John Baez
Algebraic geometric theories

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 is a symmetric monoidal finitely cocomplete linear category.

Explanding this somewhat: an algebraic geometric theory CC is, for starters, a symmetric monoidal kModk Mod-enriched category, where kModk Mod is the category of vector spaces over a fixed commutative ring kk. We also assume that CC 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)(2,1)-category with a certain property. Here we only describe the (2,1)(2,1)-category, not checking that it has the necessary property:

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

Definition: The doctrine of algebraic geometric theories, AlgGeom\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 XX is a projective algebraic variety over a field kk, and let CC be the category of coherent sheavesf kk-modules over XX. Then CC is an algebraic stack. The idea here is that we are thinking of CC as a kind of stand-in for XX. Indeed, for an affine algebraic variety XX we can use the commutative ring of algebraic functions as a stand-in for XX. For a projective variety there are not enough functions of this sort. However, there are plenty of coherent sheaves, and the category CC of such sheaves is a categorified version of a commutative ring.

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

Example: Let GG be an algebraic group over the field kk, and let CC be the category of finite-dimensional representations of GG. Then CC 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 CC is standing in for a groupoid with extra structure, namely the one-object groupoid GG.

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

Example: More generally than all the examples above, let GG be an Artin stack over the commutative ring kk, and let CC be the category of coherent sheaves of kk-modules over XX. Then CC 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 kk-linear category on one object xx. The category of morphisms from this to any algebraic geometric theory CC should be equivalent to CC itself.

Example: The theory of a strongly nn-dimensional object. Here we take the previous theory and adjoin an isomorphism Λ nx1\Lambda^n x \cong 1 where 11 is the unit object. Conjecture: working over kk, this is the same as the category of finite-dimensional algebraic representations of the affine algebraic group scheme GL(n,k)GL(n,k). This is probably called BGL(n,k)BGL(n,k) by some people.

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

Examples from Number Theory

Here are some examples from number theory:

Example: Let GG 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 GG on finitely presented kk-modules is an algebraic geometric theory, say DD. Any object of DD is a representation of some finite quotient of GG on a finitely generated kk-module. And any of these objects has an LL-function.

Revised on May 2, 2012 04:18:05 by David Roberts (203.24.207.183)