nLab mixed model structure


Let MM be a category with two closed model structures (C q,W q,F q)(C_q,W_q,F_q) and (C h,W h,F h)(C_h,W_h,F_h), and assume that F hF qF_h\subseteq F_q and W hW qW_h \subseteq W_q.


There is a (necessarily unique closed) mixed model structure (C m,W q,F h)(C_m,W_q,F_h) on MM in which the fibrations are the hh-fibrations, but the weak equivalences are the qq-equivalences.



Suppose ii and jj are m-cofibrations in the commutative diagram

  1. If ff is a q-equivalence, then it is an h-equivalence. In particular, a q-equivalence between m-cofibrant objects is an h-equivalence.

  2. If ff is an h-cofibration, then it is an m-cofibration. In particular, an h-cofibration between m-cofibrant objects is an m-cofibration.

(May–Ponto 2012, proposition 17.3.4)


A map i:AXi \colon A \to X is an m-cofibration iff it is an h-cofibration which is a composition

AjKwX A \stackrel{j}{\to} K \stackrel{w}{\to} X

of a q-cofibration jj and an h-equivalence ww. In particular, an object XX is m-cofibrant iff it is h-equivalent to a q-cofibrant object.

(May–Ponto 2012, theorem 17.3.5)


See also


The original paper is

  • Michael Cole, Mixing model structures, Top. Appl. 153 (2006) 1016–1032

There is also an exposition in

Last revised on May 30, 2022 at 13:20:46. See the history of this page for a list of all contributions to it.