Arrow categories of monoidal model categories
DOI:
https://doi.org/10.7146/math.scand.a-114968Abstract
We prove that the arrow category of a monoidal model category, equipped with the pushout product monoidal structure and the projective model structure, is a monoidal model category. This answers a question posed by Mark Hovey, in the course of his work on Smith ideals. As a corollary, we prove that the projective model structure in cubical homotopy theory is a monoidal model structure. As illustrations we include numerous examples of non-cofibrantly generated monoidal model categories, including chain complexes, small categories, pro-categories, and topological spaces.
References
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Schwede, S. and Shipley, B. E., Algebras and modules in monoidal model categories, Proc. London Math. Soc. (3) 80 (2000), no. 2, 491–511. https://doi.org/10.1112/S002461150001220X
Strøm, A., The homotopy category is a homotopy category, Arch. Math. (Basel) 23 (1972), 435–441. https://doi.org/10.1007/BF01304912
White, D., Model structures on commutative monoids in general model categories, J. Pure Appl. Algebra 221 (2017), no. 12, 3124–3168. https://doi.org/10.1016/j.jpaa.2017.03.001
White, D. and Yau, D., Smith ideals of operadic algebras in monoidal model categories, preprint arXiv:1703.05377 [math.AT], 2017.
White, D. and Yau, D., Bousfield localization and algebras over colored operads, Appl. Categ. Structures 26 (2018), no. 1, 153–203. https://doi.org/10.1007/s10485-017-9489-8
Awodey, S., A cubical model of homotopy type theory, Ann. Pure Appl. Logic 169 (2018), no. 12, 1270–1294. https://doi.org/10.1016/j.apal.2018.08.002
Barthel, T., May, J. P., and Riehl, E., Six model structures for DG-modules over DGAs: model category theory in homological action, New York J. Math. 20 (2014), 1077–1159.
Biedermann, G., Chorny, B., and Röndigs, O., Calculus of functors and model categories, Adv. Math. 214 (2007), no. 1, 92–115. https://doi.org/10.1016/j.aim.2006.10.009
Bourke, J. and Garner, R., Algebraic weak factorisation systems I: Accessible AWFS, J. Pure Appl. Algebra 220 (2016), no. 1, 108–147. https://doi.org/10.1016/j.jpaa.2015.06.002
Brown, R., Higgins, P. J., and Sivera, R., Nonabelian algebraic topology, EMS Tracts in Mathematics, vol. 15, European Mathematical Society (EMS), Zürich, 2011. https://doi.org/10.4171/083
Ching, M. and Harper, J. E., Higher homotopy excision and Blakers-Massey theorems for structured ring spectra, Adv. Math. 298 (2016), 654–692. https://doi.org/10.1016/j.aim.2016.04.025
Chorny, B., The model category of maps of spaces is not cofibrantly generated, Proc. Amer. Math. Soc. 131 (2003), no. 7, 2255–2259. https://doi.org/10.1090/S0002-9939-03-06901-6
Chorny, B. and Rosický, J., Class-combinatorial model categories, Homology Homotopy Appl. 14 (2012), no. 1, 263–280. https://doi.org/10.4310/HHA.2012.v14.n1.a13
Christensen, J. D. and Hovey, M., Quillen model structures for relative homological algebra, Math. Proc. Cambridge Philos. Soc. 133 (2002), no. 2, 261–293. https://doi.org/10.1017/S0305004102006126
Cisinski, D.-C., Univalent universes for elegant models of homotopy types, preprint arXiv:1406.0058 [math.AT], 2014.
Cohen, C., Coquand, T., Huber, S., and Mörtberg, A., Cubical type theory: a constructive interpretation of the univalence axiom, in “21st International Conference on Types for Proofs and Programs”, LIPIcs. Leibniz Int. Proc. Inform., vol. 69, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2018, Art. No. 5, 34 pp.
Dugger, D., Combinatorial model categories have presentations, Adv. Math. 164 (2001), no. 1, 177–201. https://doi.org/10.1006/aima.2001.2015
Fausk, H. and Isaksen, D. C., t-model structures, Homology Homotopy Appl. 9 (2007), no. 1, 399–438. https://doi.org/10.4310/HHA.2007.v9.n1.a16
Hirschhorn, P. S., Model categories and their localizations, Mathematical Surveys and Monographs, vol. 99, American Mathematical Society, Providence, RI, 2003.
Hovey, M., Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999.
Hovey, M., Smith ideals of structured ring spectra, preprint arXiv:1401.2850 [math.AT], 2014.
Isaacson, S. B., Cubical homotopy theory and monoidal model categories, Ph.D. thesis, Harvard University, 2009, www-home.math.uwo.ca/∼sisaacso/PDFs/diss.pdf.
Isaksen, D. C., A model structure on the category of pro-simplicial sets, Trans. Amer. Math. Soc. 353 (2001), no. 7, 2805–2841. https://doi.org/10.1090/S0002-9947-01-02722-2
Jardine, J. F., Cubical homotopy theory: a beginning, Newton Institute preprint ni02030 https://www.newton.ac.uk/files/preprints/ni02030.pdf, 2002.
Lack, S., Homotopy-theoretic aspects of $2$-monads, J. Homotopy Relat. Struct. 2 (2007), no. 2, 229–260.
Lucas, M., Cubical categories for homotopy and rewriting, Ph.D. thesis, l'Université Sorbonne Paris Cité, 2017, https://hal.archives-ouvertes.fr/tel-01668359.
Lurie, J., Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009. https://doi.org/10.1515/9781400830558
Lurie, J., Higher algebra, http://www.math.harvard.edu/∼lurie/, 2017.
May, J. P., A concise course in algebraic topology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1999.
Meadows, N. J., The local Joyal model structure, Theory Appl. Categ. 31 (2016), paper No. 24, 690–711.
Munson, B. A. and Volić, I., Cubical homotopy theory, New Mathematical Monographs, vol. 25, Cambridge University Press, Cambridge, 2015. https://doi.org/10.1017/CBO9781139343329
Østvær, P. A., Homotopy theory of $C^\ast $-algebras, Frontiers in Mathematics, Birkhäuser/Springer Basel AG, Basel, 2010. https://doi.org/10.1007/978-3-0346-0565-6
Pavlov, D. and Scholbach, J., Homotopy theory of symmetric powers, Homology Homotopy Appl. 20 (2018), no. 1, 359–397. https://doi.org/10.4310/HHA.2018.v20.n1.a20
Raptis, G., Homotopy theory of posets, Homology Homotopy Appl. 12 (2010), no. 2, 211–230.
Schwede, S. and Shipley, B. E., Algebras and modules in monoidal model categories, Proc. London Math. Soc. (3) 80 (2000), no. 2, 491–511. https://doi.org/10.1112/S002461150001220X
Strøm, A., The homotopy category is a homotopy category, Arch. Math. (Basel) 23 (1972), 435–441. https://doi.org/10.1007/BF01304912
White, D., Model structures on commutative monoids in general model categories, J. Pure Appl. Algebra 221 (2017), no. 12, 3124–3168. https://doi.org/10.1016/j.jpaa.2017.03.001
White, D. and Yau, D., Smith ideals of operadic algebras in monoidal model categories, preprint arXiv:1703.05377 [math.AT], 2017.
White, D. and Yau, D., Bousfield localization and algebras over colored operads, Appl. Categ. Structures 26 (2018), no. 1, 153–203. https://doi.org/10.1007/s10485-017-9489-8
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Published
2019-10-19
How to Cite
White, D., & Yau, D. (2019). Arrow categories of monoidal model categories. MATHEMATICA SCANDINAVICA, 125(2), 185–198. https://doi.org/10.7146/math.scand.a-114968
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