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Homotopy (Co)limits via Homotopy (Co)ends in General Combinatorial Model Categories

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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.

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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.


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S.Ø. and S.A. carried out the work and prepared the manuscript together.

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Correspondence to Sebastian Ørsted.

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Communicated by Vladimir Hinich.

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Arkhipov, S., Ørsted, S. Homotopy (Co)limits via Homotopy (Co)ends in General Combinatorial Model Categories. Appl Categor Struct 31, 47 (2023).

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