What is the spectral category? (English) Zbl 1440.18012
Facchini, Alberto (ed.) et al., Advances in rings, modules and factorizations. Selected papers based on the presentations at the international conference on rings and factorizations, Graz, Austria, February 19–23, 2018. Cham: Springer. Springer Proc. Math. Stat. 321, 135-152 (2020).
Summary: For a category \(\mathcal{C}\) with finite limits and a class \(\mathcal{S}\) of monomorphisms in \(\mathcal{C}\) that is pullback stable, contains all isomorphisms, is closed under composition, and has the strong left cancellation property, we use pullback stable \(\mathcal{S}\)-essential monomorphisms in \(\mathcal{C}\) to construct a spectral category \(\text{Spec}(\mathcal{C},\mathcal{S})\). We show that it has finite limits and that the canonical functor \(\mathcal{C}\rightarrow \text{Spec}(\mathcal{C},\mathcal{S})\) preserves finite limits. When \(\mathcal{C}\) is a normal category, assuming for simplicity that \(\mathcal{S}\) is the class of all monomorphisms in \(\mathcal{C}\), we show that pullback stable \(\mathcal{S}\)-essential monomorphisms are the same as what we call subobject-essential monomorphisms.
For the entire collection see [Zbl 1445.13002].
For the entire collection see [Zbl 1445.13002].
MSC:
| 18E10 | Abelian categories, Grothendieck categories |
| 18N40 | Homotopical algebra, Quillen model categories, derivators |
| 18A32 | Factorization systems, substructures, quotient structures, congruences, amalgams |
| 18-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to category theory |
References:
| [1] | Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and concrete categories. The joy of cats, Pure and Applied Math., John Wiley & Sons, Inc., New York (1990) · Zbl 0695.18001 |
| [2] | Adámek, J., Herrlich, H., Rosický, J., Tholen, W.: Injective hulls are not natural. Algebra Universalis 48, 374-388 (2002) · Zbl 1061.18010 · doi:10.1007/s000120200006 |
| [3] | Arroyo Paniagua, M.J., Facchini, A.: Category of \(G\)-groups and its spectral category, Comm. Algebra 45(4), 1696-1710 (2017) · Zbl 1394.20032 |
| [4] | Banaschewski, B.: Injectivity and essential extensions in equational classes of algebras. “Queen’s papers in Pure and Applied Mathematics No. 25” (Proceedings of the Conference on Universal Algebra, Kingston 1969), pp. 131-147. Kingston, Ontario (1970) · Zbl 0233.18002 |
| [5] | Bénabou, J.: Introduction to bicategories. In: 1967 Reports of the Midwest Category Seminar. Springer, Berlin, pp. 1-77 · Zbl 0165.33001 |
| [6] | Bénabou, J.: Some geometric aspects of the calculus of fractions, the European Colloquium of category theory (Tours, 1994). Appl. Categ. Struct. 4(2), 139-165 (1996) · Zbl 0874.18007 · doi:10.1007/BF00122249 |
| [7] | Carboni, A., Janelidze, G., Kelly, G.M., Paré, R.: On localization and stabilization of factorization systems. Appl. Categ. Struct. 5, 1-58 (1997) · Zbl 0866.18003 · doi:10.1023/A:1008620404444 |
| [8] | Facchini, A.: Injective modules, spectral categories, and applications. In: Jain, S.K., Parvathi, S. (eds.) Noncommutative Rings, Group Rings, Diagram Algebras and their Applications, Contemp. Math. 456, Amer. Math. Soc., Providence, RI, pp. 1-17 (2008) · Zbl 1161.16003 |
| [9] | Fichtner, K.: Eine Bermerkung über Mannigfaltigkeiten universeller Algebren mit Idealen. Monatsh. d. Deutsch. Akad. d. Wiss. (Berlin) 12, 21-25 (1970) · Zbl 0198.33601 |
| [10] | Gabriel, P.: Des catégories abéliennes. Bull. Soc. Math. Fr. 90, 323-448 (1962) · Zbl 0201.35602 · doi:10.24033/bsmf.1583 |
| [11] | Gabriel, P., Oberst, U.: Spektralkategorien und reguläre Ringe im von-Neumannschen Sinn. Math. Z. 92, 389-395 (1966) · Zbl 0136.25602 · doi:10.1007/BF01112218 |
| [12] | Gabriel, P., Zisman, M.: Calculus of Fractions and Homotopy Theory. Springer, Berlin (1967) · Zbl 0186.56802 |
| [13] | Goodearl, K.R.: Ring Theory. Nonsingular Rings and Modules. Marcel Dekker Inc., New York and Basel (1976) · Zbl 0336.16001 |
| [14] | Janelidze, G., Mac Lane, S.: Private communication |
| [15] | Janelidze, G., Tholen, W.: Strongly separable morphisms in general categories. Theory Appl. Categ. 23(7), 136-149 (2010) · Zbl 1315.18004 |
| [16] | Janelidze, G., Márki, L., Tholen, W.: Semi-abelian categories. J. Pure Appl. Algebra 168, 367-386 (2002) · Zbl 0993.18008 · doi:10.1016/S0022-4049(01)00103-7 |
| [17] | Janelidze, G., Márki, L., Ursini, A.: Ideals and clots in universal algebra and in semi-abelian categories. J. Algebra 307, 191-208 (2007) · Zbl 1110.18006 · doi:10.1016/j.jalgebra.2006.05.022 |
| [18] | Janelidze, Z.: The pointed subobject functor, \(3\times 3\) lemmas, and subtractivity of spans. Theory Appl. Categ. 23(11), 221-242 (2010) · Zbl 1234.18009 |
| [19] | Mitchell, B.: Theory of categories. Pure Appl. Math. 17 (1965). Academic Press · Zbl 0136.00604 |
| [20] | Schubert, H.: Categories. Springer, New York, Heidelberg (1972) · Zbl 0253.18002 |
| [21] | Tholen, W.: Injective objects and cogenerating sets. J. Algebra 73, 139-155 (1981) · Zbl 0487.18014 · doi:10.1016/0021-8693(81)90351-3 |
| [22] | Yoneda, N. |
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