# nLab mixed model structure

## Definition

Let $M$ be a category with two closed model structures $(C_q,W_q,F_q)$ and $(C_h,W_h,F_h)$, and assume that $F_h\subseteq F_q$ and $W_h \subseteq W_q$.

###### Theorem

There is a (necessarily unique closed) mixed model structure $(C_m,W_q,F_h)$ on $M$ in which the fibrations are the $h$-fibrations, but the weak equivalences are the $q$-equivalences.

## Properties

###### Proposition

Suppose $i$ and $j$ are m-cofibrations in the commutative diagram

1. If $f$ 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 $f$ 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)

###### Theorem

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

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

of a q-cofibration $j$ and an h-equivalence $w$. In particular, an object $X$ is m-cofibrant iff it is h-equivalent to a q-cofibrant object.

(May–Ponto 2012, theorem 17.3.5)

## Examples

• From the Quillen and Strøm model structures on topological spaces, we obtain a mixed model structure in which the weak equivalences are the weak homotopy equivalences, the fibrations are the Hurewicz fibrations, and the cofibrant objects are those of the homotopy type of a CW complex. (This example influences the notation above; the $q$-model structure is the Quillen one, the $h$-model structure is the “Hurewicz” (or “homotopy”) one.)

• Similarly, we can mix the $q$- and $h$- model structures on chain complexes.

• Let $T$ be an accessible strict 2-monad on a locally finitely presentable strict 2-category $K$. Then by a theorem of Lack, the category $T Alg_s$ of strict $T$-algebras admits a transferred model structure from the 2-trivial model structure on $K$, where the weak equivalences and fibrations are the morphisms which become internal equivalences and internal isofibrations in $K$.

On the other hand, $T Alg_s$ has its own 2-trivial model structure. Since the forgetful functor $T Alg_s \to K$ preserves equivalences and isofibrations (the latter since it has a strict left 2-adjoint), we can mix these two model structures to obtain one whose weak equivalences are the equivalences in $K$ (which are also the equivalences in the category $T Alg$ of $T$-algebras and pseudo morphisms), but whose fibrations are the isofibrations in $T Alg_s$.