# Algebraic Stacks: Progress and Prospects (12w5027)

Arriving in Banff, Alberta Sunday, March 25 and departing Friday March 30, 2012

## Organizers

Patrick Brosnan (University of Maryland, College Park)

Roy Joshua (Ohio State University)

Hsian-Hua Tseng (Ohio State University)

## Objectives

The workshop plans to bring together several of the leading experts in related fields along with several post-docs and advanced graduate students working in these areas. The talks will discuss the current state of development and explore prospects for future progress in this area. Our goal is to combine various camps of mathematicians working in aspects of geometric representation theory, differential graded algebraic geometry, algebraic topology and mathematical physics and who make use of algebraic stacks and stack theoretic techniques from possibly diverse points of view so as to promote exchange of ideas between these various camps.

We also plan to fund a number of young people, at both the post-doctoral and advanced graduate student level so that the conference will serve to stimulate future interest in these areas.

Primary subject areas: 14D23, 14D24, 14F20 (Algebraic Geometry), 55N32, 55N34 (Algebraic Topology).

Scientific Justification:

Algebraic varieties and schemes are often too restrictive to contain solutions of many important problems. The notion of algebraic stacks being a vast generalization of schemes, it is in fact possible to solve many problems in this more general framework which cannot be solved in a more classical setting. Several objects of fundamental importance in mathematics have been constructed only in the framework of stacks. For example, many moduli-spaces that cannot be constructed in the framework of schemes have been constructed in the setting of algebraic stacks as in the work of Deligne and Mumford, Mumford and Knudsen (see [MFK94], [MK76]) as well as Kontsevich [Kont95]. Construction of a virtual fundamental class associated to certain moduli-spaces of stable maps, has been carried out elegantly in stack-theoretic contexts as in the work of Behrend and Fantechi (and also Li and Tian): see [BF97] and [LT98].

An affine algebraic variety may be viewed as the zero locus of a collection of polynomial functions. For this to make sense over any commutative ring rather than a field, one needs to allow nilpotents and zero divisors in the ring. These are called affine schemes, and schemes are obtained by gluing together affine schemes making use of the Zariski topology. It is possible to consider objects that look locally like schemes, but globally are not schemes, by performing the above gluing in a different Grothendieck topology. One may view such objects as contravariant functors from the category of schemes to sets satisfying certain gluing conditions. The algebraic spaces introduced by Michael Artin are examples of such functors. Good examples of such functors are the moduli-functors appearing in moduli problems. Such a functor will associate to each scheme U, a family of certain algebraic objects over U satisfying certain conditions. Unfortunately, the fact that these are functors to sets means one has to ignore all automorphisms of these algebraic objects, or in other words one is considering only isomorphism classes of such objects rather than the objects themselves. Since the category of schemes is not closed under colimits, it means that many such functors are usually not representable by schemes or even realized as algebraic spaces.

The idea of algebraic stacks is to consider similar functors, but without modding out by the automorphisms. This makes it necessary to consider what are functors upto natural isomorphism to the category of groupoids or lax-functors to groupoids. Such lax functors that satisfy certain gluing conditions are stacks and those that look locally like affine schemes, in a suitable Grothendieck topology are algebraic stacks. The gluing conditions correspond to descent data as in Grothendieck's theory of faithfully flat descent. When the topology used is the 'etale topology, one obtains what are called Deligne-Mumford stacks: the objects called orbifolds are special cases of Deligne-Mumford stacks that are smooth (in a certain sense) and which are generically schemes. For example, if G is a finite group acting on a smooth scheme X, with stabilizers that are trivial generically, the resulting quotient stack [X/G] is an orbifold. Observe that the quotient stack [X/G] is not simply the quotient space X/G, which ignores the stabilizers: instead the quotient stack [X/G] should be viewed as an object sitting over X/G which also keeps track of all the stabilizers. The theory of orbifolds and their cohomology has been of significant interest in recent years, especially to the algebraic topologists. The last 15 years or so, saw the development of various cohomology theories for Deligne-Mumford stacks as in the work of Toen, Chen and Ruan: see [T99] and [CR04]. In addition various technical tools to study Deligne-Mumford stacks have been sharpened: for example, the Quot functor, which is extremely useful in algebraic geometry, has only been recently constructed for Deligne-Mumford stacks by Olsson and Starr: see [OS03].

These applications also promoted (in fact made it essential) further work on the foundational aspects of algebraic stacks: for example, the K-theory and intersection theory on all Artin stacks have been developed (as in the work of Kresch, Joshua and Toen: see [Kr99], [J02], [J03], [J07], [T99]) in the last 10 years partly in response to the applications to the study of virtual phenomena. The applications to virtual phenomena, and algebraic topology further led to basic work on differential graded stacks as appearing the work of Toen, Vezzosi and Lurie (see [TV05], [TV08], [Lur04]) which has introduced many new ideas and techniques.

Despite all these, the theory of algebraic stacks is rather challenging, especially to a beginner, partially because of the use of very sophisticated mathematics, which is necessary to study algebraic stacks systematically and partially because of the lack of adequate literature that proceeds at a leisurely pace and aimed at beginners in the field. Moreover, the last major program devoted to algebraic stacks seems to have been the MSRI program for the first half of 2002. In view of these, the importance and relevance of workshops like the one proposed here should be clear. By bringing together the experts working in different aspects of stacks and their applications as well as a group of beginners in the field, the organizers hope to not only promote exchange of ideas between the various groups but also equally importantly to provide a leisurely introduction to this vast and exciting field to the grad student and post-doc participants.

Scientific program:

The class of Artin stacks is much bigger than Deligne-Mumford stacks. As a simple example, if G is a linear algebraic group acting on a smooth scheme X, one obtains the quotient stack [X/G] which classifies principal G-bundles together with a G-equivariant map to X. If the stabilizers for the $G$-action are not finite, the resulting stack is an Artin stack that is not Deligne-Mumford. As the above example of quotient stacks shows, Artin stacks occur far more commonly than Deligne-Mumford stacks: yet it is only in the last 10 years or so that such stacks have begun to be studied in detail. (See for example recent work of Olsson and Laszlo (and also somewhat earlier work by Behrend and Joshua) on the l-adic derived category, perverse sheaves and t-structures associated to such stacks: see [Be03], [J93], [LO08].) The same example of quotient stacks shows the importance of such stacks in geometric approaches to representation theory, for example in the geometric Langlands correspondence. Recent work of Witten and others have already established connections of this area with mathematical physics.

In view of its importance, we will devote an important part of the program to applications in geometric representation theory.

In recent years further generalizations of the notion of algebraic stacks have begun to emerge. The notions and basic properties of higher stacks and derived stacks are by now available, thanks to works of Simpson, Toen, Vaquie, Vezzosi, Lurie, and others (see e.g. [S96], [TVa07] [TV05], [TV08], [Lur04]). When considering a moduli problem, the notion of derived stacks allows one to encode, in the moduli space, all the information about deformations/obstructions/ higher obstructions of the moduli problem. On the other hand, the notion of higher stacks allows one to encode the information about higher automorphisms of the moduli problem. In order to capture all the information of a moduli problem, it is thus necessary to work in the more generalized context of higher and derived stacks. An example showing how beneficial this can be is J. Lurie's work on interpreting the topological modular forms of Miller-Hopkins using the derived moduli stack of elliptic curves: see [Hop02], [Lur09], [Lur09a].

In light of its importance, another focus of the proposed workshop will be on the theory of higher and derived stacks and its applications to moduli problems.

Intersection theory on moduli spaces is a very important area in algebraic geometry. A notable example is the intersection theory on moduli spaces of stable pointed curves, which is known to be closely related to other subjects such as integrable systems and infinite-dimensional Lie algebras. Unlike moduli of curves, many interesting moduli spaces (for example, moduli of stable maps, of stable sheaves, e.t.c.) are usually highly singular, and one studies their intersection theory by replacing the fundamental classes with the virtual fundamental classes. Virtual fundamental classes are constructed from choices of perfect obstruction theories, which arise from the deformation/obstruction of the moduli problems. Since deformation/obstruction is better encoded in derived stacks, it is very natural to consider virtual fundamental classes in the context of higher and derived stacks. Closely related to these virtual fundamental classes (and perhaps of somewhat more intrinsic importance) are the virtual structure sheaves, a thorough understanding of which needs the K-theory and G-theory of stacks.

In this workshop we will also be interested in understanding various aspects of intersection theory, K-theory and G-theory of higher and derived stacks.

REFERENCES

[Ar74] M. Artin, Versal deformations and algebraic stacks, Invent. Math., 27, (1974), 165--189.

[BF97] K. Behrend and B. Fantechi, The Intrinsic Normal Cone, Invent. Math., 128 (1997), 45--88.

[Be03] K. Behrend, Derived l-adic categories for algebraic stacks, Mem. Amer. Math. Soc., 163 (2003) no. 774, viii+93.

[CR04] W. Chen and Y. Ruan, A new cohomology theory for orbifolds, Comm. Math. Phys. 248 (2004), 1--31.

[DM69] D. Mumford and P. Deligne, The irreducibility of the space of curves of a given genus, IHES Publ. Math. 36 (1969), 75--109.

[Gir65] J. Giraud, Cohomologie non-abeliene, Die Grundlehren der mathematischen Wissenschaften, Band 179. Springer-Verlag, Berlin-New York, 1971. ix+467 pp.

[Hop02] M. Hopkins, Algebraic Topology and Modular Forms, in ``Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002)'', 291--317, Higher Ed. Press, Beijing, 2002.

[J93] R. Joshua, The derived category and intersection cohomology of algebraic stacks, in ``Algebraic K-theory and Algebraic Topology (Lake Louise, AB, 1991)'', 91--145, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 407, Kluwer Acad. Publ., Dordrecht, 1993.

[J03] R. Joshua, Riemann-Roch for algebraic stacks: I, Compositio Math. 136 (2003), no. 2, 117--169.

[J02] R. Joshua, Higher Intersection Theory on Algebraic Stacks:I and :II, K-Theory, 27 (2002), no. 2, 134--195. and 27 (2002), no. 3, 197--244.

[J07] R. Joshua, Bredon-style homology, cohomology and Riemann-Roch for algebraic stacks, Adv. Math. 209 (2007), no. 1, 1--68.

[Kont95] M. Kontsevich, Enumeration of rational curves via torus actions, in ``The moduli space of curves (Texel Island, 1994)'', 335--368, Progr. Math., 129, Birkh"auser Boston, Boston, MA, 1995.

[Kr99] A. Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999), no. 3, 495--536.

[LO08] Y. Laszlo and M. Olsson, The six operations for sheaves on Artin stacks I, Publ. Math. IHES, 107, (2008), 109-168.

[LT98] J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), no. 1, 119--174.

[Lur04] J. Lurie, Derived Algebraic Geometry, PhD thesis, MIT, 2004.

[Lur09] J. Lurie, Higher Topos Theory, Annals of Mathematics Studies, 170. Princeton University Press, Princeton, NJ, 2009. xviii+925 pp.

[Lur09a] J. Lurie, A survey of elliptic cohomology, in ``Algebraic topology'', 219--277, Abel Symp., 4, Springer, Berlin, 2009.

[Mum65] D. Mumford, Picard groups of moduli problems, in ``Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963)'', 33--81, Harper & Row, New York, 1965.

[MK76] F. Knudsen and D. Mumford, The projectivity of the moduli space of stable curves. I: Preliminaries on "det" and "Div", Math. Scand. 39 (1976), no. 1, 19--55.

[MFK94] D. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) (Results in Mathematics and Related Areas (2)), 34. Springer-Verlag, Berlin, 1994.

[OS03] M. Olsson and J. Starr, Quot functors for Deligne-Mumford stacks, Comm. Algebra 31 (2003), no. 8, 4069--4096.

[S96] C. Simpson, Algebraic (geometric) $n$-stacks, arXiv:alg-geom/9609014.

[T99] B. Toen, Th'eor`emes de Riemann-Roch pour les champs de Deligne-Mumford, K-Theory 18 (1999), no. 1, 33--76.

[TVa07] B. Toen and M. Vaqui'e, Moduli of objects in dg-categories, Ann. Sci. 'Ecole Norm. Sup. (4) 40 (2007), no. 3, 387--444.

[TV05] B. Toen and G. Vezzosi, Homotopical algebraic geometry. I. Topos theory, Adv. Math. 193 (2005), no. 2, 257--372.

[TV08] B. Toen and G. Vezzosi, Homotopical algebraic geometry. II. Geometric stacks and applications, Mem. Amer. Math. Soc. 193 (2008), no. 902, x+224 pp.

We also plan to fund a number of young people, at both the post-doctoral and advanced graduate student level so that the conference will serve to stimulate future interest in these areas.

Primary subject areas: 14D23, 14D24, 14F20 (Algebraic Geometry), 55N32, 55N34 (Algebraic Topology).

Scientific Justification:

Algebraic varieties and schemes are often too restrictive to contain solutions of many important problems. The notion of algebraic stacks being a vast generalization of schemes, it is in fact possible to solve many problems in this more general framework which cannot be solved in a more classical setting. Several objects of fundamental importance in mathematics have been constructed only in the framework of stacks. For example, many moduli-spaces that cannot be constructed in the framework of schemes have been constructed in the setting of algebraic stacks as in the work of Deligne and Mumford, Mumford and Knudsen (see [MFK94], [MK76]) as well as Kontsevich [Kont95]. Construction of a virtual fundamental class associated to certain moduli-spaces of stable maps, has been carried out elegantly in stack-theoretic contexts as in the work of Behrend and Fantechi (and also Li and Tian): see [BF97] and [LT98].

An affine algebraic variety may be viewed as the zero locus of a collection of polynomial functions. For this to make sense over any commutative ring rather than a field, one needs to allow nilpotents and zero divisors in the ring. These are called affine schemes, and schemes are obtained by gluing together affine schemes making use of the Zariski topology. It is possible to consider objects that look locally like schemes, but globally are not schemes, by performing the above gluing in a different Grothendieck topology. One may view such objects as contravariant functors from the category of schemes to sets satisfying certain gluing conditions. The algebraic spaces introduced by Michael Artin are examples of such functors. Good examples of such functors are the moduli-functors appearing in moduli problems. Such a functor will associate to each scheme U, a family of certain algebraic objects over U satisfying certain conditions. Unfortunately, the fact that these are functors to sets means one has to ignore all automorphisms of these algebraic objects, or in other words one is considering only isomorphism classes of such objects rather than the objects themselves. Since the category of schemes is not closed under colimits, it means that many such functors are usually not representable by schemes or even realized as algebraic spaces.

The idea of algebraic stacks is to consider similar functors, but without modding out by the automorphisms. This makes it necessary to consider what are functors upto natural isomorphism to the category of groupoids or lax-functors to groupoids. Such lax functors that satisfy certain gluing conditions are stacks and those that look locally like affine schemes, in a suitable Grothendieck topology are algebraic stacks. The gluing conditions correspond to descent data as in Grothendieck's theory of faithfully flat descent. When the topology used is the 'etale topology, one obtains what are called Deligne-Mumford stacks: the objects called orbifolds are special cases of Deligne-Mumford stacks that are smooth (in a certain sense) and which are generically schemes. For example, if G is a finite group acting on a smooth scheme X, with stabilizers that are trivial generically, the resulting quotient stack [X/G] is an orbifold. Observe that the quotient stack [X/G] is not simply the quotient space X/G, which ignores the stabilizers: instead the quotient stack [X/G] should be viewed as an object sitting over X/G which also keeps track of all the stabilizers. The theory of orbifolds and their cohomology has been of significant interest in recent years, especially to the algebraic topologists. The last 15 years or so, saw the development of various cohomology theories for Deligne-Mumford stacks as in the work of Toen, Chen and Ruan: see [T99] and [CR04]. In addition various technical tools to study Deligne-Mumford stacks have been sharpened: for example, the Quot functor, which is extremely useful in algebraic geometry, has only been recently constructed for Deligne-Mumford stacks by Olsson and Starr: see [OS03].

These applications also promoted (in fact made it essential) further work on the foundational aspects of algebraic stacks: for example, the K-theory and intersection theory on all Artin stacks have been developed (as in the work of Kresch, Joshua and Toen: see [Kr99], [J02], [J03], [J07], [T99]) in the last 10 years partly in response to the applications to the study of virtual phenomena. The applications to virtual phenomena, and algebraic topology further led to basic work on differential graded stacks as appearing the work of Toen, Vezzosi and Lurie (see [TV05], [TV08], [Lur04]) which has introduced many new ideas and techniques.

Despite all these, the theory of algebraic stacks is rather challenging, especially to a beginner, partially because of the use of very sophisticated mathematics, which is necessary to study algebraic stacks systematically and partially because of the lack of adequate literature that proceeds at a leisurely pace and aimed at beginners in the field. Moreover, the last major program devoted to algebraic stacks seems to have been the MSRI program for the first half of 2002. In view of these, the importance and relevance of workshops like the one proposed here should be clear. By bringing together the experts working in different aspects of stacks and their applications as well as a group of beginners in the field, the organizers hope to not only promote exchange of ideas between the various groups but also equally importantly to provide a leisurely introduction to this vast and exciting field to the grad student and post-doc participants.

Scientific program:

The class of Artin stacks is much bigger than Deligne-Mumford stacks. As a simple example, if G is a linear algebraic group acting on a smooth scheme X, one obtains the quotient stack [X/G] which classifies principal G-bundles together with a G-equivariant map to X. If the stabilizers for the $G$-action are not finite, the resulting stack is an Artin stack that is not Deligne-Mumford. As the above example of quotient stacks shows, Artin stacks occur far more commonly than Deligne-Mumford stacks: yet it is only in the last 10 years or so that such stacks have begun to be studied in detail. (See for example recent work of Olsson and Laszlo (and also somewhat earlier work by Behrend and Joshua) on the l-adic derived category, perverse sheaves and t-structures associated to such stacks: see [Be03], [J93], [LO08].) The same example of quotient stacks shows the importance of such stacks in geometric approaches to representation theory, for example in the geometric Langlands correspondence. Recent work of Witten and others have already established connections of this area with mathematical physics.

In view of its importance, we will devote an important part of the program to applications in geometric representation theory.

In recent years further generalizations of the notion of algebraic stacks have begun to emerge. The notions and basic properties of higher stacks and derived stacks are by now available, thanks to works of Simpson, Toen, Vaquie, Vezzosi, Lurie, and others (see e.g. [S96], [TVa07] [TV05], [TV08], [Lur04]). When considering a moduli problem, the notion of derived stacks allows one to encode, in the moduli space, all the information about deformations/obstructions/ higher obstructions of the moduli problem. On the other hand, the notion of higher stacks allows one to encode the information about higher automorphisms of the moduli problem. In order to capture all the information of a moduli problem, it is thus necessary to work in the more generalized context of higher and derived stacks. An example showing how beneficial this can be is J. Lurie's work on interpreting the topological modular forms of Miller-Hopkins using the derived moduli stack of elliptic curves: see [Hop02], [Lur09], [Lur09a].

In light of its importance, another focus of the proposed workshop will be on the theory of higher and derived stacks and its applications to moduli problems.

Intersection theory on moduli spaces is a very important area in algebraic geometry. A notable example is the intersection theory on moduli spaces of stable pointed curves, which is known to be closely related to other subjects such as integrable systems and infinite-dimensional Lie algebras. Unlike moduli of curves, many interesting moduli spaces (for example, moduli of stable maps, of stable sheaves, e.t.c.) are usually highly singular, and one studies their intersection theory by replacing the fundamental classes with the virtual fundamental classes. Virtual fundamental classes are constructed from choices of perfect obstruction theories, which arise from the deformation/obstruction of the moduli problems. Since deformation/obstruction is better encoded in derived stacks, it is very natural to consider virtual fundamental classes in the context of higher and derived stacks. Closely related to these virtual fundamental classes (and perhaps of somewhat more intrinsic importance) are the virtual structure sheaves, a thorough understanding of which needs the K-theory and G-theory of stacks.

In this workshop we will also be interested in understanding various aspects of intersection theory, K-theory and G-theory of higher and derived stacks.

REFERENCES

[Ar74] M. Artin, Versal deformations and algebraic stacks, Invent. Math., 27, (1974), 165--189.

[BF97] K. Behrend and B. Fantechi, The Intrinsic Normal Cone, Invent. Math., 128 (1997), 45--88.

[Be03] K. Behrend, Derived l-adic categories for algebraic stacks, Mem. Amer. Math. Soc., 163 (2003) no. 774, viii+93.

[CR04] W. Chen and Y. Ruan, A new cohomology theory for orbifolds, Comm. Math. Phys. 248 (2004), 1--31.

[DM69] D. Mumford and P. Deligne, The irreducibility of the space of curves of a given genus, IHES Publ. Math. 36 (1969), 75--109.

[Gir65] J. Giraud, Cohomologie non-abeliene, Die Grundlehren der mathematischen Wissenschaften, Band 179. Springer-Verlag, Berlin-New York, 1971. ix+467 pp.

[Hop02] M. Hopkins, Algebraic Topology and Modular Forms, in ``Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002)'', 291--317, Higher Ed. Press, Beijing, 2002.

[J93] R. Joshua, The derived category and intersection cohomology of algebraic stacks, in ``Algebraic K-theory and Algebraic Topology (Lake Louise, AB, 1991)'', 91--145, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 407, Kluwer Acad. Publ., Dordrecht, 1993.

[J03] R. Joshua, Riemann-Roch for algebraic stacks: I, Compositio Math. 136 (2003), no. 2, 117--169.

[J02] R. Joshua, Higher Intersection Theory on Algebraic Stacks:I and :II, K-Theory, 27 (2002), no. 2, 134--195. and 27 (2002), no. 3, 197--244.

[J07] R. Joshua, Bredon-style homology, cohomology and Riemann-Roch for algebraic stacks, Adv. Math. 209 (2007), no. 1, 1--68.

[Kont95] M. Kontsevich, Enumeration of rational curves via torus actions, in ``The moduli space of curves (Texel Island, 1994)'', 335--368, Progr. Math., 129, Birkh"auser Boston, Boston, MA, 1995.

[Kr99] A. Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999), no. 3, 495--536.

[LO08] Y. Laszlo and M. Olsson, The six operations for sheaves on Artin stacks I, Publ. Math. IHES, 107, (2008), 109-168.

[LT98] J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), no. 1, 119--174.

[Lur04] J. Lurie, Derived Algebraic Geometry, PhD thesis, MIT, 2004.

[Lur09] J. Lurie, Higher Topos Theory, Annals of Mathematics Studies, 170. Princeton University Press, Princeton, NJ, 2009. xviii+925 pp.

[Lur09a] J. Lurie, A survey of elliptic cohomology, in ``Algebraic topology'', 219--277, Abel Symp., 4, Springer, Berlin, 2009.

[Mum65] D. Mumford, Picard groups of moduli problems, in ``Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963)'', 33--81, Harper & Row, New York, 1965.

[MK76] F. Knudsen and D. Mumford, The projectivity of the moduli space of stable curves. I: Preliminaries on "det" and "Div", Math. Scand. 39 (1976), no. 1, 19--55.

[MFK94] D. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) (Results in Mathematics and Related Areas (2)), 34. Springer-Verlag, Berlin, 1994.

[OS03] M. Olsson and J. Starr, Quot functors for Deligne-Mumford stacks, Comm. Algebra 31 (2003), no. 8, 4069--4096.

[S96] C. Simpson, Algebraic (geometric) $n$-stacks, arXiv:alg-geom/9609014.

[T99] B. Toen, Th'eor`emes de Riemann-Roch pour les champs de Deligne-Mumford, K-Theory 18 (1999), no. 1, 33--76.

[TVa07] B. Toen and M. Vaqui'e, Moduli of objects in dg-categories, Ann. Sci. 'Ecole Norm. Sup. (4) 40 (2007), no. 3, 387--444.

[TV05] B. Toen and G. Vezzosi, Homotopical algebraic geometry. I. Topos theory, Adv. Math. 193 (2005), no. 2, 257--372.

[TV08] B. Toen and G. Vezzosi, Homotopical algebraic geometry. II. Geometric stacks and applications, Mem. Amer. Math. Soc. 193 (2008), no. 902, x+224 pp.