# Combinatorial Homotopy Categories

• Carles Casacuberta
• Jiří Rosický
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 309)

## Abstract

A model category is called combinatorial if it is cofibrantly generated and its underlying category is locally presentable. As shown in recent years, homotopy categories of combinatorial model categories share useful properties, such as being well generated and satisfying a very general form of Ohkawa’s theorem.

## Keywords

Combinatorial model category Cofibrantly generated Locally presentable Well generated Brown representability

## Notes

### Acknowledgements

This article has been written as a contribution to the proceedings of the memorial conference for Professor Tetsusuke Ohkawa held at the University of Nagoya in 2015. The content of Sect. 4 is based on previous joint work of the authors with Javier Gutiérrez published in [9]. We also appreciate useful discussions with George Raptis. The authors were supported by the Grant Agency of the Czech Republic under grant P201/12/G028, the Agency for Management of University and Research Grants of Catalonia with references 2014 SGR 114 and 2017 SGR 585, and the Spanish Ministry of Economy and Competitiveness under AEI/FEDER research grants MTM2013-42178-P and MTM2016-76453-C2-2-P, as well as grant MDM-2014-0445 awarded to the Barcelona Graduate School of Mathematics.

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