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