Abstract
We prove and explain several classical formulae for homotopy (co)limits in general (combinatorial) model categories which are not necessarily simplicially enriched. Importantly, we prove versions of the Bousfield–Kan formula and the fat totalization formula in this complete generality. We finish with a proof that homotopy-final functors preserve homotopy limits, again in complete generality.
Similar content being viewed by others
References
Arkhipov, S., S. Ørsted: Homotopy limits in the category of dg-categories in terms of \( \rm A _{\infty } \)-comodules. Eur. J. Math. (to appear) (2018). ar**v: 1812.03583
Block, J., Holstein, J.V., Wei, Z.: Explicit homotopy limits of dg-categories and twisted complexes. Homol. Homotopy Appl. 19(2), 343–371 (2017)
Dugger, D.: A primer on homotopy colimits. http://pages.uoregon.edu/ddugger/hocolim.pdf (2008)
Gambino, N.: Weighted limits in simplicial homotopy theory. J. Pure Appl. Algebra 214(7), 1193–1199 (2010)
Hirschhorn, P.S.: Model Categories and Their Localizations, Mathematical Surveys and Monographs, vol. 99. American Mathematical Society, Philadelphia (2003)
Hovey, M.: Model Categories, Mathematical Surveys and Monographs, vol. 63. American Mathematical Society, Philadelphia (1999)
Lurie, J.: Higher Topos Theory. Annals of Mathematics Studies, vol. 170. Princeton University Press, Princeton (2009)
Riehl, E.: Categorical Homotopy Theory. Cambridge University Press, Cambridge (2014)
Acknowledgements
We would like to thank Edouard Balzin, Marcel Bökstedt, and Stefan Schwede for many fruitful discussions and for reading through a draft of this paper. Special thanks to Henning Haahr Andersen for many years of great discussions, help, and advice, and for making our cooperation possible in the first place. This paper was written mostly while the authors were visiting the Max Planck Institute for Mathematics in Bonn, Germany. We would like to express our gratitude to the institute for inviting us and for providing us with an excellent and stimulating working environment.
Funding
No funding was obtained for this study.
Author information
Authors and Affiliations
Contributions
S.Ø. and S.A. carried out the work and prepared the manuscript together.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Communicated by Vladimir Hinich.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Arkhipov, S., Ørsted, S. Homotopy (Co)limits via Homotopy (Co)ends in General Combinatorial Model Categories. Appl Categor Struct 31, 47 (2023). https://doi.org/10.1007/s10485-023-09747-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10485-023-09747-8