doctrines of algebraic geometry
this is a collaborative work-in-progress based (at least to begin with) on a series of talks that i gave in john baez’s seminar at uc riverside in the spring quarter of 2009. the talks were video-recorded and are available here:
the original title for the series of talks was “algebraic geometry for category theorists”, and although we’re aiming to reach a broader audience than just category theorists here, the original title still suggests the general thrust of the present work: to convince non-specialists in algebraic geometry that modern algebraic geometry is a lot easier to learn than you probably think it is, but that it involves using category theory in a pretty different way from what you’ve probably heard.
the collaborators involved in this project at the moment include myself (james dolan) and john baez.
i became interested in writing an exposition of algebraic geometry for category theorists when i finally learned some algebraic geometry (beyond just the “affine” case) and realized that it makes use of category theory in a way that seems significantly underappreciated.
this is also an attempt at reconciliation for me of an estrangement that i felt during my earlier and less successful experiences in trying to learn algebraic geometry. i’d heard rumors of something called “toposes” which according to an often-repeated story played a significant role in advances in algebraic geometry associated with alexander grothendieck and the proof of the “weil conjectures”. then i learned topos theory, especially through the inspiring (sometimes a little too inspiring) lectures of bill lawvere, and although i found it beautiful and highly applicable it didn’t actually seem to help me much with learning algebraic geometry.
algebraic geometry and topos theory were for me two puzzle pieces that were supposed to fit but didn’t, two cultures that were supposed to communicate but didn’t. but now i have an idea for how they fit together, with the help of a missing puzzle piece called “the doctrine of dimensional theories”, and i want to try to explain it here.
this work started out in a format similar to that of an old-fashioned research/expository article calling for a specific body of prerequisite knowledge on the part of the reader but is probably shifting towards some sort of hypertext format with hyperlinks to prerequisite material. unfortunately i have no experience with creating hypertext; fortunately i have no experience with creating old-fashioned research/expository articles either.
as originally envisioned there are two main prerequisites:
some understanding of “affine” algebraic geometry, perhaps especially from a category-theoretic viewpoint. how the category of affine schemes is equivalent to the opposite of the category of commutative rings, and how the “limit” constructions (products and equalizers) in the category of affine schemes relate to descartes’s basic idea of creating spaces as equational “varieties” in cartesian coordinate space.
some experience with “doctrines” in the beck/lawvere category-theoretic sense. how a doctrine is associated with (or perhaps simply is) a 2-category of “theories”, how a hom-category in the 2-category represents a category of “models” of the domain theory, and how theories, models, and model morphisms can be “sketched” as elaborated “diagram schemes”, “diagrams”, and “prism diagrams”, respectively. how the 2-category of theories of a “more expressive” doctrine is related by an adjunction to the 2-category of theories of a “less expressive” doctrine, as for example the doctrine of finite-limits theories is more expressive than the doctrine of finite-products theories. some experience with the doctrine of “geometric” theories (aka “toposes”) would be helpful in understanding some of the more advanced topics that we’ll discuss.
graded commutative algebras
graded commutative algebras are an important object of study in algebraic geometry (lying more on the algebraic side but admitting a geometric interpretation). we’ll offer two perspectives here on the role that they play, first a more conventional perspective (labeled here “the lowbrow story”), and then a more category-theoretic perspective (“the highbrow story”).
1 the lowbrow story
a “projective variety” x over a field k can be described by giving a “homogeneous system of equations”; that is, a list of variables v1,…,vj and a list of homogeneous polynomial equations e1,…,ek in those variables. a point p in the projective space kp^[j-1] belongs to x just in case its “homogeneous coordinates” give a solution of the system. (the homogeneous coordinates of p form a vector in the vector space k^j which is specified only up to multiplication by a scalar, but the homogeneousness of the equations ensures that the property of being a solution of the system doesn’t depend on the scalar.)
more abstractly, x can be described by giving a “graded ideal” in a polynomial algebra (namely the ideal generated by e1,…,ek in the polynomial algebra generated by v1,…,vj), or still more abstractly by giving a graded commutative k-algebra (namely the algebra obtained by modding out the polynomial algebra by the graded ideal).
(to make it easier to introduce certain motivating ideas from analysis and geometry, assume for the moment that k is the field of complex numbers and that the projective variety x is “non-singular” so that it can be thought of as a complex-analytic manifold.)
what’s the conceptual significance for the projective variety x of the elements in grade n of this graded commutative algebra? they’re not (global) holomorphic functions on x; unlike affine varieties, (irreducible) projective varieties lack non-constant holomorphic functions. rather, they’re holomorphic sections of a line bundle over x, namely of the nth tensor power of the dual of the line bundle which x tautologically carries by virtue of being embedded in a projective space.
in this way, the holomorphic sections of line bundles over a non-singular complex projective variety step forward to take over the task which the largely non-existent holomorphic functions on the variety can’t perform: to describe the variety by algebraic means.
this hints at what’s really going on: the category of (holomorphic) line bundles over the projective variety x knows everything about x and thus serves as an effective stand-in for x. in the next section we abandon the lowbrow approach and start over with this viewpoint, to found algebraic geometry on the principle that a variety is to be understood by understanding its category of line bundles (or of objects analogous to line bundles).
2 the highbrow story
definition: an object x in a symmetric monoidal category k is called a “line object” if it has an inverse object wrt tensor product, and if the canonical “switching” morphism x#x -> x#x is the identity morphism. a “section” of a line object is a morphism from the unit object. we’ll be mainly but not exclusively interested in the case where k is enriched over the category of vector spaces.
example: in a symmetric monoidal category of vector bundles, the line objects are the line bundles.
another name for the study of categories of line objects is “dimensional analysis”. in dimensional analysis, a physical theory is described by specifying an abelian group of “dimensions” (these are the line objects) together with a commutative algebra of “quantities” (these are the sections of the line objects) which is graded by the dimension group. we’ll call a physical theory described in this way a “dimensional algebra”, but the fundamental fact about a dimensional algebra is that it’s equivalent to a “dimensional category”, which is a symmetric monoidal category where all objects are line objects.
let g be the abelian group freely generated by the dimensions “mass” and “velocity”. let r be the g-graded commutative algebra generated by the six quantities “mass of particle #1”, “mass of particle #2”, “initial velocity of particle #1”, “initial velocity of particle #2”, “final velocity of particle #1”, and “final velocity of particle #2”, subject to the two relations “conservation of momentum” and “conservation of energy”.
(the names given to the dimensions, quantities, and relations here are informally meant to suggest both a physical situation and the precise dimensional algebra used to describe it. the physical context suggests taking the base field of the dimensional algebra to be the field of real numbers, but there are advantages and no disadvantages to leaving the field unspecified for now; even requiring it to be a field may be over-definite.)
for example, there are seven linearly independent quantities that live in the dimension “momentum” (defined as mass times velocity) in this dimensional algebra: multiplying the mass of either particle by the initial or final velocity of either particle gives eight quantities, but conservation of momentum cuts it down to seven. thus in the corresponding dimensional category, the hom-space from the “dimensionless” dimension to the “momentum” dimension is a 7-dimensional vector space.
as a rough first approximation, we’ll construe algebraic geometry as the study of dimensional algebras, but from the perspective that it’s better to think of them as dimensional categories. the benefit to interpreting them as categories in this way is that it’s often useful to think of categories equipped with a certain kind of extra structure as “theories” of a sort, and to think of functors that preserve the extra structure as “models of the domain theory in the environment provided by the co-domain”. (this philosophy will be explained in more detail in the next chapter, on “doctrines”.) thus we can now interpret a dimensional algebra as a “theory”, and consider “models” of this theory in our favorite “environment”, and consider isomorphism classes of such models as “points in a moduli space” (“moduli space” being a name with a certain pedigree for a set of isomorphism classes of objects of some kind). it’s in this way that a dimensional algebra gets a “geometric interpretation” as a kind of “space”, but moreover a moduli space. thus in this approach to algebraic geometry, a preoccupation with “moduli spaces” is not grafted on as an afterthought, but rather built in from the start, in that every “algebraic variety” is inherently a “moduli space” in a natural way (except that a more systematic treatment as in the next chapter (on “doctrines”) leads to replacing the idea of “moduli space” by the concept of “classifying topos”).