Connections between various generalized algebraic geometries (Toen-Vaquié, Durov, Diers, Lurie)? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T11:27:50Z http://mathoverflow.net/feeds/question/89475 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89475/connections-between-various-generalized-algebraic-geometries-toen-vaquie-durov Connections between various generalized algebraic geometries (Toen-Vaquié, Durov, Diers, Lurie)? Martin Brandenburg 2012-02-25T10:26:56Z 2012-02-29T07:10:30Z <p>As far as I know, there are four possible ways to generalize algebraic geometry by 'simply' replacing the basic category of rings with something similar but more general:</p> <p>$\bullet$ In the approach by <a href="http://arxiv.org/abs/math/0509684" rel="nofollow">Toen-Vaquié</a> we fix a nice symmetric monoidal category $C$, also called a <em>relative context</em>. An affine scheme is defined to be an algebra object in $C$, and an arbitrary scheme is a certain presheaf on affine schemes. We optain the category $\mathrm{Sch}(C)$ of schemes <em>relative</em> to $C$.</p> <p>$\bullet$ In <a href="http://arxiv.org/abs/0704.2030" rel="nofollow">Durov's theory</a> a generalized ring is an algebraic monad which is commutative in a certain sense. Then affine schemes are defined to be the spectra of generalized rings and arbitrary schemes are optained by gluing. This results in the category $\mathrm{genSch}$.</p> <p>$\bullet$ In his book <em>Categories of commutative algebras</em> Yves Diers considers Zariski categories, which seem to axiomatize familiar properties of categories of commutative algebras. If $\mathcal{A}$ is such a Zariski category, then one can develope commutative algebra <em>internal to $\mathcal{A}$</em>, construct affine schemes and then by gluing also schemes as usual. We optain the category $\mathrm{Sch}(\mathcal{A})$.</p> <p>$\bullet$ In derived algebraic geometry one replaces the category of rings with the category of simplicial rings (but I don't really know enough about that, yet).</p> <p>My <strong>question</strong> is: What are the connections between these 'generalized algebraic geometries'?</p> <p>Fortunately there is a <a href="http://arxiv.org/abs/0909.0069" rel="nofollow">map of $\mathbb{F}_1$-land</a> which draws connections between all these various approaches to schemes over $\mathbb{F}_1$. For example monoid schemes à la Deitmar/Kato are in the intersection of Toen-Vaquiè and Durov. Note, however, that the theories mentioned above are far more general.</p> <p>Specifically, one might ask the following questions: Is the category of generalized rings a Zariski category and is Durov's theory (say, with the unary localization theory) a special case of the one by Yves Diers? What is the relationship between Toen-Vaquié schemes relative to the symmetric monoidal category of simplicial rings and derived schemes? If $C$ is a relative context, is then the category of algebra objects in $C$ a Zariski category and do the corresponding schemes coincide? Probably not because Diers never mentions monoids as an example, but perhaps it's the other way round? Of course, many more questions are out there ...</p> <p>Probably I'm not the first one with this question, therefore I've also put the "reference-request" tag. It would be great if there is some paper like "Mapping AG-land".</p> http://mathoverflow.net/questions/89475/connections-between-various-generalized-algebraic-geometries-toen-vaquie-durov/89816#89816 Answer by Peter Arndt for Connections between various generalized algebraic geometries (Toen-Vaquié, Durov, Diers, Lurie)? Peter Arndt 2012-02-29T00:33:46Z 2012-02-29T07:10:30Z <p>First, here are some things about the four generalizations you mention:</p> <p>Monoids don't fall into Diers' framework: By his Proposition 1.4.1 the terminal object in his framework is strict, i.e. any morphism $1 \to A$ is an isomorphism, which is definitely not the case for monoids. I also wouldn't expect Diers' examples to be instances of Toen/Vaquie's framework in general, Diers' example 1.3.16, the category of pairs (ring, module over it), might be a counterexample. I don't know about Durov's setting.</p> <p>Durov's geometry is in no obvious way an instance of Toen/Vaquie's framework. If you want to force it into that framework, this might be an idea to go after: Monads are monoids in the monoidal category of endofunctors. Commutative monads, however, are not commutative monoids in that category; indeed it doesn't even make sense to say that since the category is not symmetric monoidal. So first you have to find a symmetric monoidal ambient category in which Durov's generalized rings live. Seeing monads as Lawvere algebraic theories or as (things presented by) sketches might do the job - a commutative theory is probably exactly a sketch with an isomorphism from its tensor square. Another idea could be to consider a category of monads where morphisms are natural monad transformations together with distributive laws...</p> <p>Derived algebraic geometry on the other hand is an instance of the homotopical version of Toen/Vaquie's framework, also contained in that article - see also below.</p> <hr> <p>Second let me point out that there are many more generalizations of algebraic geometry than those four:</p> <p>° Rings with extra structure can count as generalization, if one can endow any usual ring with such an extra structure, e.g.</p> <ul> <li><p><em>Not</em> Borger's geometry with lambda-rings: <em>Not</em> any ring can be endowed with the trivial lambda-ring structure - see his comment</p></li> <li><p>Berkovich's analytic geometry: Any ring can be endowed with the trivial metric</p></li> </ul> <p>° One can replace rings by first order structures in several ways:</p> <ul> <li><p>Several Russian authors do this in somewhat similar ways, a recent reference is <a href="http://arxiv.org/abs/1002.3562v3" rel="nofollow">this one</a> by Daniyarova, Myasnikov, Remeslennikov which has many references to other work in this direction; see also <a href="http://arxiv.org/abs/math/0210194" rel="nofollow">this one</a> by B. Plotkin.</p></li> <li><p>First order structures can be described by sketches and there is an outline of algebraic geometry along this line in <a href="http://www.math.jussieu.fr/~guitart/docpp/guitart08tteaet.pdf" rel="nofollow">this</a> text by R. Guitart.</p></li> </ul> <p>° There are hyperrings (used for algebraic geometry by Connes/Consani) and fuzzy rings (by Walter Wenzel and Andreas Dress, e.g. <a href="http://www.mis.mpg.de/de/publications/preprints/2007/prepr2007-112.html" rel="nofollow">this</a>), which are certain second order structures.</p> <p>° There are two generalizations of rings used by Shai Haran to compactify the integers, <a href="http://journals.cambridge.org/action/displayAbstract?fromPage=online&amp;aid=1019288" rel="nofollow">F-rings</a> and the one given in his <a href="http://arxiv.org/abs/0911.3522" rel="nofollow">"Non-additive prolegomena"</a>.</p> <p>° There is another generalization made by Shai Haran in his article on "dyslectic geometry" in which rings are endowed with gradings over general monoids (see <a href="http://journals1.scholarsportal.info/details.xqy?uri=/00218693/v155i0002/455_aitdg.xml" rel="nofollow">here</a>). Something quite similar seems to be going on in James Dolan's generalization of algebraic geometry (unpublished, but see <a href="http://ncatlab.org/toddtrimble/published/Notes+on+conversations+with+James+Dolan" rel="nofollow">here</a>, there also was a series of videos of talks somewhere)</p> <p>° derived algebraic geometry doesn't necessarily have to be based on simplicial rings; dg-algebras and $E_\infty$-ring spectra are equally important inputs, and there are many others, captured in</p> <ul> <li><p>Toen-Vezzosi's HAG-contexts (these are homotopically additive)</p></li> <li><p>Lurie's structured spaces from DAG 5 which capture about everything based on the idea of glueing together homotopical algebraic structure.</p></li> </ul> <p>° note that Toen/Vaquie in their <a href="http://arxiv.org/abs/math/0509684" rel="nofollow">relative geometry</a> do not stop at commutative monoids in some monoidal category but also give a homotopical version - this is something like a non-additive version of HAG-contexts and covers e.g. geometry over the "spectrum with one element" whose input are simplicial monoids.</p> <p>° replacing rings by groupoid objects in rings together with an appropriate notion of equivalence gives you stack theory. Of course you can go on to higher stacks.</p> <p>° of course there are the several approaches to non-commutative algebraic geometry - see <a href="http://arxiv.org/abs/math/0501166" rel="nofollow">Mahanta's survey</a> for some of them, many related to the next point</p> <p>° <a href="http://arxiv.org/abs/1105.3104" rel="nofollow">Chirvasitu/Johnson-Freyd's 2-schemes</a></p> <p>° <a href="http://arxiv.org/abs/1202.5085" rel="nofollow">Takagi's generalized schemes</a> </p> <p>° <a href="http://arxiv.org/abs/1102.4046" rel="nofollow">Deitmar's congruence schemes</a></p> <p>° <a href="http://arxiv.org/abs/1103.1745" rel="nofollow">Lorscheid's blue schemes</a></p> <p>° I am sure I forgot several things...</p> <hr> <p>Third, since I am at it, let me note that there are also generalizations of algebraic geometry which do not exactly build upon a generalized notion of ring, e.g.</p> <p>° Hrushovski/Zilber's Zariski Geometries (see <a href="http://people.maths.ox.ac.uk/zilber/s.pdf" rel="nofollow">here</a>) capturing the essential structure which is used in the applications of model theory to arithmetic geometry</p> <p>° Rosenberg's noncommutative geometry. It doesn't have to be non-commutative, and also not additive, as Z. Skoda pointed out <a href="http://mathoverflow.net/questions/33477/did-durovs-work-give-an-example-of-noncommutative-schemes/33514#33514" rel="nofollow">here</a></p> <p>° in particular: schemes as dg-categories (Kontsevich, Rosenberg, <a href="http://www.math.uiuc.edu/K-theory/1007/" rel="nofollow">Tabuada</a>)</p> <p>° <a href="http://www.math.ucla.edu/~balmer/research/Pubfile/SSS.pdf" rel="nofollow">Balmer's triangulated geometry</a></p> <p>...</p> <hr> <p>To summarize: Carefully mapping AG-land will keep you busy for a while. I have gathered quite some material for a rudimentary map (or maybe a low resolution satellite photo) accompanied by a few selected closer snapshots, but I won't start writing it before the second half of the year...</p>