The DG-Category of Secondary Cohomology Operations
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The DG-Category of Secondary Cohomology Operations Hans-Joachim Baues1 · Martin Frankland2 Received: 21 March 2019 / Accepted: 9 June 2020 © Springer Nature B.V. 2020
Abstract 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. Keywords Track category · DG-category · Pseudo-functor · Strictification · Secondary cohomology operation · Secondary Steenrod algebra Mathematics Subject Classification 18D05 · 55S20
1 Introduction Cohomology operations are important tools in algebraic topology. The Steenrod algebra (of primary stable mod p cohomology operations) was determined as a Hopf algebra in celebrated work of Milnor [19]. The structure of secondary cohomology operations was determined as a “secondary Hopf algebra” in [3], and via different methods in [20]. Unlike for primary operations, composition of secondary operations is not bilinear, but bilinear up to homotopy. Part of the work in [3] was to strictify the structure of secondary operations, i.e., replace it
The second author thanks the Max-Planck-Institut für Mathematik Bonn for its generous hospitality. The second author was partially funded by a grant of the Deutsche Forschungsgemeinschaft SPP 1786: Homotopy Theory and Algebraic Geometry, as well as the Natural Sciences and Engineering Research Council of Canada (NSERC), Discovery Grant RGPIN-2019-06082. Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG), subvention Découverte RGPIN-2019-06082.
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Martin Frankland [email protected] Hans-Joachim Baues [email protected]
1
Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
2
University of Regina, 3737 Wascana Parkway, Regina, Saskatchewan S4S 0A2, Canada
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H.-J. Baues, M. Frankland
with a weakly equivalent differential bigraded algebra, in which composition is bilinear. The purpose of this paper is to revisit this strictification step, simplify it, and generalize it. Here is the motivating example in more detail. For a fixed prime number p, mod p cohomology operations correspond to maps between finite products of Eilenberg–MacLane spaces K (F p , n), the representing objects. Stable operations correspond to maps between finite products of Eilenberg–MacLane spectra Σ n H F p . Primary operations are encoded by homotopy classes of such maps. More precisely, the Steenrod algebra A is given by homotopy classes of maps A n = [H F p , Σ n H F p ].
For higher order cohomology operations, one needs more than homotopy classes. One way to encode higher order operations is the topolo
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