# nLab line object

Contents

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

topos theory

## Theorems

#### Algebra

higher algebra

universal algebra

# Contents

## 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.

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

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)

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

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.

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

$n$Calabi-Yau n-foldline n-bundlemoduli of line n-bundlesmoduli of flat/degree-0 n-bundlesArtin-Mazur formal group of deformation moduli of line n-bundlescomplex oriented cohomology theorymodular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
$n = 0$unit in structure sheafmultiplicative group/group of unitsformal multiplicative groupcomplex K-theory
$n = 1$elliptic curveline bundlePicard group/Picard schemeJacobianformal Picard groupelliptic cohomology3d Chern-Simons theory/WZW model
$n = 2$K3 surfaceline 2-bundleBrauer groupintermediate Jacobianformal Brauer groupK3 cohomology
$n = 3$Calabi-Yau 3-foldline 3-bundleintermediate JacobianCY3 cohomology7d Chern-Simons theory/M5-brane
$n$intermediate Jacobian