The concept of a (left/right) cancellative category is the generalization of the concept of cancellative monoid from monoids to categories.
In category theory, a category $\mathcal{C}$ is left cancellative if all morphisms in $\mathcal{C}$ are monomorphisms (for arbitrary morphisms $f,h_0,h_1$ of $\mathcal{C}$, if $f\circ h_0=f\circ h_1$, then $h_0=h_1$). $\mathcal{C}$ is right cancellative if all morphisms in $\mathcal{C}$ are epimorphisms (for arbitrary morphisms $f,h_0,h_1$ of $\mathcal{C}$, if $h_0 \circ f=h_1\circ f$, then $h_0=h_1$). $\mathcal{C}$ is cancellative if it is both left cancellative and right cancellative.
any cancellative monoid, regarded as a category with a single object
any groupoid
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, βOrdered Groupoids and Left Cancellative Categoriesβ Semigroup Forum, Volume 68, Issue 3, (2004), 458β-476
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