jointly epimorphic family


Given an object XX in some category, a family (f i:U iX) i(f_i\colon U_i \to X)_i of morphisms to XX is an epic sink, or a jointly epic family if, given any two morphisms g,h:XYg, h\colon X \to Y such that gf i=hf i g \circ f_i= h\circ f_i for all ii, it follows that g=hg = h.

Dually, a family (f i:XU i) i(f_i\colon X \to U_i)_i of morphisms from XX is a monic source, or a jointly monic family if, given any two morphisms g,h:YXg, h\colon Y \to X such that f ig=f ihf_i \circ g = f_i \circ h for all ii, it follows that g=hg = h.

Sometimes we are interested only in small families of morphisms, but if so then it is best to say so explicitly.

A single morphism UXU \to X is an epimorphism if and only it forms an epic sink by itself; conversely, a sink (f i:U iX) i(f_i\colon U_i \to X)_i is epic iff the induced map iU iX\coprod_i U_i \to X is an epimorphism, assuming that the coproduct iU i\coprod_i U_i exists. (Note, though, that for a large family of morphisms, this coproduct might not exist even if the category has all small coproducts.) Dual results hold for monomorphisms and products.

Finally, the empty family of morphisms with domain XX is a monic source iff XX is a subterminal object (and dually).


If a functor F:JCF \colon J \to C has a colimit colimF\mathrm{colim}F, with maps ι i:FicolimF\iota_i \colon F i \to \mathrm{colim}F for iJ i \in J, then the family (ι i) iJ(\iota_i)_{i \in J} is jointly epic. Similarly, the maps (π i:limFFi) iJ(\pi_i \colon \mathrm{lim}F \to F i)_{i\in J} are jointly monic, when the limit exists.

Last revised on April 21, 2021 at 13:50:47. See the history of this page for a list of all contributions to it.