This entry discusses line objects, their multiplicative groups and additive groups in generality. For the traditional notions see at affine line.
In linear algebra over a field $k$, the line is the field $k$ regarded as a vector space over itself. More generally, a line is a vector space isomorphic to this, i.e. any 1-dimensional $k$-vector space.
The real line $\mathbb{R}$ models the naive intuition of the geometric line in Euclidean geometry. See also at complex line.
In many contexts of modern mathematics, however, line implicitly refers to the complex line $\mathbb{C}$ (which as a real vector space is the complex plane!). For instance this is the line usually meant when speaking of line bundles.
We discuss here how in the context of spaces modeled on duals of algebras over an algebraic theory $T$, there is a canonical space $\mathbb{A}_T$ which generalizes the real line $\mathbb{R}$.
For $T$ (the syntactic category of) any Lawvere theory we have that Isbell conjugation
relates $T$-algebras to the sheaf topos over duals $T \hookrightarrow C \subset T Alg^{op}$ of $T$-algebras, for $C$ a small full subcategory with subcanonical coverage.
By the free T-algebra adjunction
we have the free $T$-algebra $F_T(*) \in TAlg$ on a single generator.
The $T$-line object is
For $\mathcal{Ab}$ the Lawvere theory of abelian groups, say that a morphism $ab\colon \mathcal{Ab} \to T$ of Lawvere theories is an abelian Lawvere theory. Algebras over abelian Lawvere theories have underlying abelian groups
For $T$ an abelian Lawvere theory, by its underlying abelian group we have that $\mathbb{A}_T$ inherits the structure of an abelian group object in $Sh(C)$. Write
for this group object on $\mathbb{A}_T$.
For $\mathbb{A}_T$ a line object, write
be the maximal subobject of the line on those elements that have inverses under the multiplication $\mathbb{A}_T \times \mathbb{A}_T \to \mathbb{A}_T$.
This is called the multiplicative group of the line object, often denoted $\mathbb{G}_m$.
For $T$ the theory of ordinary commutative associative algebras over a ring $R$, we have that
$\mathbb{A}_T = \mathbb{A}_R$ is what is the affine line over $R$;
$\mathbb{G}_m$ is the standard algebraic multiplicative group;
$\mathbb{G}_a$ is the standard algebraic additive group.
For $T \coloneqq Smooth \coloneqq$ CartSp the theory of smooth algebras, we have that $\mathbb{A}_{Smooth} = \mathbb{R}$ is the real line regarded as a diffeological space.
The additive group in this case the the additive Lie group of real numbers. The multiplicative group is the Lie group $\mathbb{R}^\times = \mathbb{R} - \{0\}$ of non-zero real numbers under multiplication.
See also analytic affine line.
For $R$ a ring and $H^n_{et}(-,-)$ the etale cohomology, $\mathbb{G}_m$ the multiplicative group of the affine line; then
$H^0_{et}(R, \mathbb{G}_m) = R^\times$ (group of units)
$H^1_{et}(R, \mathbb{G}_m) = Pic(R)$ (Picard group: iso classes of invertible $R$-modules)
$H^2_{et}(R, \mathbb{G}_m) = Br(R)$ (Brauer group Morita classes of Azumaya $R$-algebras)
The notion of a line object over general abelian Lawvere theories has been considered in
in the context of function algebras on ∞-stacks.
Given a monoidal category $C$, one may define a line object in $C$ to be an object $L$ such that the tensoring functor $- \otimes L : C \to C$ has an inverse.