nLab line object

This entry discusses line objects, their multiplicative groups and additive groups in generality. For the traditional notions see at affine line.

In linear algebra

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.

Over an algebraic theory

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 $\mathbb{R}$.

Definition

For $T$ (the syntactic category of) any Lawvere theory we have that Isbell conjugation

$$(\mathcal{O} \dashv Spec)\colon T Alg^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}} Sh(C)$$

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

$$(F_T \dashv U_T)\colon T Alg \stackrel{\overset{F_T}{\leftarrow}}{\underset{U_T} {\to}} Set$$

we have the free $T$-algebra $F_T(*) \in TAlg$ on a single generator.

Definition

The $T$-line object is

$$\mathbb{A}_T \coloneqq Spec F_T(*) \in Sh(C) \,.$$

The additive group object

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

$$(ab_* \dashv ab^*)\colon T Alg \stackrel{\overset{ab_*}{\leftarrow}}{\underset{ab^*}{\to}} Ab \,.$$

Definition

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

$$\mathbb{G}_T \in Ab(Sh(C))$$

for this group object on $\mathbb{A}_T$.

The multiplicative group object

Definition

For $\mathbb{A}_T$ a line object, write

$$(\mathbb{A}_T^\times \hookrightarrow \mathbb{A}_T) \in Sh(C)$$

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 $\mathbb{G}_m$.

The group of roots of unity

See at roots of unity.

Examples

• 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.

Properties

Cohomology

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)

Related concepts

• function algebra

• formal multiplicative group

• exponential exact sequence

• affine modality

• Picard scheme, Picard modality?

References

The notion of a line object over general abelian Lawvere theories has been considered in

• Herman Stel, ∞-Stacks and their Function Algebras – with Applications to ∞-Lie Theory (2010)

in the context of function algebras on ∞-stacks.

In a monoidal category

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.

Related concepts

• 2-line

moduli spaces of line n-bundles with connection on $n$-dimensional $X$

$n$	Calabi-Yau n-fold	line n-bundle	moduli of line n-bundles	moduli of flat/degree-0 n-bundles	Artin-Mazur formal group of deformation moduli of line n-bundles	complex oriented cohomology theory	modular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
$n = 0$		unit in structure sheaf	multiplicative group/group of units		formal multiplicative group	complex K-theory
$n = 1$	elliptic curve	line bundle	Picard group/Picard scheme	Jacobian	formal Picard group	elliptic cohomology	3d Chern-Simons theory/WZW model
$n = 2$	K3 surface	line 2-bundle	Brauer group	intermediate Jacobian	formal Brauer group	K3 cohomology
$n = 3$	Calabi-Yau 3-fold	line 3-bundle		intermediate Jacobian		CY3 cohomology	7d Chern-Simons theory/M5-brane
$n$				intermediate Jacobian