×

The DG-category of secondary cohomology operations. (English) Zbl 1460.18003

Summary: We study track categories (i.e., groupoid-enriched categories) endowed with additive structure similar to that of a 1-truncated DG-category, except that composition is not assumed right linear. We show that if such a track category is right linear up to suitably coherent correction tracks, then it is weakly equivalent to a 1-truncated DG-category. This generalizes work of the first author on the strictification of secondary cohomology operations. As an application, we show that the secondary integral Steenrod algebra is strictifiable.

MSC:

18D20 Enriched categories (over closed or monoidal categories)
18N10 2-categories, bicategories, double categories
55S20 Secondary and higher cohomology operations in algebraic topology
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Baues, H.-J., Jibladze, M., Pirashvili, T.: Strengthening track theories, (2003) Preprint, available at arXiv:math/0307185
[2] Baues, H.-J., Pirashvili, T.: Shukla cohomology and additive track theories, (2004) Preprint, available at arXiv:math/0401158
[3] Baues, H-J, The Algebra of Secondary Cohomology Operations, Progress in Mathematics (2006), Basel: Birkhäuser Verlag, Basel · Zbl 1091.55001
[4] Baues, H-J; Pirashvili, T., Comparison of Mac Lane, Shukla and Hochschild cohomologies, J. Reine Angew. Math., 598, 25-69 (2006) · Zbl 1116.18009
[5] Baues, H-J; Muro, F., The homotopy category of pseudofunctors and translation cohomology, J. Pure Appl. Algebra, 211, 3, 821-850 (2007) · Zbl 1124.18004 · doi:10.1016/j.jpaa.2007.04.008
[6] Baues, H-J; Jibladze, M.; Pirashvili, T., Third Mac Lane cohomology, Math. Proc. Cambridge Philos. Soc., 144, 2, 337-367 (2008) · Zbl 1145.18007 · doi:10.1017/S030500410700076X
[7] Baues, H.-J., Frankland, M.: Eilenberg-MacLane mapping algebras and higher distributivity up to homotopy. New York J. Math. 23, (2017) · Zbl 1403.55010
[8] Borceux, F., Handbook of Categorical Algebra 1: Basic Category Theory, Encyclopedia of Mathematics and its Applications, 50 (1994), Cambridge: Cambridge University Press, Cambridge · Zbl 0803.18001
[9] Borceux, F., Handbook of Categorical Algebra 2: Categories and Structures, Encyclopedia of Mathematics and its Applications, 51 (1994), Cambridge: Cambridge University Press, Cambridge · Zbl 0843.18001
[10] Bourn, D., Another denormalization theorem for abelian chain complexes, J. Pure Appl. Algebra, 66, 3, 229-249 (1990) · Zbl 0716.18003 · doi:10.1016/0022-4049(90)90029-H
[11] Bourn, D.: Moore normalization and Dold-Kan theorem for semi-abelian categories, Categories in algebra, geometry and mathematical physics, Contemp. Math., 431, Amer. Math. Soc., Providence, RI, 105-124, (2007), 10.1090/conm/431/08268 · Zbl 1143.18013
[12] Dwyer, WG; Kan, DM, Simplicial localizations of categories, J. Pure Appl. Algebra, 17, 3, 267-284 (1980) · Zbl 0485.18012 · doi:10.1016/0022-4049(80)90049-3
[13] Elmendorf, AD; Mandell, MA, Permutative categories, multicategories and algebraic \(K\)-theory, Algebr. Geom. Topol., 9, 4, 2391-2441 (2009) · Zbl 1205.19003 · doi:10.2140/agt.2009.9.2391
[14] Gaudens, G., The \(\Gamma \)-structure of an additive track category, J. Homotopy Relat. Struct., 5, 1, 63-95 (2010) · Zbl 1278.18030
[15] Gurski, N., The monoidal structure of strictification, Theory Appl. Categ., 28, 1, 1-23 (2013) · Zbl 1273.18012
[16] Lack, S.: A Quillen model structure for 2-categories. \(K\)-Theory 26(2), 171-205 (2002). 10.1023/A:1020305604826 · Zbl 1017.18005
[17] Lack, S., Codescent objects and coherence, note=Special volume celebrating the 70th birthday of Professor Max Kelly, J. Pure Appl. Algebra, 175, 1-3, 223-241 (2002) · Zbl 1142.18301 · doi:10.1016/S0022-4049(02)00136-6
[18] Lack, S.: A Quillen model structure for bicategories. \(K\)-Theory 33(3), 185-197 (2004). 10.1007/s10977-004-6757-9 · Zbl 1069.18008
[19] Milnor, J., The Steenrod algebra and its dual, Ann. Math., 2, 67, 150-171 (1958) · Zbl 0080.38003 · doi:10.2307/1969932
[20] Nassau, C., On the secondary Steenrod algebra, New York J. Math., 18, 679-705 (2012) · Zbl 1254.55012
[21] Power, AJ, A general coherence result, J. Pure Appl. Algebra, 57, 2, 165-173 (1989) · Zbl 0668.18010 · doi:10.1016/0022-4049(89)90113-8
[22] Schwede, S.; Shipley, B., Equivalences of monoidal model categories, Algebr. Geom. Topol., 3, 287-334 (2003) · Zbl 1028.55013 · doi:10.2140/agt.2003.3.287
[23] Shulman, Michael A., Not every pseudoalgebra is equivalent to a strict one, Adv. Math., 229, 3, 2024-2041 (2012) · Zbl 1242.18010 · doi:10.1016/j.aim.2011.01.010
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.