Model structures for coalgebras

Drummond-Cole, Gabriel C.; Hirsh, Joseph

doi:10.1090/proc/12823

Model structures for coalgebras
HTML articles powered by AMS MathViewer

by Gabriel C. Drummond-Cole and Joseph Hirsh PDF

Proc. Amer. Math. Soc. 144 (2016), 1467-1481 Request permission

Abstract:

Classically, there are two model category structures on coalgebras in the category of chain complexes over a field. In one, the weak equivalences are maps which induce an isomorphism on homology. In the other, the weak equivalences are maps which induce a weak equivalence of algebras under the cobar functor. We unify these two approaches, realizing them as the two extremes of a partially ordered set of model category structures on coalgebras over a cooperad satisfying mild conditions.

References

Marc Aubry and David Chataur, Cooperads and coalgebras as closed model categories, J. Pure Appl. Algebra 180 (2003), no. 1-2, 1–23. MR 1966520, DOI 10.1016/S0022-4049(02)00174-3
David Ayala and John Francis, Poincaré/Koszul duality, Preprint, http://arxiv.org/abs/1409.2478, 2014.
A. L. Agore, Limits of coalgebras, bialgebras and Hopf algebras, Proc. Amer. Math. Soc. 139 (2011), no. 3, 855–863. MR 2745638, DOI 10.1090/S0002-9939-2010-10542-7
Jiří Adámek and Jiří Rosický, Locally presentable and accessible categories, London Mathematical Society Lecture Note Series, vol. 189, Cambridge University Press, Cambridge, 1994. MR 1294136, DOI 10.1017/CBO9780511600579
Marzieh Bayeh, Kathryn Hess, Varvara Karpova, Magdalena Kędziorek, Emily Riehl, and Brooke Shipley, Left-induced model structures and diagram categories, Proceedings of the Women in Topology Workshop at Banff International Research Station, Banff, Alberta, Canada, (Maria Basterra, Kristine Bauer, André Leroy, Kathryn Hess and Brenda Johnson, eds.) Contemporary Mathematics, vol. 641, 2015, American Mathematical Society, Providence, RI.
C. Barwick and D.M̃. Kan, Relative categories: another model for the homotopy theory of homotopy theories, Indag. Math. (N.S.) 23 (2012).
C. Barwick and D. M. Kan, Relative categories: another model for the homotopy theory of homotopy theories, Indag. Math. (N.S.) 23 (2012), no. 1-2, 42–68. MR 2877401, DOI 10.1016/j.indag.2011.10.002
Denis-Charles Cisinski, Les préfaisceaux comme modèles des types d’homotopie, Astérisque 308 (2006), xxiv+390 (French, with English and French summaries). MR 2294028
Michael Ching and Emily Riehl, Coalgebraic models for combinatorial model categories, Homology Homotopy Appl. 16 (2014), no. 2, 171–184. MR 3257572, DOI 10.4310/HHA.2014.v16.n2.a9
Sjoerd E. Crans, Quillen closed model structures for sheaves, J. Pure Appl. Algebra 101 (1995), no. 1, 35–57. MR 1346427, DOI 10.1016/0022-4049(94)00033-F
John Francis and Dennis Gaitsgory, Chiral Koszul duality, Selecta Math. (N.S.) 18 (2012), no. 1, 27–87. MR 2891861, DOI 10.1007/s00029-011-0065-z
Benoit Fresse, Operadic cobar constructions, cylinder objects and homotopy morphisms of algebras over operads, Alpine perspectives on algebraic topology, Contemp. Math., vol. 504, Amer. Math. Soc., Providence, RI, 2009, pp. 125–188. MR 2581912, DOI 10.1090/conm/504/09879
Ezra Getzler and Paul Goerss, A model category structure for differential graded coalgebras, Preprint, http://www.math.northwestern.edu/~pgoerss/papers/model.ps, 1999.
Vladimir Hinich, Homological algebra of homotopy algebras, Comm. Algebra 25 (1997), no. 10, 3291–3323. MR 1465117, DOI 10.1080/00927879708826055
Vladimir Hinich, DG coalgebras as formal stacks, J. Pure Appl. Algebra 162 (2001), no. 2-3, 209–250. MR 1843805, DOI 10.1016/S0022-4049(00)00121-3
Dale Husemoller, John C. Moore, and James Stasheff, Differential homological algebra and homogeneous spaces, J. Pure Appl. Algebra 5 (1974), 113–185. MR 365571, DOI 10.1016/0022-4049(74)90045-0
Kathryn Hess and Brooke Shipley, The homotopy theory of coalgebras over a comonad, Proc. Lond. Math. Soc. (3) 108 (2014), no. 2, 484–516. MR 3166360, DOI 10.1112/plms/pdt038
Kenji Lefèvre-Hasegawa, Sur les A-infini catégories, Unpublished Ph.D. thesis, 2003.
Jacob Lurie, Higher algebra, Preprint, http://www.math.harvard.edu/~lurie/papers/higheralgebra.pdf, 2003.
Jacob Lurie, Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009. MR 2522659, DOI 10.1515/9781400830558
Jean-Louis Loday and Bruno Vallette, Algebraic operads, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 346, Springer, Heidelberg, 2012. MR 2954392, DOI 10.1007/978-3-642-30362-3
Michael Makkai and Jiří Rosický, Cellular categories, Preprint, http://arxiv.org/abs/1304.7572, 2013.
Mauro Porta, Model structure for cooperads and for coalgebras, mathoverflow question, http://mathoverflow.net/questions/148626, 2013.
Leonid Positselski, Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence, Mem. Amer. Math. Soc. 212 (2011), no. 996, vi+133. MR 2830562, DOI 10.1090/S0065-9266-2010-00631-8
Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295. MR 258031, DOI 10.2307/1970725
Justin R. Smith, Model-categories of coalgebras over operads, Theory Appl. Categ. 25 (2011), No. 8, 189–246. MR 2805750
Bruno Vallette, Homotopy theory of homotopy algebras, Preprint, http://arxiv.org/abs/1411.5533, 2014.