Algebraic & Geometric Topology

On the construction of functorial factorizations for model categories

Tobias Barthel and Emily Riehl

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We present general techniques for constructing functorial factorizations appropriate for model structures that are not known to be cofibrantly generated. Our methods use “algebraic” characterizations of fibrations to produce factorizations that have the desired lifting properties in a completely categorical fashion. We illustrate these methods in the case of categories enriched, tensored and cotensored in spaces, proving the existence of Hurewicz-type model structures, thereby correcting an error in earlier attempts by others. Examples include the categories of (based) spaces, (based) G–spaces and diagram spectra among others.

Article information

Algebr. Geom. Topol., Volume 13, Number 2 (2013), 1089-1124.

Received: 10 May 2012
Revised: 29 November 2012
Accepted: 2 December 2012
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55U35: Abstract and axiomatic homotopy theory 55U40: Topological categories, foundations of homotopy theory
Secondary: 18A32: Factorization of morphisms, substructures, quotient structures, congruences, amalgams 18G55: Homotopical algebra

functorial factorizations Hurewicz fibrations algebraic weak factorization systems


Barthel, Tobias; Riehl, Emily. On the construction of functorial factorizations for model categories. Algebr. Geom. Topol. 13 (2013), no. 2, 1089--1124. doi:10.2140/agt.2013.13.1089.

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