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.
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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.
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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.
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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
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DOI: https://doi.org/10.1007/s10485-017-9485-z