topos theory

# 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 $ℝ$ 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 $ℂ$ (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 ${𝔸}_{T}$ which generalizes the real line $ℝ$.

### Definition

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

$\left(𝒪⊣\mathrm{Spec}\right):T{\mathrm{Alg}}^{\mathrm{op}}\stackrel{\stackrel{𝒪}{←}}{\underset{\mathrm{Spec}}{\to }}\mathrm{Sh}\left(C\right)$(\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 $T↪C\subset T{\mathrm{Alg}}^{\mathrm{op}}$ of $T$-algebras, for $C$ a small full subcategory with subcanonical coverage.

$\left({F}_{T}⊣{U}_{T}\right):T\mathrm{Alg}\stackrel{\stackrel{{F}_{T}}{←}}{\underset{{U}_{T}}{\to }}\mathrm{Set}$(F_T \dashv U_T)\colon T Alg \stackrel{\overset{F_T}{\leftarrow}}{\underset{U_T} {\to}} Set

we have the free $T$-algebra ${F}_{T}\left(*\right)\in \mathrm{TAlg}$ on a single generator.

###### Definition

The $T$-line object is

${𝔸}_{T}≔\mathrm{Spec}{F}_{T}\left(*\right)\in \mathrm{Sh}\left(C\right)\phantom{\rule{thinmathspace}{0ex}}.$\mathbb{A}_T \coloneqq Spec F_T(*) \in Sh(C) \,.

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

$\left({\mathrm{ab}}_{*}⊣{\mathrm{ab}}^{*}\right):T\mathrm{Alg}\stackrel{\stackrel{{\mathrm{ab}}_{*}}{←}}{\underset{{\mathrm{ab}}^{*}}{\to }}\mathrm{Ab}\phantom{\rule{thinmathspace}{0ex}}.$(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 ${𝔸}_{T}$ inherits the structure of an abelian group object in $\mathrm{Sh}\left(C\right)$. Write

${𝔾}_{T}\in \mathrm{Ab}\left(\mathrm{Sh}\left(C\right)\right)$\mathbb{G}_T \in Ab(Sh(C))

for this group object on ${𝔸}_{T}$.

### The multiplicative group object

###### Definition

For ${𝔸}_{T}$ a line object, write

$\left({𝔸}_{T}^{×}↪{𝔸}_{T}\right)\in \mathrm{Sh}\left(C\right)$(\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 ${𝔸}_{T}×{𝔸}_{T}\to {𝔸}_{T}$.

This is called the multiplicative group of the line object, often denoted ${𝔾}_{m}$.

### Examples

• For $T$ the theory of ordinary commutative associative algebras over a ring $R$, we have that

• ${𝔸}_{T}={𝔸}_{R}$ is what is the affine line over $R$;

• ${𝔾}_{m}$ is the standard multiplicative group;

• ${𝔾}_{a}$ is the standard additive group.

• For $T≔\mathrm{Smooth}≔$ CartSp the theory of smooth algebras, we have that ${𝔸}_{\mathrm{Smooth}}=ℝ$ 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 ${ℝ}^{×}=ℝ-\left\{0\right\}$ of non-zero real numbers under multiplication.

#### Properties

##### Cohomology

For $R$ a ring and ${H}_{\mathrm{et}}^{n}\left(-,-\right)$ the etale cohomology, ${𝔾}_{m}$ the multiplicative group of the affine line; then

• ${H}_{\mathrm{et}}^{0}\left(R,{𝔾}_{m}\right)={R}^{×}$ (group of units)

• ${H}_{\mathrm{et}}^{1}\left(R,{𝔾}_{m}\right)=\mathrm{Pic}\left(R\right)$ (Picard group: iso classes of invertible $R$-modules)

• ${H}_{\mathrm{et}}^{2}\left(R,{𝔾}_{m}\right)=\mathrm{Br}\left(R\right)$ (Brauer group Morita classes of Azumaya $R$-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 $C$, 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.

Revised on February 5, 2013 02:45:05 by Urs Schreiber (89.204.154.134)