category theory

# Contents

## Idea

In the same way that internal homs in a cartesian monoidal category behave as an internal version of hom-sets between two objects, the object of isomorphisms should behave as an internal version of the sub-hom-set of isomorphisms between two objects.

These generalize bijection sets in the category of sets Set.

## Definition

Given a cartesian monoidal category $C$ and two objects $A \in C$ and $B \in C$, there is a functor

$X \mapsto \mathrm{Hom}_{\mathrm{Core}(C/X)}(A \times X, B \times X)$

which takes an object $X \in C$ to a morphism between objects $A \times X$ and $B \times X$ in the core of the slice category $C/X$. A representing object of the above functor, if it exists, is the object of isomorphisms between $A$ and $B$ and is denoted as $[A, B]_\mathrm{iso}$.

## Properties

If the category is a cartesian closed category, then there is a monomorphism $i:\mathrm{Mono}_C([A, B]_\mathrm{iso}, [A, B])$ between the object of isomorphisms between $A$ and $B$ and the internal hom from $A$ to $B$.