Homology, Homotopy and Applications

Volume 24 (2022)

Number 1

Algebraically cofibrant and fibrant objects revisited

Pages: 271 – 298

DOI: https://dx.doi.org/10.4310/HHA.2022.v24.n1.a14


John Bourke (Department of Mathematics and Statistics, Masaryk University, Brno, Czech Republic)

Simon Henry (Department of Mathematics and Statistics, University of Ottawa, STEM complex, Ottawa, Ontario, Canada)


We extend all known results about transferred model structures on algebraically cofibrant and fibrant objects by working with weak model categories. We show that for an accessible weak model category there are always Quillen equivalent transferred weak model structures on both the categories of algebraically cofibrant and algebraically fibrant objects. Under additional assumptions, these transferred weak model structures are shown to be left, right or Quillen model structures. By combining both constructions, we show that each combinatorial weak model category is connected, via a chain of Quillen equivalences, to a combinatorial Quillen model category in which all objects are fibrant.

2010 Mathematics Subject Classification

18C35, 18G55, 55U35

The first-named author acknowledges the support of the Grant Agency of the Czech Republic under the grant 19-00902S.

Received 14 May 2020

Received revised 23 May 2021

Accepted 1 June 2021

Published 13 April 2022