For now, the dominant theme of our discussions has been on developing aspects of Jim’s approach to various styles of doing algebraic geometry as explained in terms of various categorical doctrines.
A key motto, one which might not be immediately comprehensible on first reading, but which is anyway a concentrated expression of our program, is:
Thus, under a logical completeness theorem which allows us to retrieve a theory from its “space of models”, the motto means we ought to be able to understand spaces of algebro-geometric type by considering them as theories of various sorts. The word ‘theory’ is here understood in the sense of categorical logic: there are for example coherent theories, geometric theories, left exact theories, finite product theories, etc., as described either by syntactic categories which bear enough structure in which to interpret the language of the theory, or otherwise by classifying-category constructions (e.g., Grothendieck toposes seen as classifying geometric theories). Such structured categories naturally group together to form a doctrine, in a sense we describe more precisely below.
Our point of view is that various approaches to algebraic geometry are usefully described in terms of appropriately chosen categorical doctrines. Some quick examples to indicate some assorted flavors:
Classical algebraic geometry, say over a field $k$, has to a large degree been the study of projective varieties and their properties. A principal technique for considering the geometry of a variety $X$ as projective algebraic geometry, in other words for constructing suitable maps $X\to {\mathbb{P}}^{n}(k)$, is by considering line bundles on $X$ and their sections. In a sense we may consider the category of line bundles as “knowing all about” the projective geometry of $X$. Now, the category of line bundles over a variety forms a special type of $k$-linear tensor category in which every object is invertible. These sorts of tensor categories are actually interesting for their own sake; we call them “dimensional categories” or “dimensional theories”, and we think of them as collectively forming a doctrine useful for the study of projective algebraic geometry.
Branching out from just line objects or dimensional theories, algebraic geometers are intensely interested in more general “vector bundles” over schemes, i.e., in coherent and quasicoherent sheaves of modules over them, their cohomology, and so on. Such sheaves form nice (finitely) cocomplete abelian categories equipped with a nice tensor product, and in some sense they know all about the intrinsic geometry of a scheme. Such structured categories are interesting for their own sake, and may be regarded as “theories” of a “doctrine”; we call them AG theories. In fact, one can often characterize or understand a scheme in terms of logical properties of an AG theory, or indeed see the scheme itself as a moduli space associated with an AG theory.
Toric algebraic geometry on the other hand is about sheaves of permutation representations of schemes which locally are spectra of commutative monoid algebras that are glued together by certain toric localizations. (We call these sheaves of representations “toric quasicoherent sheaves”.) Such sheaf categories are naturally viewed as cocomplete toposes equipped with a nice tensor product, and these once again are objects of a yet another doctrine.
The emphasis on doctrines is not often taught in the schools (even if many aspects of it are known to cognoscenti), but our experience is that it helps provide motivated conceptual explanations for the precise form of the basic objects of study in algebraic geometry. Thus, this point of view may be well-suited for categorically minded readers who wonder whether there are conceptual categorical reasons for the precise definition of ‘scheme’, as opposed to any of a dozen other notions one might get by fiddling with definitional knobs. And why, conceptually, it makes logical sense to use the methods that are in fact used in the study of schemes.
However, because this point of view might be seen as pretty novel or off-the-wall, and carries with it a danger of looking like stratospheric abstract nonsense, it might be wise to proceed somewhat in a bottom-up style, building intuition on special cases and illustrative examples. Perhaps what we mean is that we will adopt the standpoint of a classroom lecturer who is aiming to teach the real flavor of a subject, rather than by wimping out and just presenting an abstract theory.
But, in fact as teachers we are really teaching ourselves, and part of the measure of our understanding this approach to algebraic geometry is how far we succeed in the hard conceptual analysis involved in making clear to ourselves the precise description of the doctrines involved.
Here are a few quick examples of doctrines which will be useful in our development. They are all 2-categories (or (2, 1)-categories) whose objects are structured categories of one kind or another.
The doctrine of Grothendieck toposes and geometric morphisms,
The doctrine of categories with filtered colimits and functors that preserve them,
The doctrine of symmetric monoidal $k$-linear categories, with some cocompleteness properties built in (e.g., finite cocompleteness, full cocompleteness; in each case the tensor should preserve colimits in each argument),
The doctrine of symmetric monoidal categories, again with cocompleteness assumption built in.
There will be more examples as we proceed. In each of these cases, we may think of the objects as types of theories, with 1-cells being the models of the theory and the 2-cells as model homomorphisms. (The models can also be considered “points” of a “spectrum”, a tradition that goes back to Stone duality. Often we are rather more in the mood for stacks, particularly when we take the model isomorphisms seriously, which will be much of the time.)
Some of the doctrines involved have a more syntactic flavor, and others have a semantic flavor, and often there is a kind of adjunction between the syntax and the semantics. A typical example might be the doctrine of finitely complete categories and left exact functors (as an example of a doctrine with a ‘syntactic flavor’, where theories are often presented by limit sketches), whereas a typical example of a doctrine with a more semantic flavor might be the doctrine of locally finitely presentable categories and finitely accessible continuous functors. These two doctrines are connected by Gabriel-Ulmer duality.
Frequently, when we use the word ‘theory’, we are placing ourselves on the side of a syntactic doctrine. There can be ‘big’ theories and ‘small theories’. The big theories often play the role of ‘backgrounds’ in which models of small theories are interpreted. For example, in the doctrine of categories with finite products (Lawvere theories), the category $\mathrm{Set}$ is a typical background theory in which models of small Lawvere theories are valued. Another way of thinking of this is that some objects of a doctrine play the role of ‘tautological’ theories; for example, for the doctrine of props, there are tautological endomorphism props whose values are ${X}_{m,n}=\mathrm{hom}({X}^{\otimes m},{X}^{\otimes n})$ for some object $X$ in a symmetric monoidal category, and then an algebra of the prop $P$ will be such an object $X$ together with a prop morphism $P\to {X}_{\u2022}$.
We may as well give one example of a tautological theory and the doctrine in which it lives; this will play a key role throughout.
A (finitary) algebro-geometric theory (AG theory for short) over a commutative ring $k$ is a symmetric monoidal finitely cocomplete $k$-$\mathrm{Mod}$-enriched category. (The words ‘symmetric monoidal’ include the condition that the tensor preserves colimits in separate arguments.) If $R$ is a $k$-algebra, the tautological AG theory of $R$ is the category of $R$-modules, with the usual tensor product.
Of course, $\mathrm{Mod}(R)$ can be a receiver or background for theories of other doctrines as well, such as for example small-cocomplete abelian categories and left exact left adjoints. The particular background doctrine in which $\mathrm{Mod}(R)$ is considered to live will generally be clear from context. Here are some formal definitions.
A doctrine is a locally presentable (2, 1)-category, i.e., a locally presentable groupoid-enriched category.
(N.B.: Any 2-category can be considered groupoid-enriched by restricting to its invertible 2-cells.)
Given a theory $T$ (in a doctrine $\mathcal{D}$) and a commutative algebra $R$ over $k$, a model of $T$ over $R$ (or an $R$-point of $T$) is a morphism $T\to \mathrm{Mod}(R)$ in $\mathcal{D}$.
One of the central concepts of our development is the following spectrum construction. Consider the (2, 1)-category whose objects are prestacks on affine spectra, i.e., groupoid-valued pseudofunctors
(We are undecided at the moment whether we want all commutative $k$-algebras, or just the finitely presented ones. We’ll let that pass for the moment.)
The spectrum (or moduli prestack) of a theory $T$ in a doctrine $\mathcal{D}$ is the pseudofunctor $\mathrm{Spec}(T):{\mathrm{Alg}}_{k}\to \mathrm{Gpd}$ defined by $\mathrm{Spec}(T)(R)=\mathcal{D}(T,\mathrm{Mod}(R))$.
Thus, the $\mathcal{D}$-spectrum defines a pseudofunctor
where $\mathrm{Aff}={\mathrm{Alg}}_{k}^{\mathrm{op}}$ is the category of affine spectra. (Depending on doctrinal needs, $\mathrm{Aff}$ may be replaced by some other category, but this should give the basic flavor of what we mean by a “moduli stack” or spectrum.)
In the other direction, we have a pseudofunctor
which we call “globalization”. Intuitively, it is the left Kan extension of the pseudofunctor
along the Yoneda embedding $y:\mathrm{Aff}\to \mathrm{Prestack}(\mathrm{Aff})$. Slightly more precisely, if we consider $\mathrm{Prestack}(\mathrm{Aff})$ as a free cocompletion of $\mathrm{Aff}$, so that any prestack $S$ is canonically given as the (weak or pseudo) colimit of representables,
then $\mathcal{O}(S)$ is the corresponding colimit in ${\mathcal{D}}^{\mathrm{op}}$:
By construction, then, we have a (bi)adjunction
and often we will be interested in the fixed points of this and other adjunctions.
Let us consider as our first test case the geometry of a projective variety or projective scheme $X$. Recall that projective varieties in projective space ${\mathbb{P}}^{n}$ are zero sets of homogeneous polynomials in projective coordinates ${x}_{0},{x}_{1},\dots ,{x}_{n}$. The homogeneous polynomials which vanish on a projective variety $X$ form a homogeneous ideal $I$; the ring of polynomials modulo $I$ form a graded algebra $A$, and $X$ is obtained by applying a “Proj” construction to $A$ ($X=\mathrm{Proj}(A)$; see for instance Hartshorne for details).
(Note that this is really an extrinsic description: projective varieties as subvarieties $X\hookrightarrow {\mathbb{P}}^{n}$, corresponding to graded algebra quotients $A\to A/I$, where $A$ is the graded polynomial algebra $k[{x}_{0},\dots ,{x}_{n}]$. One is also interested in intrinsic geometry of $X$, not tied to a particular embedding.)
Projective varieties are in marked contrast to affine varieties. To study affine varieties $X$ over a field $k$, we may consider global functions $f:X\to k$ and understand $X$ in terms of the ring of global functions $\mathcal{O}(X)$. Indeed, an affine variety is retrieved from the ring of global functions essentially by taking the spectrum of the ring; from the modern point of view, the category of affine schemes is simply the dual or opposite of the category of commutative $k$-algebras, thus affording a complete dictionary between the algebra and the geometry.
In the case of projective varieties $X$, this approach does not work because usually there aren’t many global functions. (The case of algebraic geometry over the complex numbers is illustrative: we mean that there are few holomorphic global functions. Indeed, projective varieties over $\u2102$ are compact, hence a global holomorphic function, having bounded image in $\u2102$, would be constant.) Therefore, to describe geometrically the homogeneous functions whose zero sets give rise to a projective variety, one must proceed a bit more subtly.
There are various ways to describe the process (involving the language of divisors, linear systems, etc.), but in a nutshell, it turns out that modules of homogeneous polynomials on $X$ can be described as modules of sections of certain line bundles over $X$. (Global functions on $X$ would correspond to sections of the trivial line bundle.) Thus, it is the category of line bundles on $X$ which knows all about $X$, and we can retrieve the graded commutative algebra $A$ for which $X=\mathrm{Proj}(A)$ by studying sections of line bundles.
In the study of algebraic geometry, the notion of line bundle (aka “invertible sheaf”) is definable in terms of the intrinsic geometry of a scheme $X$. Line bundles can be tensored to form new line bundles, and the category of line bundles on $X$ forms a very special sort of symmetric monoidal category $\mathrm{Line}(X)$. Each line bundle $L$ has a dual line bundle ${L}^{*}$, making $\mathrm{Line}(X)$ a compact closed category, but moreover
The adjunctions $L\u22a3{L}^{*}$ are adjoint equivalences (this captures what we mean by “invertible”), and
The symmetry or braiding $\sigma :V\otimes V\to V\otimes V$ is strict in the sense that the component ${\sigma}_{v}:v\otimes v\to v\otimes v$ is an identity for each object $v$ of $V$ . (This has to do with commutativity of multiplication of scalars.)
This makes the category of line bundles rather special as a symmetric monoidal category, and provides a particular doctrine useful for projective algebraic geometry.
A dimensional category over $k$ is a symmetric monoidal ${\mathrm{Mod}}_{k}$-enriched category in which every object is invertible and the symmetry is strict. Objects of a dimensional category are called line objects or dimensions. A trivial line object is one isomorphic to the monoidal unit $I$, and a section of a line object $L$ is a morphism $\varphi :I\to L$.
Dimensional categories are the objects of a doctrine whose morphisms are symmetric monoidal functors. Every symmetric monoidal $k$-$\mathrm{Mod}$-enriched category $M$ has an underlying dimensional category; the objects are simply the invertible objects for which the self-symmetry is strict, and with all morphisms in $M$ between them. (It may be shown that such objects are closed under tensor products.)
To each dimensional category $C$ over a ground ring $k$, we may associate a graded object, which we call a dimensional algebra, as follows. The grades themselves are isomorphism classes of line objects, forming a group (which we often write as an additive group). If $g=[L]$ is a grade, the homogeneous component ${C}_{g}$ is simply the $k$-linear object of sections of $L$. Note that there is a multiplication which restricts to homogeneous components as
so we do get a graded object. More precisely,
A dimensional algebra (with coefficients in $k$) consists of an abelian group $G$ together with a $k$-module $A$ in the symmetric monoidal category ${\mathrm{Ab}}^{G}$, where the tensor product is Day convolution induced by the group structure of $G$.
Here $k$ (which is a commutative monoid in $\mathrm{Ab}$) is regarded as a commutative monoid in ${\mathrm{Ab}}^{G}$ via the symmetric monoidal functor $\mathrm{Ab}\to {\mathrm{Ab}}^{G}$ induced by the homomorphism $1\to G$. Modules over monoids in a monoidal category are defined in the usual way.
As our naming suggests, we can also think of the grades $g$ as dimensions, as in dimensional analysis. For example, we can form linear combinations of quantities $x,y\in {C}_{g}$ having the same dimension. We can multiply dimensions, and also quantities of different dimensions. At a purely formal level, we can add quantities of different dimensions, but of course the result will not have sense as a quantity of some dimension. Thus, we may view dimensional categories or dimensional algebras as formalizing ordinary dimensional analysis.
Dimensional categories and dimensional algebras are in essence equivalent concepts. Given a dimensional algebra $(G,A:G\to \mathrm{Ab})$, we may form a dimensional category, defining objects to be elements $g$ of $G$, and
The rest is a routine verification. The equivalence can be lifted to a 2-equivalence between 2-categories (notice that our definition of dimensional algebra is manifestly 2-categorical).
If $E$ is a Grothendieck topos, then the category of models (= points, = geometric morphisms $\mathrm{Set}\to E$) is accessible. If $E$ is a coherent topos, then the category of models is finitely accessible (i.e., $\omega $-accessible).
Every Grothendieck topos may be regarded as a classifying topos of a small geometric sketch; see Makkai and Paré, proposition 3.1.2. The category of models of a small sketch is always accessible; see Makkai and Paré, theorem 3.3.4. I’m not quite sure about the last statement.
Every $\lambda $-accessible category $C$ is the free cocompletion of a small category with respect to $\lambda $-filtered colimits.
Let $K\hookrightarrow C$ be the full subcategory of $\lambda $-presentable objects, i.e., objects $k$ that represent functors $\mathrm{hom}(k,-):C\to \mathrm{Set}$ that preserve any $\lambda $-filtered colimits that exist in $C$. Then in fact $C$ can be reconstructed as the $\lambda $-filtered cocompletion of $K$. See for instance Adamek and Rosicky, theorem 2.26, page 83.
The doctrine whose algebras are filtered-cocomplete (locally small) categories is a KZ doctrine.
The monad itself takes a category $C$ to the category of flat functors $F:{C}^{\mathrm{op}}\to \mathrm{Set}$, where a functor is flat if it is a small filtered colimit of representables. (It is also called the $\mathrm{Ind}$-completion, and is locally small if $C$ is.) The crucial thing to check is that if $C$ is filtered-cocomplete, then the structure map $\mathrm{Flat}({C}^{\mathrm{op}},\mathrm{Set})\to C$ is left adjoint to the Yoneda embedding $C\to \mathrm{Flat}({C}^{\mathrm{op}},\mathrm{Set})$. I’ll write this up later.
Let $C$ be a small category with filtered colimits. Then the category of filtered-colimit preserving functors $\mathrm{Filt}(C,\mathrm{Set})$ is a Grothendieck topos.
Intuitively, the idea is clear: the full inclusion $i:\mathrm{Filt}(C,\mathrm{Set})\hookrightarrow {\mathrm{Set}}^{C}$ is closed under finite limits and arbitrary colimits, and therefore one expects the left exact inclusion $i$ to have a right adjoint, making $\mathrm{Filt}(C,\mathrm{Set})$ a coreflective subcategory. In particular, $\mathrm{Filt}(C,\mathrm{Set})$ would thereby be a category of coalgebras for a left exact comonad on ${\mathrm{Set}}^{C}$, and hence a Grothendieck topos. In order to prove a right adjoint to $i$ exists, one might hope to apply the special adjoint functor theorem. However, it is not obvious (to us) that $\mathrm{Filt}(C,\mathrm{Set})$ has a small set of generators.
This problem is akin to a similar difficulty in proving the (known) theorem that the 2-category of Grothendieck toposes and geometric morphisms has weak colimits, and in fact the present theorem can be proved by appeal to that result. To be continued.
The following is adapted from the Wikipedia article on toric varieties.
Suppose that $V$ is a finitely generated free abelian group. A cone in $V$ is a submonoid $C$ such that $v\in C$ and $-v\in C$ implies $v=0$. A fan (in $V$) is a collection of cones in $V$ closed under taking intersections and faces.
Let ${V}^{*}$ denote the dual $\mathrm{hom}(V,\mathbb{Z})$, and define a relation $R\subset {V}^{*}\times V$:
The dual of a cone $C\subseteq V$, denoted ${C}^{*}$, is the value of $C$ under the Galois connection induced by $R$. Explicitly:
In other words, ${C}^{*}$ is formed as a pullback in the category of commutative monoids:
where $i:\mathbb{N}\to \mathbb{Z}$, $j:C\to V$ are the inclusions. We caution that ${C}^{*}$ need not be a cone in ${V}^{*}$: if $f\in {V}^{*}$ is nonzero but its restriction to $C$ is zero, then of course both $f\in {C}^{*}$ and $-f\in {C}^{*}$.
Given a fan $F$ in $V$, we define a category as follows. The objects are the cones in $F$; $\mathrm{hom}(C,D)$ consists of elements $f\in {V}^{*}$ such that $g\in {C}^{*}$ implies $f+g\in {D}^{*}$. We make some remarks.