The concept of a (left/right) cancellative category is the generalization of the concept of (left/right) cancellative monoid from monoids to categories.

Definition

In category theory, “right cancellative” is a synonym for all arrows are epic. Thus the typical way for right cancellative categories to be constructed to take a category $C$ and then restrict to a class of epimorphisms closed under composition, such as all epimomorphisms, or extremal epimorphisms, etc.

In fact every right cancellative $C$ arises this way (in the tautological sense of applying this consideration to $C$ itself): a category$\mathcal{C}$ being right cancellative means all its morphisms are epis.

Equivalently, for arbitrary morphisms $f,h_0,h_1$ of $\mathcal{C}$, if $h_0 \circ f=h_1\circ f$, then $h_0=h_1$.