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Keywords:
Homological algebra; characteristic one; monads; Kleisli and Eilenberg-Moore categories; homological category
Homological algebra; characteristic one; monads; Kleisli and Eilenberg-Moore categories; homological category
Summary:
This article develops several main results for a general theory of homological algebra in categories such as the category of idempotent semimodules. In the analogy with the development of homological algebra for abelian categories the present paper should be viewed as the analogue of the development of homological algebra for abelian groups. Our selected prototype, the category $\Bbb B$mod of semimodules over the Boolean semifield $\Bbb B:=\{0,1\}$ is the replacement for the category of abelian groups. We show that the semi-additive category $\Bbb B$mod fulfills analogues of the axioms AB1 and AB2 for abelian categories. By introducing a precise comonad on $\Bbb B$mod we obtain the conceptually related Kleisli and Eilenberg-Moore categories. The latter category $\Bbb B{\rm mod}^{\frak s}$ is simply $\Bbb B$mod in the topos of sets endowed with an involution and as such it shares with $\Bbb B$mod most of its abstract categorical properties. The three main results of the paper are the following. First, when endowed with the natural ideal of null morphisms, the category $\Bbb B{\rm mod}^{\frak s}$ is a semiexact, homological category in the sense of M. Grandis. Second, there is a far reaching analogy between $\Bbb B{\rm mod}^{\frak s}$ and the category of operators in Hilbert spaces, and in particular results relating null kernel and injectivity for morphisms. The third fundamental result is that, even for finite objects of $\Bbb B{\rm mod}^{\frak s}$, the resulting homological algebra is non-trivial and gives rise to a computable Ext functor. We determine explicitly this functor in the case provided by the diagonal morphism of the Boolean semiring into its square.
References:
[1] Akian, M., Gaubert, S., Guterman, A.: Tropical Cramer determinants revisited. Tropical and idempotent mathematics and applications, 1–45, Contemp. Math., 616, Amer. Math. Soc., Providence, RI
[2] Amadio, R. M., Curien, P-L.: Domains and lambda-calculi. Cambridge Tracts in Theoretical Computer Science, 46. Cambridge University Press, Cambridge
[3] Barr, M., Wells, C.: Toposes, triples and theories. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 278. Springer-Verlag, New York
[4] Bourbaki, N.: Topological vector spaces. Chapter I–V. Springer
[5] Buhler, T.: Exact categories. Expo. Math. 28, no. 1, 1–69 DOI 10.1016/j.exmath.2009.04.004
[6] Cartan, H., Eilenberg, S.: Homological algebra. Princeton University Press, Princeton
[7] Cohen, G., Gaubert, S., Quadrat, J.P.: Duality and separation theorems in idempotent semimodules. Tenth Conference of the International Linear Algebra Society. Linear Algebra Appl. 379, 395–422
[8] Connes, A., Consani, C.: The Arithmetic Site. Comptes Rendus Mathematique Ser. I 352, 971–975
[9] Connes, A., Consani, C.: Geometry of the Arithmetic Site. Advances in Mathematics 291 274–329 DOI 10.1016/j.aim.2015.11.045
[10] Connes, A., Consani, C.: The Scaling Site. C.R. Mathematique, Ser. I 354 1–6
[11] Connes, A., Consani, C.: Geometry of the Scaling Site. Selecta Math. (N.S.) 23, no. 3, 1803–1850 DOI 10.1007/s00029-017-0313-y
[12] Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order. Cambridge University Press Zbl 1002.06001
[13] Ehresmann, C.: Sur une notion générale de cohomologie. C. R. Acad. Sci. Paris 259 2050–2053
[14] Gabriel, P., Ulmer, F.: Lokal Praesentierbare Kategorien. Springer Lecture Notes in Mathematics 221, Berlin
[15] Gaubert, S.: Théorie des systèmes linéaires dans les diodes. Thèse, École des mines de Paris,
[16] Gierz, G., Hofmann, K., Keimel, K., Lawson, J., Mislove, M., Scott, D.: Continuous lattices and domains. Encyclopedia of Mathematics and its Applications, 93. Cambridge University Press, Cambridge
[17] Grandis, M.: A Categorical Approach to Exactness in Algebraic Topology. V International Meeting on Topology in Italy (Italian) (Lecce, 1990/Otranto, 1990)
[18] Grandis, M.: Homological algebra in strongly non-abelian settings. World Scientific Publishing, Hackensack, xii+343 pp.
[19] Grothendieck, A.: Sur quelques points d’algèbre homologique. I. Tohoku Math. J. (2) 9, no. 2, 119–221
[20] Lavendhomme, R.: La notion d’idéal dans la théorie des catégories. Ann. Soc. Sci. Bruxelles, Sér. 1 79 5–25
[21] Lane, S. Mac: Categories for the working mathematician. Graduate Texts in Mathematics, 5. Springer-Verlag, New York
[22] Lane, S. Mac, Moerdijk, I: Sheaves in geometry and logic. A first introduction to topos theory. Universitext. Springer-Verlag, New York
[23] Moller, P.: Théorie algébrique des Systèmes à évènements Discrets. Thèse, Ecole des Mines de Paris
[24] Patchkoria, A.: Extensions of semimodules by monoids and their cohomological characterization. Bull. Georgian Acad. Sci. 86, No.1, 21–24 (in Russian).
[25] Patchkoria, A.: Cohomology of monoids with coefficients in semimodules. Bull. Georgian Acad. Sci. 86, No.3, 545–548 (in Russian).
[26] Patchkoria, A.: On monoid cohomology. Proc. A.Razmadze Math. Inst. 91, 36–43 (in Russian).
[27] Rowen, L.: Algebras with a negation map. arXiv:1602.00353
[28] Weibel, C.: An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, 38. Cambridge University Press, Cambridge
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