# Extending model structure from a full subcategory

Say I have a category $\mathcal{C}$ and a full subcategory $\mathcal{D}$, and suppose I have a model category structure on $\mathcal{D}$. Under what conditions is it possible to extend the model structure to the whole of $\mathcal{C}$? And when/if this can be done, is it possible to characterize the fibrations, cofibrations and, most importantly, the weak equivalences?

Remark 1: For the first part of the question, I know that if I have an adjunction between $\mathcal{C}$ and $\mathcal{D}$ (where the left adjoint is most probably the inclusion, i.e. a forgetful functor of some kind), then often it is possible to transfer the model structure along the adjunction. Are there other ways to do this?

Remark 2: You are allowed to use any of the various sets of axioms of model categories. If you really want, you can suppose that the $\mathcal{D}$ is a simplicial model category (but I'd rather not, as it seem a bit restrictive).

## 1 Answer

There is a sense in which you question becomes trivial, but I bet that's not what you want :-)

There is another way in which you can use the transport theorem, but there are a lot of things to check; let $i : {\cal D} \to {\cal C}$ be your functor; it is necessary that $\cal C$ is cocomplete to become a model category. So there is an adjunction $L\dashv N$, $$\text{Lan}_y i \dashv \hom(i,1) : [{\cal D}^\text{op}, {\bf sSets}] \rightleftarrows \cal C$$ between the Yoneda extension of $i$ and the functor $N = \lambda d.\lambda c.\hom(i(d),c)$.

$[{\cal D}^\text{op}, {\bf sSets}]$ has the projective model structure that you can (left) Bousfield-localize to take into account that $\cal D$ has one too. This results in adding weak equivalences (you want each $\hom(-,d)\to \hom(-,d')$ to become a weak equivalence when $d\to d'$ is) and gives you a model category ${\bf L}_y\cal D = {\cal D}^\odot$.

Now let's establish a notation and a few assumptions:

1. Let's assume that $\cal D$ is cofibrantly generated by $I$ (generating cofibrations) and $J$ (generating trivial cofibrations). $\cal D^\odot$ is cofibrantly generated by $I^\odot, J^\odot$.

2. Let's call $Wk_{\cal C}, Fib_{\cal C}$ the classes of maps $N^\leftarrow(Wk_\odot), N^\leftarrow(Fib_\odot)$ respectively

Now, the Crans transfer machinery says that there is a model structure $(Wk_{\cal C}, Fib_{\cal C}, Cof_{\cal C} = \textsf{llp}(Wk_{\cal C}\cap Fib_{\cal C}))$ on $\cal C$ if

1. $\hom(i,1)$ commutes with filtered colimits (this boils down to assumptions on $i$, which "behaves like" the inclusion of a subcategory of compact objects) and
2. the closure under sequential colimits and pushouts of $\{L(j) \mid j\in J^\odot\}$ is a subset of $Wk_{\cal C}$.

If all this works, I bet I'm only rewriting in an ignorant way some results in Rezk's Model topoi.

Hope it helps!

• Thanks! It will take me a while to absorb all of this. I'll try to see if it works in the concrete example I'm considering. – Daniel Robert-Nicoud Oct 18 '16 at 18:17
• I've been told that there is a flaw I'm not able to avoid: if $\cal D$ is model, it is cocomplete, and hence small iff it is a poset. This is a size-issue preventing $[{\cal D}^\text{op}, {\bf sSets}]$ from having the projective model structure. If there isn't a way to corcumvent this problem this idea probably doesnt work. – Fosco Oct 18 '16 at 18:26
• Sorry, but I don't see why a small cocomplete category must be a poset. What would be the order on the elements? If it's given by the arrows in some way (which would be kind of logic to me), then isomorphic objects should pose a problem (the ordering would not be antisymmetric)... – Daniel Robert-Nicoud Oct 21 '16 at 5:56