Contents

category theory

# Contents

## Idea

The notion of an internal antisymmetric relation is the generalization of that of antisymmetric relations as one passes from the ambient category of sets into more general ambient categories with suitable properties.

## Definitions

In a finitely complete category $C$, an internal antisymmetric relation is an internal relation $R\stackrel{(s,t)}\hookrightarrow X \times X$ on an object $X$ with a monomorphism $\alpha:R \times_X R^\op \hookrightarrow X$ into the diagonal subobject $X$, where $R \times_X R^\op$ is the pullback of the internal relation $(s,t)$ and its opposite internal relation $(t,s)$.