nLab intermediate model structure

Intermediate model structures

Intermediate model structures


Let MM be a category with two model structures (C 1,F 1,W)(C_1,F_1,W) and (C 2,F 2,W)(C_2,F_2,W) having the same class of weak equivalences. Moreover, let (L,R)(L,R) be a weak factorization system such that C 1LC 2C_1 \subseteq L \subseteq C_2.

The following theorem is essentially due to Jardine.


There is a (necessarily unique) intermediate model structure (L,F i,W)(L,F_i,W) with the same weak equivalences and LL as the cofibrations.


Define F iF_i to be the class of morphisms having the right lifting property with respect to LWL\cap W. Now we observe the following:

  • The assumption C 1LC 2C_1 \subseteq L \subseteq C_2 implies F 2WRF 2WF_2 \cap W \subseteq R \subseteq F_2\cap W. In particular, RWR\subseteq W.
  • We also have C 1WLWC 2WC_1\cap W \subseteq L\cap W \subseteq C_2\cap W, hence F 2F iF 1F_2 \subseteq F_i \subseteq F_1.
  • Finally, LWLL\cap W\subseteq L implies RF iR\subseteq F_i. Thus, RF iWR\subseteq F_i \cap W.

Now one of the weak factorization systems in the desired model structure is just (L,R)(L,R). The other will be (LW,F i)(L\cap W, F_i), for which it suffices to verify the existence of factorizations. Given ff, we first factor it as pip i with pF 2p\in F_2 (hence also pF ip\in F_i) and iC 2Wi\in C_2\cap W. Now factor ii as qjq j with jLj\in L and qRq\in R, hence also qF iWq\in F_i \cap W. Thus, by the 2-out-of-3 property, jWj\in W, so we have f=(pq)jf = (p q) j with jLWj\in L\cap W and pqF ip q\in F_i.

It remains only to show that F iWRF_i \cap W \subseteq R, and this follows by a standard retract argument. Given fF iWf\in F_i \cap W, factor it as pip i with pRp\in R and iLi\in L. Then pWp\in W, so by 2-out-of-3 iLWi\in L\cap W as well. Hence ff has the right lifting property with respect to ii, so it is a retract of pp, hence lies in RR.

We also have:


If in the above situation MM is a locally presentable category and either (C 1,F 1,W)(C_1,F_1,W) or (C 2,F 2,W)(C_2,F_2,W) is cofibrantly generated (hence combinatorial), and (L,R)(L,R) is cofibrantly generated, then (L,F i,W)(L,F_i,W) is also combinatorial.


The assumption about (C 1,F 1,W)(C_1,F_1,W) or (C 2,F 2,W)(C_2,F_2,W) ensures that Arr W(C)Arr_W(C) is an accessibly embedded accessible full subcategory of Arr(C)Arr(C). The rest of the hypotheses of Smith’s theorem about combinatorial model categories are automatically satisfied because we already know that (L,F i,W)(L,F_i,W) is a model category.


Model structures on diagram categories

Let CC be a small category and MM a combinatorial model category. Then M CM^C has a projective model structure (C proj,F proj,W)(C_{proj},F_{proj},W) with F projF_{proj} the objectwise fibrations, and an injective model structure (C inj,F inj,W)(C_{inj},F_{inj},W) with C injC_{inj} the objectwise cofibrations, and WW in both cases the objectwise weak equivalences.

Thus, from any weak factorization system (L,R)(L,R) on M CM^C in which (1) every LL-map is an objectwise cofibration and (2) every RR-map is an objectwise acyclic fibration, we obtain a new model structure (L,F i,W)(L,F_i,W) on M CM^C, and dually.

Specific examples include:

  • If SS is a set of cofibrations in M CM^C containing the generating projective cofibrations, then by the small object argument it generates a weak factorization system (L,R)(L,R) with C projLC injC_{proj} \subseteq L \subseteq C_{inj}. This is the original example.

  • If CC is a Reedy category (or a generalized Reedy category), then we have a weak factorization system consisting of Reedy cofibrations and Reedy trivial fibrations, giving rise to the Reedy model structure. The dual of the theorem, applied to the weak factorization system of Reedy trivial cofibrations and Reedy fibrations, produces the same Reedy model structure.

  • By the algebraic small object argument, we can enhance the two weak factorization systems of MM to algebraic weak factorization systems, and then simply apply them objectwise to obtain two weak factorization systems on M CM^C to which the theorem and its dual can be applied.

See also


Last revised on July 26, 2022 at 01:55:16. See the history of this page for a list of all contributions to it.