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category theory

# Contents

## Idea

The concept of a cancellative category is the generalization of the concept of cancellative monoid from monoids to categories.

## Definition

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

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

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

## Examples

Last revised on May 25, 2021 at 08:15:23. See the history of this page for a list of all contributions to it.