Abstract
We give an account of Bousfield localisation and colocalisation for one-dimensional model categories—ones enriched over the model category of 0-types. A distinguishing feature of our treatment is that it builds localisations and colocalisations using only the constructions of projective and injective transfer of model structures along right and left adjoint functors, and without any reference to Smith’s theorem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adámek, J., Rosický, J.: Locally Presentable and Accessible Categories, Volume 189 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1994)
Barthel, T., Riehl, E.: On the construction of functorial factorizations for model categories. Algebr. Geom. Topol. 13(2), 1089–1124 (2013)
Bayeh, M., Hess, K., Karpova, V., Kȩdziorek, M., Riehl, E., Shipley, B.: Left-induced model structures and diagram categories. In: Women in Topology: Collaborations in Homotopy Theory, Volume 641 of Contemporary Mathematics, pp. 49–81. American Mathematical Society (2015)
Beke, T.: Sheafifiable homotopy model categories. Math. Proc. Camb. Philos. Soc. 129(3), 447–475 (2000)
Berthelot, P., Grothendieck, A., Illusie, L.: Théorie des intersections et théorème de Riemann-Roch (SGA 6), volume 225 of Lecture Notes in Mathematics. Springer, New York (1971)
Bird, G.: Limits in 2-categories of locally-presented categories. Ph.D. thesis, University of Sydney (1984)
Blumberg, A.J., Riehl, E.: Homotopical resolutions associated to deformable adjunctions. Algebr. Geom. Topol. 14(5), 3021–3048 (2014)
Bourke, J., Garner, R.: Algebraic weak factorisation systems I: accessible AWFS. J. Pure Appl. Algebra 220(1), 108–147 (2016)
Bousfield, A.K.: Constructions of factorization systems in categories. J. Pure Appl. Algebra 9(2–3), 207–220 (1977)
Bousfield, A.K., Friedlander, E.M.: Homotopy theory of \(\Gamma \)-spaces, spectra, and bisimplicial sets. In: Geometric Applications of Homotopy Theory (Proceedings of Conference, Evanston, IL, 1977), II, Volume 658 of Lecture Notes in Mathematics, pp. 80–130. Springer, Berlin (1978)
Cassidy, C., Hébert, M., Kelly, G.M.: Reflective subcategories, localizations and factorization systems. J. Aust. Math. Soc. Ser. A 38(3), 287–329 (1985)
Ching, M., Riehl, E.: Coalgebraic models for combinatorial model categories. Homol. Homotopy Appl. 16(2), 171–184 (2014)
Cole, M.: Mixing model structures. Topol. Appl. 153(7), 1016–1032 (2006)
Dubuc, E.J.: Logical opens and real numbers in topoi. J. Pure Appl. Algebra 43(2), 129–143 (1986)
Dugger, D.: Combinatorial model categories have presentations. Adv. Math. 164(1), 177–201 (2001)
Freyd, P.J., Kelly, G.M.: Categories of continuous functors I. J. Pure Appl. Algebra 2(3), 169–191 (1972)
Gabriel, P., Ulmer, F.: Lokal präsentierbare Kategorien, Volume 221 of Lecture Notes in Mathematics. Springer, New York (1971)
Grothendieck, A., Raynaud, M.: Revêtements étales et groupe fondamental (SGA 1), Volume 224 of Lecture Notes in Mathematics. Springer, New York (1971)
Hirschhorn, P.S.: Model Categories and Their Localizations, Volume 99 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2003)
Johnstone, P.T.: Calibrated toposes. Bull. Belg. Math. Soc. Simon Stevin 19(5), 891–909 (2012)
Johnstone, P.T.: Sketches of an Elephant: A Topos Theory Compendium, Volume 44 of Oxford Logic Guides, vol. 2. Oxford University Press, Oxford (2002)
Kock, A.: Synthetic Differential Geometry, Volume 51 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1981)
Lawvere, F.W.: Functorial semantics of algebraic theories. Ph.D. thesis, Columbia University (1963). Republished as: Reprints in Theory and Applications of Categories 5 (2004)
Makkai, M., Paré, R.: Accessible Categories: The Foundations of Categorical Model Theory, Volume 104 of Contemporary Mathematics. American Mathematical Society, Providence (1989)
Makkai, M., Rosický, J.: Cellular categories. J. Pure Appl. Algebra 218(9), 1652–1664 (2014)
Nikolaus, T.: Algebraic models for higher categories. K. Ned. Akad. van Wet. Indag. Math. N. Ser. 21(1–2), 52–75 (2011)
Pultr, A., Tholen, W.: Free Quillen factorization systems. Georgian Math. J. 9(4), 807–820 (2002)
Riehl, E.: Algebraic model structures. N. Y. J. Math. 17, 173–231 (2011)
Ringel, C.M.: Diagonalisierungspaare. I. Math. Z. 117, 249–266 (1970)
Rosický, J., Tholen, W.: Factorization, fibration and torsion. J. Homotopy Relat. Struct. 2(2), 295–314 (2007)
Salch, A.: The Bousfield localizations and colocalizations of the discrete model structure. Topol. Appl. 219, 78–89 (2017)
Stanculescu, A.E.: Note on a theorem of Bousfield and Friedlander. Topol. Appl. 155(13), 1434–1438 (2008)
Sweedler, M.E.: Hopf Algebras. Mathematics Lecture Note Series. W. A. Benjamin Inc, New York (1969)
Univalent Foundations Program: Homotopy Type Theory: Univalent Foundations of Mathematics. http://homotopytypetheory.org/book, Institute for Advanced Study (2013)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M. M. Clementino.
The work described here was carried out during a visit by the first author to Sydney supported by Macquarie University Research Centre funding; both authors express their gratitude for this support. The second author also acknowledges, with equal gratitude, the support of Australian Research Council Grants DP160101519 and FT160100393.
Rights and permissions
About this article
Cite this article
Balchin, S., Garner, R. Bousfield Localisation and Colocalisation of One-Dimensional Model Structures. Appl Categor Struct 27, 1–21 (2019). https://doi.org/10.1007/s10485-018-9537-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-018-9537-z