×

The theory of half derivators. (English) Zbl 1499.18042

Following pioneering works of Daniel Kan, Peter Gabriel and Michel Zisman, and Daniel Quillen, the subject of homotopy inticately involved category theory. By 1998, with the work of Alex Heller and Alexander Grothendieck, a homotopy theory was seen as a category \(\mathbb{D}\) parametrized by (or varying over) the 2-category \(\mathrm{Cat}\) of small categories. Under some conditions, involving completeness, cocompleteness and a strong generating role for the terminal category \(\mathbf{1}\), Grothendieck used the term derivator for such a \(\mathbb{D}\). It was pleasing to see that ideas of R. Street [J. Pure Appl. Algebra 21, 307–338 (1981; Zbl 0469.18007)] were being applied in this field. In particular, note that cocompleteness for a variable category involves a Beck-Chevalley-like condition on comma squares.
While the paper under review does not explicitly define half derivator, it seems to mean that either the completeness or cocompleteness is dropped. Keeping the bicategorical viewpoint to a minimum, the author shows that many of the results for derivators hold under the weaker conditions. The author concludes “by defining the maximal domain for a \(K\)-theory of derivators generalising Waldhausen \(K\)-theory”. (As a sign of a new generation, the author seems apologetic for referring to a paper published in typewritten form before LATEX.)

MSC:

18N40 Homotopical algebra, Quillen model categories, derivators
55U35 Abstract and axiomatic homotopy theory in algebraic topology

Citations:

Zbl 0469.18007
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Michael Artin, Alexander Grothendieck, and Jean-Louis Verdier. Theorie de topos et cohomologie \'etale des schemas I, II, III, Lecture Notes in Math., 269, 270, 305. Springer, 1972 and 1973, DOI 10.1007/BFb0081551(vol.269), zbl 0234.00007(vol.269), MR0354652(vol.269) DOI 10.1007/BFb0061319(vol.270), zbl 0237.00012vol.270), DOI 10.1007/BFb0070714(vol.305), zbl 0245.00002(vol.305) · doi:10.1007/BFb0081551(vol.269)
[2] Francis Borceux. Handbook of categorical algebra. 1. Basic category theory, Encyclopedia of Mathematics and its Applications, 50. Cambridge University Press, Cambridge, 1994, zbl 0803.18001, MR1291599 · Zbl 0803.18001
[3] Denis-Charles Cisinski. Les pr\'efaisceaux comme mod\`eles des types d’homotopie. Ast\'erisque (308):xxiv+390, 2006, zbl 1111.18008, MR2294028 · Zbl 1111.18008
[4] Denis-Charles Cisinski. Propri\'et\'es universelles et extensions de Kan d\'eriv\'ees. Theory Appl. Categ. 20(17):605-649, 2008, zbl 1188.18009, MR2534209 · Zbl 1188.18009
[5] Denis-Charles Cisinski. Cat\'egories d\'erivables. Bull. Soc. Math. France 138(3):317-393, 2010, DOI 10.24033/bsmf.2592, zbl 1203.18013, MR2729017 · Zbl 1203.18013 · doi:10.24033/bsmf.2592
[6] Ian Coley. The Stabilization and K-theory of Pointed Derivators. PhD thesis, University of California, Los Angeles, 2019. https://iancoley.org/pub/doctoral-thesis.pdf · Zbl 1423.18014
[7] Ian Coley. The K-theory of left pointed derivators. Preprint, 2020, arxiv 2009.09063 · Zbl 1423.18014
[8] Jens Franke. Uniqueness theorems for certain triangulated categories with an Adams spectral sequence, 1996. http://www.math.uiuc.edu/K-theory/0139/Adams.pdf
[9] Alexandre Grothendieck. Les d\'erivateurs, 1990, https://webusers.imj-prg.fr/ georges.maltsiniotis/groth/Derivateurs.html
[10] Moritz Groth. Derivators, pointed derivators and stable derivators. Algebr. Geom. Topol. 13(1):313-374, 2013, DOI 10.2140/agt.2013.13.313, zbl 1266.55009, MR3031644, arxiv 1112.3840 · Zbl 1266.55009 · doi:10.2140/agt.2013.13.313
[11] Moritz Groth. Characterizations of abstract stable homotopy theories. Preprint, 2016, unpublished, arxiv 1602.07632 · Zbl 1337.55024
[12] Moritz Groth. Revisiting the canonicity of canonical triangulations. Theory Appl. Categ. 33:350-389, 2018, zbl 1394.55014, MR3806332, arxiv 1602.04846 · Zbl 1394.55014
[13] Moritz Groth. The theory of derivators, 2019, http://www.math.uni-bonn.de/ mgroth/monos.htmpl · Zbl 1302.18001
[14] Alex Heller. Homotopy theories. Mem. Amer. Math. Soc. 71(383):vi+78, 1988, DOI 10.1090/memo/0383, zbl 0643.55015, MR0920963 · Zbl 0643.55015 · doi:10.1090/memo/0383
[15] Bernhard Keller. Appendice: Le d\'erivateur triangul\'e associ\'e \`a une cat\'egorie exacte. In Categories in algebra, geometry and mathematical physics, Contemp. Math., 431, 369-373. Amer. Math. Soc., Providence, RI, 2007, MR2342837
[16] G. M. Kelly and Ross Street. Review of the elements of \(2\)-categories. In Category Seminar (Proc. Sem., Sydney, 1972/1973), 75-103. Lecture Notes in Math., 420. Springer, Berlin, 1974, zbl 0334.18016, MR0357542 · Zbl 0334.18016
[17] Ioannis Lagkas-Nikolos. Levelwise modules over separable monads on stable derivators. J. Pure Appl. Algebra 222(7):1704-1726, 2018, DOI 10.1016/j.jpaa.2017.08.001, zbl 1420.18018, MR3763278, arxiv 1608.06340 · Zbl 1420.18018 · doi:10.1016/j.jpaa.2017.08.001
[18] Jacob Lurie. Higher topos theory, Annals of Mathematics Studies, 170. Princeton University Press, Princeton, NJ, 2009, DOI 10.1515/9781400830558, zbl 1175.18001, MR2522659, arxiv math/0608040 · Zbl 1175.18001 · doi:10.1515/9781400830558
[19] Saunders MacLane. Categories for the working mathematician. Graduate Texts in Mathematics, 5. Springer, New York-Berlin, 1971, zbl 0232.18001, MR0354798 · Zbl 0232.18001
[20] Georges Maltsiniotis. \(La K\)-th\'eorie d’un d\'erivateur triangul\'e. In Categories in algebra, geometry and mathematical physics, Contemp. Math., 431, 341-368. Amer. Math. Soc., Providence, RI, 2007, zbl 1136.18002, MR2342836 · Zbl 1136.18002
[21] Georges Maltsiniotis. Carr\'{e}s exacts homotopiques et d\'{e}rivateurs. Cah. Topol. G\'{e}om. Diff\'{e}r. Cat\'{e}g. 53(1):3-63, 2012, zbl 1262.18014, MR2951712, arxiv 1101.4144 · Zbl 1262.18014
[22] Saunders Mac Lane. Categories for the working mathematician, Graduate Texts in Mathematics, 5. Springer, New York, second edition, 1998, zbl 0906.18001, MR1712872 · Zbl 0906.18001
[23] Fernando Muro and George Raptis. \(K\)-theory of derivators revisited. Ann. K-Theory 2(2):303-340, 2017, DOI 10.2140/akt.2017.2.303, zbl 1364.19002, MR3590348, arxiv 1402.1871 · Zbl 1364.19002 · doi:10.2140/akt.2017.2.303
[24] Rev\^etements \'{e}tales et groupe fondamental ({SGA} 1), S\'{e}minaire de g\'{e}om\'{e}trie alg\'{e}brique du Bois Marie 1960-61. [Algebraic Geometry Seminar of Bois Marie 1960-61], directed by A. Grothendieck, With two papers by M. Raynaud. Updated and annotated reprint of the 1971 original [Lecture Notes in Math., 224. Springer, Berlin]. Documents Math\'{e}matiques (Paris), 3. Soci\'{e}t\'{e} Math\'{e}matique de France, Paris, 2003, zbl 1039.14001, MR2017446 · Zbl 1039.14001
[25] Michael A. Shulman. Set theory for category theory. Preprint, 2008, unpublished, arxiv 0810.1279
[26] Friedhelm Waldhausen. Algebraic \(K\)-theory of spaces. In Algebraic and geometric topology (New Brunswick, NJ, 1983), Lecture Notes in Math., 1126, 318-419. Springer, Berlin, 1985, zbl 0579.18006, MR0802796 · Zbl 0579.18006
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.