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

Sergey Arkhipov; Sebastian Ørsted

doi:10.1007/s10485-023-09747-8

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

^*Corresponding author for this work

Department of Mathematics

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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.

Original language	English
Article number	47
Journal	Applied Categorical Structures
Volume	31
Issue	6
ISSN	0927-2852
DOIs	https://doi.org/10.1007/s10485-023-09747-8
Publication status	Published - Nov 2023

Keywords

Bousfield–Kan formula
Category theory
Derived functors
Homotopy limit
Model categories
Simplicial sets

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10.1007/s10485-023-09747-8

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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.",

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N2 - 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.

AB - 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.

KW - Bousfield–Kan formula

KW - Category theory

KW - Derived functors

KW - Homotopy limit

KW - Model categories

KW - Simplicial sets

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

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Keywords

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