Abstract
Consider a Quillen adjunction of two variables between combinatorial model categories from \(\mathcal {C}\times \mathcal {D}\) to \(\mathcal {E}\), a set \(\mathcal {S}\) of morphisms in \(\mathcal {C}\) and a set \(\mathcal {K}\) of objects in \(\mathcal {C}\). We prove that there is a localised model structure \(L_{\mathcal {S}}\mathcal {E}\) on \(\mathcal {E}\), where the local objects are the \(\mathcal {S}\)-local objects in \(\mathcal {E}\) described via the right adjoint. Dually, we show that there is a colocalised model structure \(C_{\mathcal {K}}\mathcal {E}\) on \(\mathcal {E}\), where the colocal equivalences are the \(\mathcal {K}\)-colocal equivalences in \(\mathcal {E}\) described via the right adjoint. These localised and colocalised model structures generalise left and right Bousfield localisations of simplicial model categories, Barnes and Roitzheim’s familiar model structures, and Barwick’s enriched left and right Bousfield localisations.
Similar content being viewed by others
References
Adámek, J., Rosický, J.: Locally Presentable and Accessible Categories, London Math. Soc. Lecture Note Ser., vol. 189. Cambridge University Press, Cambridge (1994)
Barnes, D., Roitzheim, C.: Local Framings. New York J. Math. 17, 513–552 (2011)
Barnes, D., Roitzheim, C.: Stable left and right Bousfield localisations. Glasg. Math. J. 56(1), 13–42 (2014)
Barnes, D., Roitzheim, C.: Homological localisation of model categories. Appl. Categ. Struct. 23(3), 487–505 (2015)
Barwick, C.: On left and right model categories and left and right Bousfield localizations. Homology, Homotopy Appl. 12(2), 245–320 (2010)
Beke, T.: Sheafifiable homotopy model categories. Math. Proc. Camb. Phil. Soc. 129(3), 447–475 (2000)
Dundas, B.I., Röndings, O., Østvær, P.A.: Motivic functors. Doc. Math. 8, 489–525 (2003)
Dugger, D.: Combinatorial model categories have presentations. Adv. Math. 164, 177–201 (2001)
Dwyer, W.G., Spalinski, J.: Homotopy theories and model categories, pp. 73–126. Handbook of algebraic topology, North-Holland, Amsterdam (1995)
Gabriel, P., Ulmer, F.: Lokal präsentierbare Kategorien, Lecture Notes in Math., vol. 221. Springer-Verlag, Berlin, Heidelberg (1971)
Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory, Progress in Math., vol. 174. Basel, Birkhäuser (1999)
Gutiérrez, J.J., Roitzheim, C.: Towers and fibered products of model structures. Mediterr. J. Math. 13(6), 3863–3886 (2016)
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)
Hirschhorn, P.S.: Model Categories and Their Localizations, Math. Surveys and Monographs, vol. 99, Amer. Math. Soc., Providence (2003)
Hovey, M.: Model Categories, Math. Surveys and Monographs, vol. 63. Amer. Math. Soc., Providence (1999)
Hovey, M., Shipley, B., Smith, J.H.: Symmetric spectra. J. Amer. Math. Soc. 13(1), 149–208 (2000)
Jardine, J.F.: Motivic symmetric spectra. Doc. Math. 5, 445–553 (2000)
Joyal, A., Tierney, M.: Quasi-categories vs Segal spaces. In: Categories in algebra, geometry and mathematical physics, 277–326, Contemp. Math., 431, Amer. Math. Soc., Providence, RI (2007)
Lurie, J.: Higher Topos Theory, Annals of Math. Studies, vol. 170. Princeton University Press, Princeton (2009)
Makkai, M., Paré, R.: Accessible Categories: The Foundations of Categorical Model Theory, Contemp. Math., vol. 104, Amer. Math. Soc., Providence (1989)
Morel, F., Voevodsky, V.: 𝔸1-homotopy theory of schemes. Publ. Math. IHÉS 90(1), 45–143 (1999)
Schwede, S., Shipley, B.: Algebras and modules in monoidal model categories. Proc. Lond. Math. Soc. 80(3), 491–511 (2000)
Acknowledgments
The first author would like to thank Dimitri Ara for many useful conversations on some of the topics of this paper. The second author would like to thank David Barnes for motivating discussions and the Radboud Universiteit Nijmegen for their hospitality.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author was supported by the NWO (SPI 61-638) and the MEC-FEDER grants MTM2010-15831 and MTM2013-42178-P. Both authors received support from the LMS Scheme 4 grant no. 41360.
Rights and permissions
About this article
Cite this article
Gutiérrez, J.J., Roitzheim, C. Bousfield Localisations along Quillen Bifunctors. Appl Categor Struct 25, 1113–1136 (2017). https://doi.org/10.1007/s10485-017-9485-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-017-9485-z