A functor between small categories is a strong epimorphism exactly when its image generates its codomain as a subcategory. A monomorphism between small categories is injective on objects and faithful, so its image can be identified with a subcategory. In a lifting square, all image generators lift to that subcategory; their composites lift too, and faithfulness forces their relations to agree. This gives the diagonal filler. Conversely, factoring through the generated image and applying the lifting property makes that inclusion an isomorphism.
A regular epimorphism need not be a full functor. For example, the unique functor from a two-object discrete category to the terminal category is a coequalizer of the two functors selecting its objects. A functor equalizing those selections has exactly one choice of factorization through the terminal category. Nevertheless the original functor is not full: there is no arrow between the two distinct source objects, whereas the target has its identity arrow.
Every regular epimorphism is strong, but the converse fails. The nonregular extremal epimorphism in the category of categories gives an image-generating functor for which imposing only its kernel pair identifications does not impose all target relations. Since small categories have pullbacks in a category, a regular epimorphism would have to be the coequalizer of that kernel pair.
Articles by others on the same topic
There are currently no matching articles.