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category theory

# Contents

## Idea

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

## Definition

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.

## 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

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