Zoran Škoda, Bogotá 2012 lectures, plan, version 0.0
Some background on category theory and spaces is sketched at essay geotop, it also includes an excerpt from spectral cookbook (with a little bit more detail than needed). Later the materials will be together merged into a lectures page which is not created so far. The distribution of material within lectures will be approximately as follows:
P1. General introduction into noncommutative geometry
I will start here with general overview of geometry with emphasis on the duality between fields of quantities on a space and the space itself. First part will be about the basic duality and reconstruction theorems: Gelfand-Naimark theorem, Gabriel-Rosenberg reconstruction of schemes from the categories of quasicoherent sheaves, analogues using tensor category structure and the Tannaka duality. All of these will be quoted at the surface here and further insight will be given later.
Then I will go toward sketching main features of several principal worlds of noncommutative geometry like quantum defromations, semiquantum schemes, derived geometry, operator algebraic version and version based on abelian categories of quasifoherent sheaves. I will give some examples.
P2. Yoneda and functionals
It is a triviality that points of a space provide functions on the space of functions, namely via evaluation functionals. If we can characterize the evaluation functionals abstractly within abstract algebra of functions, we can reconstruct the space. This general principle has many reincarnations and it is parallel to the many usages of Yoneda lemma in abstract geometry. See below in the text for some explanation on this viewpoint.
In this part I will sketch how Yoneda embedding can be used to embed local models of spaces into bigger category of generalized spaces, a la Grothendieck. Here one can use Grothendieck topologies, or more generally, Q-categories of Rosenberg. I will also mention about similar formal perspective on basic of categorical model theory following ideas of Makkai.
Points form spectra. I will compare several cases and veiwpoints on creating points from quantity data. Points as functionals, as ideals, as special objects in categories of sheaves, as solutions to spectral equations etc.
P3. Spectra, orders and localizations
Sometimes we localize to parts of a space, not to a point. This is formalized using localization theory, which is very important for noncommutative geometry. I will explain the localization theory at several levels.
In categorical framework spectral constructions are typically exploring certain partial orders. Interestingly enough, orders appear also in many other dualities of similar kind like Stone duality, which involves much simpler kind of spectrum.
I will quickly mention few principal kinds of spectra like Gabriel spectrum of indecomposable injectives, Rosenberg spectrum of an abelian category and the Ziegler spectrum of an associative algebra. Then I will show the importance of partial orders, skecthing the simplest version of Rosenberg’s spectral cookbook.
This will go a bit deeper into some reconstruction situations: Tannaka reconstruction, descent theory (including Beck style monadicity), Barr embedding (for regular and exact categories) and Giraud’s theorem (reconstructing a site from Grothendieck topos). I will follow some ideas of Szlachanyi in comparing Tannaka to ideas of Giraud reconstruction.
Tannaka reconstruction is about realizing some monoidal category (a category with tensor products of objects and of morphisms) as representations of a symmetry object (e.g. group, Hopf algebra, groupoid, Hopf algebroid) by objects which are objects in a simpler monoidal category, together with the additional structure of a representation. For example, representations of algebraic groups on vector spaces form a complicated monoidal category, which is equipped with the forgetful functor to vector spaces. This functor, called the fiber functor is part of the data. Roughly speaking, the idea is that the elements in the symmetry group are reconstructed as endomorphisms of the fiber functor.
Here I will present what the categorification can give to geometry and possibly model theory. I will sketch the meaning of derived geometry and mention (in relevance to logic and model theory) homotopy type theory.
I will explain why the noncommutative geometry inevitably leads to categorification, and the relevance of Morita equivalence and anamorphisms.
Tannaka and spectral reconstructions mix the categorified and noncategorified level. hence the definability at the level of algebras should correspond to a categorified notion of definability at the level of categories of modules. This gives an interesting research topic, on which I could (and will) mainly speculate. A result of Hrushovski on a certain reconstruction theorem about imaginaries and internal covers will be interpreted via comparison to non-definable analogues.
If time permits I will say few words on categorification of Q-categorical generalization of sheaf theory. It is partly overlooked by higher categorical community because they look basically at homotopy theory and commutative geometry where one can restrict to Grothendieck topologies.
Outline main questions in the foundations of noncommutative algebraic geometry, and perhaps some other related frameworks of modern (generalized) geometry. By foundation I mean
Classical geometries were at first developed as incidence geometries: one defines some primitive objects, like points, lines etc. as the main objects of geometry and introduces their “incidence” relations, like 2 point determine a straight line, 2 straight lines intersect in one point and so on. In projective planar geometry, for example, the lines and points are treated on the equal footing (formalized via projective duality). Gradually, points take primacy and with the emphasis on naive set theory in 19th century, each space is thought of as a set of points with some additional structure. It is however very important that points were somewhat earlier coordinated (Descartes and then more generally Lagrange), i.e. the tuples of numbers were related to points in the space of in a part of space (local coordinate system). Coordinates can be viewed as functions on a space; one could consider more complicated functions which can be expressed in terms of basic coordinate functions (with value in a field). Functions multiply and add pointwise, giving rise to a commutative ring/algebra; depending on the class of functions which are suitable to some type of geometry the algebra can have an additional structure (topology, norm, involution…).
While the metric is a function of (pairs of) points, in mid 20th century, more complicated structures on geometries became relevant. For examples, algebraic schemes of Rosenberg are at first introduced as (a class of examples of locally) ringed spaces, i.e. topological spaces equipped with a sheaf of rings (“structure ring”). Even if the underlying topological space is the same two schemes can be quite different. Thus in addition to points and relations among the points one needs to look at functions, algebras of functions, sheaf versions of those, and in later more general examples, categories of sheaves of modules over them, categorified sheaves (stacks) and so on.
Slogan: Noncommutative geometry is the approach to generalized geometry in which a space is defined in terms of a collection of objects which are living over the space.
The individual objects (over the space) may be various observables (functions or, say, fields of operators, some cocycles), geometric objects like vector bundles (possibly equipped with structure like choice of a connection), quasicoherent sheaves, complexes of abelian sheaves (complexes in the sense of homological algebra) etc.
The objects over the space are typically organized into a structure: an algebra (of functions), a category (of sheaves), category with structure (enriched category, abelian catgory, triangulated category, category with a distinguished object), a higher category etc.
The basic philosophy that the functions or more general fields of observables (“quantities”) on a space define the object, may be taken not only as a matter of definition, but also as a statement with content. Indeed, if there is elementary, more set-theoretic way to define the underlying object, then one try to reconstruct it from a structure made out of fields of quantity. Gel’fand-Naimark theorem is an example of that principle: the category of commutative unital $C^*$-algebras is antiequivalent to the category of compact Hausdorff topological spaces. This example has some of the more general features of the theory, namely the points of the space recovered are the (continuous) characters into the ground field which are then identified with the evaluation functionals in points of the original space. In commutative algebra there is similar duality between commutative unital rings and affine schemes, extending the duality between affine algebraic varieties over a field and commutative Noetherian rings with no nilpotent elements. For algebraic varieties the points come from maximal ideals which are again evaluation functionals; for the scheme case one includes prime ideals which are a bigger class and which has functorial behaviour, unlike the case of varieties. This role of evaluation functionals in geometry is similar to the most important triviality in mathematics, the Yoneda lemma, where the objects of the original category are obtained as the analogues of the evaluation functionals – the representable presheaves.
Now, in noncommutative algebraic geometry, it appears that the the analogues of prime ideals, which are in the noncommutative case the one sided completely prime ideals, are insufficient to supply a rich geometry for most noncommutative rings. Thus one soon realizes that the lattice of left ideals should be replaced by the full category of left modules. Inside one again stumbles into objects which appear in some sense the most primitive; which can not be approached from below in certain finiteness-involving technical sense, very much alike the limit cardinals do. This leads to the whole machinery of spectral theories. Good spectra of (a suitable class of) abelian categories were devised and may be equipped with a topology and structure stack of categories. If the abelian category if the category of quasicoherent sheaves on an algebraic scheme, then one can go a step further and take a construction of a center of a fiber of such a stack to simplify it into a set and as a result one will get a locally ringed space, which under mild conditions, for a suitable variant of a spectrum, reconstructs the scheme. But in noncommutative case, one really can not pass with allowing only for a sheaf, one must work at the stack level.
I will explain some of the heuristics why the raise in categorical level comes with noncommutativity and explain how higher categorical analogues of this picture are also interesting from the point of view of geometry.
I should also explain in the talk a relation (in some special cases) to some variants of reconstruction which involve the tensor product structure; most notably the Tannakian reconstruction and some related dualities of relevance for (the applications in) model theory itself (I should mention here Gabriel-Ulmer duality and some common generalizations with a version of Tannaka, well used in categorical logic). I think that the common mechanisms in different dualities should be looked together when such applications are in mind. My partial understanding is that one of the theorems of Hrushovski related to the elimination of imaginaries is essentially a variant of some embedding theorems for regular/exact categories, and suspect some a direction where to look for higher categorical analogues. The embedding theorems mentioned are weaker structure analogues of a Giraud reconstruction theorem in the context of Grothendieck topoi.
Much geometry is possible without ever going to the spectral level. In noncommutative algebraic geometry, open sets are replaced by localizations, which are quite unlike the usual Zariski topology, but from the point of view of the descent theory not that much. I will put much emphasis in the course on the thinking in these terms and the potential in creation of nonaffine spaces which comes from it.
It is likely a challenge for model theory to develop the variant which will work for noncommutative geometry governed by categories of sheaves. Namely instead of a classifying structures, from the point of view of noncommutative geometry, one should classify structured categories of objects over thought-of structure. In good cases, the reconstruction really reveals a structure in standard sense, either at global level, or at least at local level. In the sense of categorical logic, one expects (typically nonalgebraic) 2-theories whose models locally look as models of corresponding (typically algebraic) 1-theories. Similar reasoning is not unknown in categorical logic, but its extension to the noncommutative case is not considered so far, to my knowledge. First of all, in Grothendieck’s approach to algebraic spaces (generalizing algebraic schemes), a space is a sheaf of sets on a site of local shapes (or models, in geometry sense, think of legos) with respect to some subcanonical Grothendieck topology, which is locally representable (“locally affine”). I will explain this standard phrase from the geometry (including few words on how many nonalgebraic kinds of geometric examples also fit like rigid analytic geometry and so on). In noncommutative geometry, one needs to extend the notion of the sheaf appropriately, as most candidates for the Grothendieck topology are not topologies in fact; this point is hard to find in the literature and I will give a quick introduction. Unlike the picture of noncommutative localizations as generating an analogue of a small Grothendieck site, this is an approach akin to the one via the big site.
Fortunately, in the commutative case, one does not need to go to the level of 2-theories. Namely the locally affine spaces, considered as sheaves still form certain subcategory of a Grothendieck topos, so one can in principle look at them as models in a topos, rather than models in $Set$. The big site version has this feature in noncommutative case as well, except that the subcategory has worse properties, making it difficult. However, if one works at the reconstruction levels (representing by categories of quasicoherent sheaves) then one can not remove the need for a good categorified definability theory; and if in place to restate the relevant reconstruction theorem by relating the categorified definability theory of a 2-theory with definability theory for a 1-theory, so to say, the definable reconstruction theorem. To say it more precisely, this is not about looking at definable subcategories in the usual sense, as this is too strict. Instead, one is likely going to need a 2-categorical weakening of definability, when interpreting a 2-theory in categorified model. In particular the notions like a closed subscheme in the setup of abelian categories will have a 2-definable variant.
Finally, we interpret mentioned theorem of Hrushovski on imaginaries and internal groupoids as a reconstruction theorem (which is helping understanding 3- and 4-amalgamation in terms of internal groupoids), and propose some ideas toward a generalization where higher amalgamation would be eventually related to higher internal groupoids.
(under reconstruction, in principle the references at the beginning of the list are much simpler than the ones at the end of the list)
Pierre Cartier, A mad day’s work: from Grothendieck to Connes and Kontsevich The evolution of concepts of space and symmetry, Bull. Amer. Math. Soc. 38 (2001), 389-408, pdf
entries in $n$Lab: definable set, Tannaka duality, Q-category, algebraic scheme, quasicoherent sheaf
D. Kaledin, Tokyo lectures “Homological methods in non-commutative geometry”, pdf, TeX; and related but different Seoul lectures
Zoran Škoda, Noncommutative localization in noncommutative geometry, London Math. Society Lecture Note Series 330, ed. A. Ranicki; pp. 220–313, math.QA/0403276.
Z. Škoda, Some equivariant constructions in noncommutative algebraic geometry, Georgian Mathematical Journal 16 (2009), No. 1, 183–202, arXiv:0811.4770
Kornél Szlachányi, Fiber functors, monoidal sites and Tannaka duality for bialgebroids, arxiv/0907.1578
Alexander Rosenberg, Noncommutative schemes, Compos. Math. 112 (1998) 93–125 (doi)
M. Kontsevich, A. Rosenberg, Noncommutative smooth spaces, The Gelfand Mathematical Seminars, 1996–1999, 85–108, Gelfand Math. Sem., Birkhäuser Boston, Boston, MA, 2000; (arXiv:math/9812158), Noncommutative spaces, preprint MPI-2004-35 (dvi,ps), Noncommutative spaces and flat descent, MPI-2004-36 dvi,ps, Noncommutative stacks, MPI-2004-37 dvi,ps
Ehud Hrushovski, Groupoids, imaginaries and internal covers, arxi/math.LO/0603413
Last revised on April 11, 2013 at 20:00:57. See the history of this page for a list of all contributions to it.