John Baez
Algebraic Stacks

Preface

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

Definition and Examples

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

An algebraic stack is a symmetric monoidal finitely cocomplete linear category.

Explanding this somewhat: an algebraic stack $C$ is, for starters, a symmetric monoidal $k\mathrm{Mod}$-enriched category, where $k\mathrm{Mod}$ 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 are more elegant ways to say the same thing, but let us forego them. More important is to notice that:

• An algebraic stack 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 stack can be seen as a kind of ‘theory’. This will have a moduli stack of models, in the more usual sense of the word ‘stack’ — and this is part of why the term ‘algebraic stack’ is justified.

• The category of coherent sheaves on a stack, in the more usual sense of the word ‘stack’, is an example of an algebraic stack as defined above — and this is another justification for our terminology.

To see how the definition ties 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 stack. (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?)