×

The hitchhiker guide to categorical Banach space theory. II. (English) Zbl 1501.46060

Summary: What has category theory to offer to Banach spacers? In this second part of our survey-like paper, we will focus on very much needed advanced categorical and homological elements, such as Kan extensions, derived category and derived functor or Abelian hearts of Banach spaces.
For Part I, see [the author, ibid. 25, No. 2, 103–149 (2010; Zbl 1235.46072)].

MSC:

46M15 Categories, functors in functional analysis
46M18 Homological methods in functional analysis (exact sequences, right inverses, lifting, etc.)
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
18-02 Research exposition (monographs, survey articles) pertaining to category theory

Citations:

Zbl 1235.46072
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] J. Ad´amek, H. Herrlich, G.E. Strecke,Abstract and Concrete Categories. The Joy of Cats. Available athttp://katmat.math.uni-bremen. de/acc. · Zbl 1113.18001
[2] A.K. Austin,Miscellanea: Modern Research in Mathematics,Amer. Math. Monthly84(1977), 566.
[3] M. Barr,Exact Categories, in “ Exact Categories and Categories of Sheaves ”, Lecture Notes in Mathmatics, 236, Springer-Verlag, Berlin, 1971. · Zbl 0223.18009
[4] A. Beilinson, J. Bernstein, P. Deligne,Faisceaux pervers, in “ Analysis and Topology on Singular Spaces. I ”, (Proceedings of the colloquium held at Luminy, July 6 - 11, 1981),Ast´erisque100(1982), 5 - 171.
[5] T. B¨uhler,Exact categories,Expo. Math.28(2010), 1 - 69. · Zbl 1192.18007
[6] F. Cabello, J.M.F. Castillo,The long homology sequence for quasi Banach spaces, with applications,Positivity8(2004), 379 - 394. · Zbl 1086.46046
[7] F. Cabello S´anchez, J.M.F. Castillo,Stability constants and the homology of quasi-Banach spaces,Israel J. Math.198(2013), 347 - 370. · Zbl 1284.46002
[8] F. Cabello S´anchez, J.M.F. Castillo,Homological methods in Banach space theory, Cambridge Studies in Advanced Mathematics, 2021, Online ISBN: 9781108778312.
[9] F. Cabello S´anchez, J.M.F. Castillo, W.H.G. Corrˆea, V. Ferenczi, R. Garc´ıa,On theExt2-problem in Hilbert spaces, J. Funct. Anal. 280(2021), paper no. 108863, 36 pp. · Zbl 1466.46070
[10] F. Cabello S´anchez, J.M.F. Castillo, R. Garc´ıa,Homological dimensions of Banach spaces,Mat. Sbornik.212(2021), 91 - 112. · Zbl 1478.46073
[11] F. Cabello S´anchez, J.M.F. Castillo, F. S´anchez,Nonlinear metric projections in twisted twilight,RACSAM94(2000), 473 - 483. · Zbl 1278.46003
[12] F. Cabello S´anchez, J. Garbuli´nska-We¸grzyn, W. Kubi´s,QuasiBanach spaces of almost universal disposition,J. Funct. Anal.267(2014), 744 - 771. · Zbl 1321.46003
[13] J.M.F. Castillo,The hitchhiker guide to categorical Banach space theory. Part I,Extracta Math.25(2010), 103 - 149. · Zbl 1235.46072
[14] J.M.F. Castillo, M. Cho, M. Gonz´alez,Three-operator problems in Banach spaces,Extracta Math.33(2018), 149 - 165. · Zbl 07046829
[15] J.M.F. Castillo, R. Garc´ıa, J. Su´arez,Extension and lifting of operators and polynomials,Mediterr. J. Math.9(2012), 767 - 788. · Zbl 1262.46016
[16] J.M.F. Castillo, Y. Moreno,The category of exact sequences between Banach spaces, in “ Methods in Banach Space Theory ” (Proceedings of the V Conference in Banach spaces, C´aceres, 2004; J.M.F. Castillo and W.B. Johnson, eds.), LN London Math. Soc. Lecture Note Ser., 337, Cambridge Univ. Press, Cambridge, 2006, 139 - 158. · Zbl 1135.46042
[17] J.M.F. Castillo, Y. Moreno,Twisted dualities in Banach space theory, in “ Banach Spaces and their Applications in Analysis ”, (B. Radrianantoanina and N. Radrianantoanina, eds.), Walter de Gruyter, Berlin, 2007, 59 - 76. · Zbl 1139.46048
[18] J.M.F. Castillo, Y. Moreno,Extensions by spaces of continuous functions, Proc. Amer. Math. Soc.136(2008), 2417 - 2423. · Zbl 1160.46047
[19] J.M.F. Castillo, Y. Moreno,Sobczyk’s theorem and the bounded approximation property,Studia Math.201(2010), 1 - 19. · Zbl 1216.46063
[20] J. Cigler,Tensor products of functors on categories of Banach spaces, in “ Categorical Topology (Proc. Conf., Mannheim, 1975) ”, Lecture Notes in Math., 540, Springer, Berlin, 1976, 164 - 187. · Zbl 0333.46047
[21] J. Cigler, V. Losert, P.W. Michor,“ Banach Modules and Functors on Categories of Banach Spaces ”, Lecture Notes in Pure and Applied Mathematics, 46, Marcel Dekker, Inc., New York, 1979. · Zbl 0411.46044
[22] S. Eilenberg, S. MacLane,General theory of natural equivalences,Trans. Amer. Math. Soc.58(1945), 231 - 294. · Zbl 0061.09204
[23] P. Enflo, J. Lindenstrauss, G. Pisier,On the “three space problem”, Math. Scand.36(1975), 199 - 210. · Zbl 0314.46015
[24] A. Grothendieck,Une caract´erisation vectorielle-m´etrique des espacesL1, Canadian J. Math.7(1955), 552 - 561. · Zbl 0065.34503
[25] L.Frerick,D.Sieg,Exactcategoriesinfunctionalanalysis, Script 2010. Available athttps://www.math.uni-trier.de/abteilung/ analysis/HomAlg.pdf.
[26] P. Gabriel, M. Zisman,“ Calculus of Fractions and Homotopy Theory ”, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, SpringerVerlag New York, Inc., New York, 1967. · Zbl 0186.56802
[27] J. Garbuli´nska-We¸grzyn, W. Kubi´s,A note on universal operators between separable Banach spaces,RACSAM114(2020), paper 114, 11 pp. · Zbl 1506.47027
[28] S.I. Gelfand, Yu I. Manin,“ Methods of Homological Algebra ”, Second Edition, Springer Monographs in Math., Springer-Verlag, Berlin, 2003. · Zbl 1006.18001
[29] C. Herz, J.W. Pelletier,Dual functors and integral operators in the category of Banach spaces,J. Pure Appl. Algebra8(1976), 5 - 22. · Zbl 0331.46064
[30] P. Hilton, U. Stammbach,“ A Course in Homological Algebra ”, Graduate Texts in Mathematics, 4, Springer-Verlag, New York-Berlin, 1971. · Zbl 0238.18006
[31] M.ˇI. Kadec,On complementably universal Banach spaces,Studia Math.40 (1971), 85 - 89. · Zbl 0218.46015
[32] S. Kaijser, J.W. Pelletier,“ A Categorical Framework for Interpolation Theory ”, Lecture Notes in Math., 962, Springer, Berlin-New York, 1982. · Zbl 0525.46041
[33] S. Kaijser, J.W. Pelletier,“ Interpolation Functors and Duality ”, Lecture Notes in Mathematics, 1208, Springer-Verlag, Berlin, 1986. · Zbl 0552.46041
[34] N.J. Kalton, N.T. Peck,Twisted sums of sequence spaces and the three space problem,Trans. Amer. Math. Soc.255(1979) 1 - 30. · Zbl 0424.46004
[35] D.M. Kan,Adjoint functors,Trans. Amer. Math. Soc.87(1958), 294 - 329. categorical banach space theory ii55 · Zbl 0090.38906
[36] G. Kato,“ The Heart of Cohomology ”, Springer, Dordrecht, 2006. · Zbl 1101.18001
[37] G. K¨othe,Hebbare lokalkonvexe R¨aume,Math. Ann.165(1966), 181-195. · Zbl 0141.11605
[38] W. Kubi´s,Fra¨ıss´e sequences: category-theoretic approach to universal homogeneous structures,Ann. Pure Appl. Logic165(2014), 1755 - 1811. · Zbl 1329.18002
[39] M.Ch. Lehner,All Concepts are Kan Extensions: Kan Extensions as the Most Universal of the Universal Constructions, Undergraduate senior thesis, Harvard College, 2014. Available athttps://www.math.harvard.edu/ media/lehner.pdf.
[40] S. Lubkin,Imbedding of Abelian categories,Trans. Amer. Math. Soc.97 (1960), 410 - 417. · Zbl 0096.25501
[41] S. MacLane,“ Categories for the Working Mathematician ”, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York-Berlin, 1971. · Zbl 0705.18001
[42] S. Mac Lane,“ Homology ”, Die Grundlehren der mathematischen Wissenschaften, 114, Springer-Verlag, Berlin-New York, 1975. · Zbl 0328.18009
[43] Y. Moreno,“ Theory ofz-Linear Maps ”, Ph.D. Thesis, Universidad de Extremadura, Badajoz, 2003.
[44] G. No¨el,Une immersion de la cat´egorie des espaces bornologiques convexes s´epar´es dans une cat´egorie ab´elienne (French),C. R. Acad. Sci. Paris S´er. A-B269(1969), A195 - A197. · Zbl 0177.15902
[45] A. Orty´nski,On complemented subspaces of‘p(Γ) for 0< p <1,Bull. Acad. Polon. Sci. S´er. Sci. Math. Astronom. Phys.26(1978), 31 - 34. · Zbl 0367.46009
[46] A. Pe lczy´nski,Universal bases,Studia Math.32(1969), 247 - 268. · Zbl 0185.37401
[47] A. Pe lczy´nski and P. Wojtaszczyk,Banach spaces with finite dimensional expansions of identity and universal bases of finite dimensional spaces, Studia Math.40(1971), 91 - 108. · Zbl 0221.46014
[48] J.W. Pelletier,Dual functors and the Radon-Nikod´ym property in the category of Banach spaces,J. Austral. Math. Soc. Ser. A27(1979), 479 - 494. · Zbl 0402.46042
[49] J.W. Pelletier,Kan extensions of the Hom functor in the category of Banach spaces,Ann. Sci. Math. Qu´ebec4(1980) 59 - 71. · Zbl 0432.46065
[50] K. Pothoven,Projective and injective objects in the category of Banach spaces,Proc. Amer. Math. Soc.22(1969), 437 - 438. · Zbl 0176.42903
[51] D.G. Quillen,Higher algebraic K-theory. I, in “ Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Washington,1972), Lecture Notes in Math., 341, Springer, Berlin, 1973, 85 - 147. · Zbl 0292.18004
[52] E. Riehl,“ Category Theory in Context ”, Dover Publications, 2016. · Zbl 1348.18001
[53] P. Scholze,“ Lectures on Analytic Geometry ”. Available atwww.math. uni-bonn.de/people/scholze/Analytic.pdf. · Zbl 1475.14002
[54] Z. Semadeni, H. Zidenberg,Inductive and inverse limits in the category of Banach spaces,Bull. Acad. Polon. Sci. S´er. Sci. Math. Astronom. Phys. 13(1965), 579 - 583. · Zbl 0144.16704
[55] Z. Semadeni,The Banach Mazur functor and related functors,Comment. Math. Prace Mat.14(1970), 173 - 182. · Zbl 0237.46074
[56] Z. Semadeni,“ Banach Spaces of Continuous Functions, Vol. I ”, Monografie Matematyczne, Tom 55, PWN-Polish Scientific Publishers, Warsaw, 1971. · Zbl 0225.46030
[57] K. Vonnegut,Slaughterhouse five.
[58] L. Waelbroeck,Quotient Banach spaces, in “ Spectral Theory (Warsaw 1977) ”, Banach Center Publ., 8, PWN, Warsaw, 1982, 553 - 562. · Zbl 0492.46012
[59] L. Waelbroeck,Quotient Banach spaces; multilinear theory, in “ Spectral Theory (Warsaw 1977) ”, Banach Center Publ., 8, PWN, Warsaw, 1982, 563 - 571. · Zbl 0492.46013
[60] L. Waelbroeck,The Taylor spectrum and quotient Banach spaces, in “ Spectral Theory (Warsaw 1977) ”, Banach Center Publ., 8, PWN, Warsaw, 1982, 573ˆu578. · Zbl 0492.46014
[61] L. Waelbroeck,The category of quotient bornological spaces, in “ Aspects of Mathematics and its Applications ”, North-Holland Math. Library, 34, North-Holland, Amsterdam, 1986, 873 - 894. · Zbl 0633.46071
[62] L. Waelbroeck,Around the quotient bornological spaces, Dedicated to the memory of Professor Gottfried K¨othe,Note Mat.11(1991), 315 - 329. · Zbl 0802.46009
[63] L. Waelbroeck,Quotient Fr´echet spaces,Rev. Roumaine Math. Pures Appl. 34(1989), 171 - 179. · Zbl 0696.46052
[64] L. Waelbroeck,“ Bornological Quotients ”, with the collaboration of Guy No¨el, M´emoire de la Classe des Sciences, Collection in-4o, 3e S´erie [Memoir of the Science Section, Collection in-4o, 3rd Series], VII, Acad´emie Royale de Belgique, Classe des Sciences, Brussels, 2005. · Zbl 1084.46001
[65] S.A. Wegner,The heart of the Banach spaces,J. Pure Appl. Algebra221 (2017), 2880 - 2909. · Zbl 1377.46053
[66] Ch.A. Weibel,“ An Introduction to Homological Algebra ”, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, Cambridge, 1994. · Zbl 0797.18001
[67] J. Wengenroth,“ Derived Functors in Functional Analysis ”, Lecture Notes in Mathematics, 1810, Springer-Verlag, Berlin, 2003. · Zbl 1031.46001
[68] M. Wodzicki,Homological dimensions of Banach spaces, in “ Linear and Complex Analysis Problem Book 3, Part I ”, V.P. Havin and N.K. Nikolskii (eds), Notes in Mathematics, 1573, Springer-Verlag, Berlin, 1994, 34 - 35 · Zbl 0893.30036
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.