

It is easy to see that in any category, any regular epimorphism is strict. In a category without pullbacks, it seems that not every strict epimorphism need be regular. However, every strict epimorphism is strong, and hence extremal, for the same reason that any regular epimorphism is.


If for an epimorphism ff, the composition gfg \circ f is a strict epimorphism then gg is a strict epimorphism.


See also Definition 2.1 in

Textbook accounts:

Last revised on December 12, 2023 at 03:22:00. See the history of this page for a list of all contributions to it.