#
nLab

unit object

### Context

#### Category theory

## Concepts

## Universal constructions

## Theorems

## Extensions

## Applications

# Contents

## Idea

In a monoidal category $(C, \otimes, I)$, the **unit object** (or *tensor unit*) $I$ is the object which plays the role of the unit for the tensor product $\otimes$, in that for any other object $a$ there are isomorphism $a \otimes I \simeq a$ and $I \otimes a \simeq a$ (the unitors).

Dually, in a closed category there is a *unit object* which is such that maps out of it into an internal hom correspond to the external hom.

Last revised on February 4, 2014 at 02:57:15.
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