The concept of a (left/right) cancellative category is the generalization of the concept of (left/right) cancellative monoid from monoids to categories.
In category theory, “left cancellative” is a synonym for all arrows are monic. Thus the typical way for left cancellative categories to be constructed to take a category $C$ and then restrict to a class of monomorphisms closed under composition, such as all monomomorphisms, or regular monomorphisms if the category is regular, etc.
In fact every left cancellative $C$ arises this way (in the tautological sense of applying this consideration to $C$ itself): a category $\mathcal{C}$ being left cancellative means all its morphisms are monos.
Equivalently, for arbitrary morphisms $f,h_0,h_1$ of $\mathcal{C}$, if $f\circ h_0=f\circ h_1$, then $h_0=h_1$.
Tautological examples:
any (left) cancellative monoid, regared as a category with a single object
any groupoid
Non-tautological examples:
the category of fields (with ring homomorphisms as the morphisms)
the category of nontrivial vector spaces (over the field of real numbers or complex numbers) equipped with positive definite inner products
M. V. Lawson and A. R. Wallis, A categorical description of Bass-Serre theory (arXiv:1304.6854v5)
M. V. Lawson, “Ordered Groupoids and Left Cancellative Categories” Semigroup Forum, Volume 68, Issue 3, (2004), 458–-476
Last revised on May 23, 2021 at 13:53:16. See the history of this page for a list of all contributions to it.