# Preface

This is a writeup of some material developed mainly by James Dolan, with help from me.

## The idea

We’ll tentatively use the following new definition of the existing term ‘algebraic stack’:

Definition: An algebraic geometric theory, or for short an algebraic stack, is a symmetric monoidal finitely cocomplete linear category.

Explanding this somewhat: an algebraic geometric theory $C$ is, for starters, a symmetric monoidal $k Mod$-enriched category, where $k Mod$ is the category of vector spaces over a fixed commutative ring $k$. We also assume that $C$ has finite colimits, and that the operation of taking tensor product distributes over finite colimits.

There should be a doctrine of algebraic geometric theories. In our sense, a doctrine is a $(2,1)$-category with a certain property. Here we only describe the $(2,1)$-category, not checking that it has the necessary property:

Definition: Given algebraic geometric theories $C$ and $D$, a morphism of algebraic geometric theories $f: C \to D$ is a symmetric monoidal linear functor preserving finite colimits. We also call this a model of $C$ in the environment $D$. Given two such morphisms $f, g : C \to D$, a 2-morphism is a natural isomorphism.

Definition: The doctrine of algebraic geometric theories, $\Alg\Geom$ is the 2-category of algebraic geometric theories, morphisms between these, and 2-morphisms between those.

Notice that:

• An algebraic geometric theory can be seen as a categorified version of a commutative ring: the finite colimits play the role of ‘addition’, the tensor product plays the role of ‘multiplication, and the fact that this tensor product is symmetric plays the role of commutativity.

• An algebraic geometric theory can be seen as a close relative of an algebraic stack. In particular, it will have a moduli stack of models, which should often be an ‘algebraic stack’, e.g. an Artin stack.

• The category of coherent sheaves on a stack, in the more usual sense of the word ‘stack’, should be an example of an algebraic geometric theory.

## Examples

To see how the above definitions tie in to existing notions, it is good to consider some examples:

Example: Suppose $X$ is a projective algebraic variety over a field $k$, and let $C$ be the category of coherent sheavesf $k$-modules over $X$. Then $C$ is an algebraic stack. The idea here is that we are thinking of $C$ as a kind of stand-in for $X$. Indeed, for an affine algebraic variety $X$ we can use the commutative ring of algebraic functions as a stand-in for $X$. For a projective variety there are not enough functions of this sort. However, there are plenty of coherent sheaves, and the category $C$ of such sheaves is a categorified version of a commutative ring.

Example: More generally suppose $X$ is a scheme over the commutative ring $k$, and let $C$ be the category of coherent sheaves of $k$-modules over $X$. Then $C$ is an algebraic geometric theory. (Need to check that this is really true: reference?)

Example: Let $G$ be an algebraic group over the field $k$, and let $C$ be the category of finite-dimensional representations of $G$. Then $C$ is an algebraic stack. Unlike the previous examples, this example is really ‘stacky’. In other words: instead of standing in for a set with extra structure, now $C$ is standing in for a groupoid with extra structure, namely the one-object groupoid $G$.

Example: More generally, let $G$ be an group scheme over the commutative ring $k$, and let $C$ be the category of representations of $G$ on finitely generated $k$-modules. Then $C$ is again an algebraic stack. (Need to check that this is really true: reference?)

Example: More generally than all the examples above, let $G$ be an Artin stack over the commutative ring $k$, and let $C$ be the category of coherent sheaves of $k$-modules over $X$. Then $C$ is an algebraic stack. (Need to check that this is really true: reference?)

Here are some examples of a more syntactic nature, where it would be nice to describe an algebraic geometric theory using a ‘sketch’.

Example: The theory of an object should be the free symmetric monoidal cocomplete $k$-linear category on one object $x$. The category of morphisms from this to any algebraic geometric theory $C$ should be equivalent to $C$ itself.

Example: The theory of a strongly $n$-dimensional object. Here we take the previous theory and adjoin an isomorphism $\Lambda^n x \cong 1$ where $1$ is the unit object. Conjecture: working over $k$, this is the same as the category of finite-dimensional algebraic representations of the affine algebraic group scheme $GL(n,k)$. This is probably called $BGL(n,k)$ by some people.

Example: The theory of nothing is the category of finitely generated $k$-modules, $fg k \Mod$. This is the initial algebraic geometric theory, since for any algebraic geometric theory $C$ there is a morphism $f : fg k \Mod \to C$ sending $k$ to the unit object in $C$.

## Examples from Number Theory

Here are some examples from number theory:

Example: Let $G$ be the profinite completion of the absolute Galois group of $\mathbb{Q}$. The category of torsors of this is equivalent to the category of algebraic closures of $\mathbb{Q}$.

The category of representations of $G$ on finitely presented $k$-modules is an algebraic geometric theory, say $D$. Any object of $D$ is a representation of some finite quotient of $G$ on a finitely generated $k$-module. And any of these objects has an $L$-function.

Revised on September 25, 2017 16:55:54 by John Baez (2602:306:30b9:cf40:92b:5cb:4102:d610)