Singular superspaces. (English) Zbl 1300.32004
Summary: We introduce a wide category of superspaces, called locally finitely generated, which properly includes supermanifolds, but enjoys much stronger permanence properties, as are prompted by applications. Namely, it is closed under taking finite fibre products (i.e. is finitely complete) and thickenings by spectra of Weil superalgebras. Nevertheless, in this category, morphisms with values in a supermanifold are still given in terms of coordinates. This framework gives a natural notion of relative supermanifolds over a locally finitely generated base. Moreover, the existence of inner homs, whose source is the spectrum of a Weil superalgebra, is established; they are generalisations of the Weil functors defined for smooth manifolds.
References:
| [1] | Alldridge, A., Laubinger, M.: Infinite-dimensional supermanifolds over arbitrary base fields. Forum Math. 24(3), 565-608 (2012). doi:10.1515/form.2011.074 · Zbl 1252.58002 · doi:10.1515/form.2011.074 |
| [2] | Balduzzi, L., Carmeli, C., Fioresi, R.: The local functors of points of supermanifolds. Expo. Math. 28(3), 201-217 (2010). doi:10.1016/j.exmath.2009.09.005 · Zbl 1209.58005 · doi:10.1016/j.exmath.2009.09.005 |
| [3] | Bouarroudj, S., Grozman, PYa., Leites, D.A., Shchepochkina, I.M.: Minkowski superspaces and superstrings as almost real-complex supermanifolds. Theor. Math. Phys. 173(3), 1687-1708 (2012). doi:10.1007/s11232-012-0141-3 · Zbl 1282.81142 · doi:10.1007/s11232-012-0141-3 |
| [4] | Bredon, G.E.: Sheaf Theory, Graduate Texts in Mathematics, vol. 170, 2nd edn. Springer, New York (1997) · Zbl 0874.55001 |
| [5] | Carchedi, D., Roytenberg, D.: On theories of superalgebras of differentiable functions. Theory Appl. Categ. 28(30), 1022-1098 (2013) · Zbl 1291.18007 |
| [6] | Deligne, P., Morgan, J. W.: Notes on supersymmetry. Quantum Fields and Strings: A Course for Mathematicians, pp. 41-98 (1999) |
| [7] | Demazure, M., Gabriel, P.: Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs. Masson & Cie, Éditeur, Paris (1970) · Zbl 0203.23401 |
| [8] | Gelfand, S.I., Manin, Y.I.: Methods of Homological Algebra. Springer, Berlin (1996) · Zbl 0855.18001 · doi:10.1007/978-3-662-03220-6 |
| [9] | Grothendieck, A., Dieudonné, J.A.: Éléments de géométrie algébrique. I., Grundlehren der mathematischen Wissenschaften, vol. 166. Springer, Berlin (1971) · Zbl 0203.23301 |
| [10] | Hohnhold, H., Kreck, M., Stolz, S., Teichner, P.: Differential forms and 0-dimensional supersymmetric field theories. Quantum Topol. 2(1), 1-41 (2011). doi:10.4171/QT/12 · Zbl 1236.19008 · doi:10.4171/QT/12 |
| [11] | Iversen, B.: Cohomology of Sheaves, Universitext. Springer, Berlin (1986) · Zbl 1272.55001 · doi:10.1007/978-3-642-82783-9 |
| [12] | Kashiwara, M., Schapira, P.: Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften, vol. 332. Springer, Berlin (2006) · Zbl 1118.18001 |
| [13] | Kolář, I., Michor, P.W., Slovák, J.: Natural Operations in Differential Geometry. Springer, Berlin (1993) · Zbl 0782.53013 |
| [14] | Kontsevich, M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66(3), 157-216 (2003). doi:10.1023/B:MATH.0000027508.00421.bf · Zbl 1058.53065 · doi:10.1023/B:MATH.0000027508.00421.bf |
| [15] | Leites, D.A.: Introduction to the Theory of Supermanifolds, Uspekhi Mat. Nauk, (1), 3-57 (1980) (Russian); English transl. Russian Math. Surveys 35(1), 1-64 (1980) · Zbl 0439.58007 |
| [16] | Mac Lane, S.: Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5, 2nd edn. Springer, New York (1998) · Zbl 0906.18001 |
| [17] | Manin, Y.I.: Gauge Field Theory and Complex Geometry, Grundlehren der Mathematischen Wissenschaften, vol. 289, 2nd edn. Springer, Berlin (1997) · Zbl 0884.53002 · doi:10.1007/978-3-662-07386-5 |
| [18] | Moerdijk, I., Reyes, G.E.: Models for Smooth Infinitesimal Analysis. Springer, New York (1991) · Zbl 0715.18001 · doi:10.1007/978-1-4757-4143-8 |
| [19] | Reichard, K.: Nichtdifferenzierbare Morphismen differenzierbarer Räume. Manuscripta Math. 17, 243-250 (1975) · Zbl 0305.58003 · doi:10.1007/BF01168676 |
| [20] | Varadarajan, V.S.: Supersymmetry for mathematicians: an introduction, Courant Lecture Notes in Mathematics, vol. 11. New York University, Courant Institute of Mathematical Sciences. New York; American Mathematical Society, Providence, RI (2004) · Zbl 1142.58009 |
| [21] | Weil, A.: Théorie des points proches sur les variétés différentiables, Géométrie différentielle. Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourg, 1953, Centre National de la Recherche Scientifique, Paris, pp. 111-117 (1953) (French) · Zbl 0053.24903 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
