The hitchhiker guide to Categorical Banach space theory. Part II.

Main Article Content

Jesús M.F. Castillo


What has category theory to offer to Banach spacers? In this second part 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.


Download data is not yet available.


Metrics Loading ...

Article Details

How to Cite
Castillo, J. (2022). The hitchhiker guide to Categorical Banach space theory. Part II. Extracta Mathematicae, 37(1), 1-56.
Banach Spaces and Algebras


[1] J. Adámek, H. Herrlich, G.E. Strecke, Abstract and Concrete Categories. The Joy of Cats. Available at
[2] A.K. Austin, Miscellanea: modern research in mathematics, Amer. Math. Monthly 84 (1977) 566.
[3] M. Barr, Exact Categories, in Exact Categories and Categories of Sheaves, Lecture Notes in Math. 236, Springer 1971.
[4] A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, in Analysis and topology on singular spaces, I. (Luminy 1981), Astérisque 100 (1982) 5–171.
[5] T. Bühler, Exact categories, Expo. Math. 28 (2010) 1-69.
[6] F. Cabello and J.M.F. Castillo, The long homology sequence for quasi Banach spaces, with applications, Positivity, 8 (2004) 379-394.
[7] F. Cabello Sánchez, J.M.F. Castillo, Stability constants and the homology of quasi-Banach spaces, Israel J. Math. 198 (2013) 347-370.
[8] F. Cabello Sánchez, J.M.F. Castillo, Homological methods in Banach space theory, Cambridge Studies in Advanced Mathematics, 2021. Online ISBN. 9781108778312.
[9] F. Cabello Sánchez, J.M.F. Castillo, W.H.G. Corrêa, V. Ferenczi, R. Garcı́a, On the Ext2 -problem in Hilbert spaces, J. Functional Anal. 280 (2021) 108863.
[10] F. Cabello Sánchez, J.M.F. Castillo, R. Garcı́a, Homological dimensions of Banach spaces, Mat. Sbornik. 212 (2021) 91-112.
[11] F. Cabello Sánchez, J.M.F. Castillo, F. Sánchez, Nonlinear metric projections in twisted twilight, RAC- SAM 94 (2000) 473-483.
[12] F. Cabello Sánchez, J. Garbulińska-Wȩgrzyn, W. Kubiś, Quasi-Banach spaces of almost universal disposition, J. Funct. Anal. 267 (2014) 744–771.
[13] J.M.F. Castillo, The Hitchhiker Guide to Categorical Banach Space Theory. Part I, Extracta Math. 25 (2010) 103–149.
[14] J.M.F. Castillo, M. Cho, M. González, Three-operator problems in Banach spaces, Extracta Math. 33 (2018) 149–165.
[15] J.M.F. Castillo, R. Garcı́a, J. Suárez, Extension and lifting of operators and polynomials, Mediterranean J. Math. 9 (2012) 767-788.
[16] J.M.F. Castillo, Y. Moreno, The category of exact sequences between Banach spaces, in Banach space methods, Proceedings of the V Conference in Banach spaces, Cáceres, 2004, (J.M.F. Castillo and W.B. Johnson eds.), LN London Math. Society 337, Cambridge University Press. 2006, pp. 139-158.
[17] J.M.F. Castillo, Y. Moreno, Twisted dualities in Banach space theory in “Banach spaces and their applications in Analysis”, (B. Radiantoanina and N. Radrianantoanina, eds.) Walter de Gruyter Proceedings in Maths. 59–76 (2007).
[18] J.M.F. Castillo, Y. Moreno, Extensions by spaces of continuous functions, Proc. Amer. Math. Soc. 136 (2008) 2417–2424.
[19] J.M.F. Castillo, Y. Moreno, Sobczyk’s theorem and the Bounded Approximation Property, Studia Math. 201 (2010) 1–19.
[20] J. Cigler, Tensor products of functors on categories of Banach spaces, in “Categorical Topology”, Proceedings of the 1975 Mannheim Conference, Lecture Notes in Math. 540, Springer, Berlin, 1976, 164 – 187.
[21] J. Cigler, V. Losert, P.W. Michor, Banach Modules on Categories of Banach Spaces, Lecture Notes in Pure and Appl. Math. 46, Marcel-Dekker, Inc., New York, 1979.
[22] S. Eilenberg, S. Mac Lane, General theory of natural equivalences, Trans. Amer. Math. Soc. 58 (1945) 231-294.
[23] P. Enflo, J. Lindenstrauss, G. Pisier, On the three space problem, Math. Scand. 36 (1975) 199–210.
[24] L. Frerick, D. Sieg, Exact categories in functional analysis, Script 2010. Available at:
[25] P. Gabriel, M. Zisman, Calculus of fractions and homotopy theory, Ergeb. der Math. und ihrer Grenzgebiete, Band 35, Springer-Verlag New York, Inc., New York, 1967.
[26] J.Garbulińska-Wȩgrzyn, W. Kubiś, A note on universal operators between separable Banach spaces, RACSAM (2020) 114–148.
[27] S.I. Gelfand, Yu I. Manin, Methods of Homological Algebra. Second Edition. Springer Monographs in Math., Springer, 2010.
[28] A. Grothendieck, Une caractérisation vectorielle-métrique des espaces L1, Canad. J. Math. 7 (1955) 552-561.
[29] C. Herz, J.W. Pelletier, Dual functors and integral operators in the category of Banach spaces, J. Pure Appl. Algebra 8 (1976) 5 – 22.
[30] P. Hilton, U. Stammbach, A course in homological algebra, GTM 4, Springer-Verlag 1970.
[31] M. I. Kadec, On complementably universal Banach spaces, Studia Math. 40 (1971) 85–89.
[32] S. Kaijser, J.W. Pelletier, A Categorical Framework for Interpolation Theory, Lecture Notes in Math. 962, Springer, Berlin, 1982.
[33] S. Kaijser, J.W. Pelletier, Interpolation Functors and Duality, Lecture Notes in Math. 1208, Springer 1986.
[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.
[35] D.M. Kan, Adjoint functors, Trans. Amer. Math. Soc. 87 (1958) 294–329.
[36] G. Kato, The heart of cohomology, Springer 2006.
[37] G. Köthe, Hebbare lokalkonvexe Räume, Math. 165 (1966) 181-195.
[38] W. Kubiś, Fraı̈ssé sequences - a category-theoretic approach to universal homogeneous structures, Ann. Pure Appl. Logic 165 (2014) 1755–1811.
[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 at www.math.harvard. edu/theses/senior/lehner/lehner.pdf
[40] S. Lubkin, Imbedding of Abelian categories, Trans. Amer. Math. Soc. 97 (1960) 410 – 417.
[41] S. Mac Lane, Categories for the working mathematician. GTM 5, Springer 1971.
[42] S. Mac Lane, Homology, Grundlehren der mathematischen Wissenschaften 114, Springer-Verlag 1975.
[43] Y. Moreno, Theory of z-linear maps, Ph.D. Thesis, Univ. Extremadura, 2003.
[44] G.Noël, Une immersion de la catégorie des espaces bornologiques convexes séparés dans une catégorie abélienne, (French) C. R. Acad. Sci. Paris Sér. A-B 269 (1969), A195–A197.
[45] A. Ortyński, On complemented subspaces of lp (Γ) for 0 < p < 1, Bull. Acad. Pol. Sci. 26 (1978) 31–34.
[46] A. Pelczyński, Universal bases, Studia Math. 32 (1969) 247–268.
[47] A. Pelczyński and P. Wojtaszczyk, Banach spaces with finite dimensional expansions of identity and universal bases of finite dimensional spaces, Studia Math. XL (1971) 91–108.
[48] J.W. Pelletier, Dual functors and the Radon-Nikodým property in the category of Banach spaces, J. Austral. Math. Soc. Ser A, 27 (1979) 479 – 494.
[49] J.W. Pelletier, Kan extensions of the Hom functor in the category of Banach spaces, Ann. Sci. Québec IV. Austral. Math. Soc. Ser A, 27 (1979) 479 – 494.
[50] K. Pothoven, Projective and injective objects in the category of Banach spaces, Proc. Amer. Math. Soc. 22 (1969) 437-438.
[51] D.G. Quillen, Higher algebraic K-theory. I, in Algebraic K-theory,I : Higher K-theories, Proceedings of the Conference, Battelle Memorial Institute, Seattle, Washington,1972, Lecture Notes in Mathematics 341, Springer 1973, 85–147.
[52] E. Riehl, Category Theory in Context, Dover (2016).
[53] P. Scholze, Lectures on Analytic Geometry,
[54] Z. Semadeni, H. Zidenberg, Inductive limits in the category of Banach spaces, Bull. Acad. Sci Pol. 13 (1965) 579-583
[55] Z. Semadeni, The Banach Mazur functor and related functors, Comment. Math. Prace Math 14 (1970) 173–182.
[56] Z. Semadeni, Banach spaces of continuous functions, vol. 1 PWN Warszawa 1971.
[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.
[59] L. Waelbroeck, Quotient Banach spaces: the multilinear theory, in Spectral Theory (Warsaw 1977), Banach Center Publ. 8, PWN Warsaw 1982, 563–571.
[60] L. Waelbroeck, The Taylor spectrum and quotient Banach spaces. Spectral theory (Warsaw, 1977), 573-578, Banach Center Publ., 8, PWN, Warsaw, 1982.
[61] L. Waelbroeck, The category of quotient bornological spaces in “Aspects of Math and its Applications”, J. Barroso (ed.) 1986, pp. 873–894.
[62] L. Waelbroeck, Around the quotient bornological spaces. Dedicated to the memory of Professor Gottfried Köthe. Note Mat. 11 (1991), 315–329.
[63] L. Waelbroeck, Quotient Fréchet spaces, Rev. Roumaine Math. Pures et Appl. 34 (1999) 171–179
[64] L. Waelbroeck, Bornological quotients, with the collaboration of Guy Noël. Mémoire de la Classe des Sciences. Collection in-4o. 3e Série [Memoir ofthe Science Section. Collection in-4o. 3rd Series], VII, Académie Royale de Belgique. Classe des Sciences, Brussels, 2005.
[65] S.A. Wegner, The heart of the Banach spaces, J. Pure and Appl. Algebra 221 (2017) 2880–2909
[66] Ch. A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38. (1995) Cambridge Univ. Press.
[67] J. Wengenroth, Derived Functors in Functional Analysis, Lecture Notes in Math. 1810, Springer 2003.
[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), Lecture Notes in Math. 1573, pp. 34–35, Springer 1994