Contents

category theory

Contents

Idea

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, “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 $h_0=h_1$.

Examples

Tautological examples:

Non-tautological examples:

References

• 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.