nLab
line object

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

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

In linear algebra

In linear algebra over a field kk, the line is the field kk regarded as a vector space over itself. More generally, a line is a vector space isomorphic to this, i.e. any 1-dimensional kk-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 TT, there is a canonical space 𝔸 T\mathbb{A}_T which generalizes the real line \mathbb{R}.

Definition

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

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

relates TT-algebras to the sheaf topos over duals TCTAlg opT \hookrightarrow C \subset T Alg^{op} of TT-algebras, for CC a small full subcategory with subcanonical coverage.

By the free T-algebra adjunction

(F TU T):TAlgU TF TSet (F_T \dashv U_T)\colon T Alg \stackrel{\overset{F_T}{\leftarrow}}{\underset{U_T} {\to}} Set

we have the free TT-algebra F T(*)TAlgF_T(*) \in TAlg on a single generator.

Definition

The TT-line object is

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

The additive group object

For 𝒜𝒷\mathcal{Ab} the Lawvere theory of abelian groups, say that a morphism ab:𝒜𝒷Tab\colon \mathcal{Ab} \to T of Lawvere theories is an abelian Lawvere theory. Algebras over abelian Lawvere theories have underlying abelian groups

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

For TT an abelian Lawvere theory, by its underlying abelian group we have that 𝔸 T\mathbb{A}_T inherits the structure of an abelian group object in Sh(C)Sh(C). Write

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

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

The multiplicative group object

Definition

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

(𝔸 T ×𝔸 T)Sh(C) (\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 𝔸 T×𝔸 T𝔸 T\mathbb{A}_T \times \mathbb{A}_T \to \mathbb{A}_T.

This is called the multiplicative group of the line object, often denoted 𝔾 m\mathbb{G}_m.

Examples

See also analytic affine line.

Properties

Cohomology

For RR a ring and H et n(,)H^n_{et}(-,-) the etale cohomology, 𝔾 m\mathbb{G}_m the multiplicative group of the affine line; then

  • H et 0(R,𝔾 m)=R ×H^0_{et}(R, \mathbb{G}_m) = R^\times (group of units)

  • H et 1(R,𝔾 m)=Pic(R)H^1_{et}(R, \mathbb{G}_m) = Pic(R) (Picard group: iso classes of invertible RR-modules)

  • H et 2(R,𝔾 m)=Br(R)H^2_{et}(R, \mathbb{G}_m) = Br(R) (Brauer group Morita classes of Azumaya RR-algebras)

References

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 CC, one may define a line object in CC to be an object LL such that the tensoring functor L:CC- \otimes L : C \to C has an inverse.

Revised on November 7, 2013 12:38:33 by Urs Schreiber (82.169.114.243)