Unit object

Last updated January 27, 2022

In a monoidal category, a unit object is the unit of the tensor product, i.e. $a\otimes u\simeq a$ and $u\otimes a\simeq a$ for any $a$ in the category.

Examples

• In $\text{Vect}_K$, the unit object is $K$, regarded as a vector space over itself.
• The unit object of the category of linear representations over $\text{Vect}_K$ is the trivial representation which sends each $g\in G$ to the identity automorphism on $K$. Representation of a group