symmetric monoidal (∞,1)-category of spectra
For $R$ a ring, its group of units, denoted $R^\times$ or $GL_1(R)$, is the group whose elements are the elements of $R$ that are invertible under the product, and whose group operation is the multiplication in $R$.
The group of units of $R$ is equivalently the collection of morphisms from $Spec R$ into the group of units $\mathbb{G}_m$
There is an adjunction
between the category of associative algebras over $R$ and that of groups, where $R[-]$ forms the group algebra over $R$ and where $(-)^\times$ assigns to an $R$-algebra its group of units.
The group of units of the ring of adeles is the group of ideles.
group of units/multiplicative group, Picard group, Brauer group