# Model structures on diagrams indexed by a Reedy category

I'm interested in the way to put a model structure on the category of functors $F : P^{op} \rightarrow Ch(\mathbf{k})$ where $\mathbf{k}$ is a field of characteristic zero, $Ch(\mathbf{k})$ the (co)chain complexes and $P$ a finite poset using the formalism of Reedy category.

To me, it seems that there is a priori, two different ways to put a model structure on $Fun(P^{op}, Ch(\mathbf{k}))$ using Reedy categories.

## First Attempt

Let $P$ be a finite poset, one can endows $P$ with a structure of Reedy category by defining

• $P_{+} := P$,
• $P_{-} := \mathrm{Disc}(P)$ where $\mathrm{Disc}(P)$ is the underlying discrete category of the finite poset $P$.

and then following Hovey, model categories or Riehl and Verity, The theory and practice of Reedy categories we have a Reedy category on $P^{op}$ by considering

• $(P^{op})_{+} := (P_{-})^{op} = \mathrm{Disc}(P)$,
• $(P^{op})_{-} := (P_{+})^{op} = P^{op}$.

Then we know that the weak equivalences are the objectwise weak equivalences and (by considering the relative latching maps) the cofibrations are the objectwise cofibrations in $Ch(\mathbf{k})$. This model structure is then close to the injective model structure on $Fun(P^{op}, Ch(\mathbf{k}))$ induced by the injective one on $Ch(\mathbf{k})$.

## Second Attempt

$P^{op}$ is also a finite poset and we can consider it as a Reedy category with

• $(P^{op})_{+} := P^{op}$,
• $(P^{op})_{-} := \mathrm{Disc}(P^{op})= \mathrm{Disc}(P)$.

Then we know that the weak equivalences are the objectwise weak equivalences and (by considering the relative matching maps) the fibrations are the objectwise fibrations in $Ch(\mathbf{k})$. This model structure is then close to the projective model structure on $Fun(P^{op}, Ch(\mathbf{k}))$ induced by the projective one on $Ch(\mathbf{k})$.

## Question

Is there one attempt that is more natural than the other ?

My guess is that, well it depends of the model structure you consider on $Ch(\mathbf{k})$. If I'm not wrong $Ch(\mathbf{k})$ is endowed with a combinatorial model category with both, the projective and the injective, model structures, so we have Quillen equivalences between the projective, Reedy and injective model structure on $Fun(P^{op}, Ch(\mathbf{k}))$ and the three structures give the same notion of homotopy. But again, I might be wrong and there is only one way to think.

• Please tell me 'totally legit' is a technical term that defines some poset property :-) – David White Aug 18 '15 at 12:16
• Well no, sorry about that :-). It was just to point out that $P^{op}$ was also a finite poset. I'll edit my question. – Vorph Aug 18 '15 at 12:21

## 1 Answer

Your last paragraph is correct. For any ring $R$, Ch(R) is combinatorial because it's a Grothendieck category. Also, the injective and projective model structures on Ch(R) are Quillen equivalent. If $k$ is a field of characteristic zero then the projective and injective model structures coincide in fact. Now, even if you have two ways to put a Reedy structure on some indexing category, that doesn't matter because both ways are guaranteed to be Quillen equivalent to the projective model structure on $Fun(P^{op},Ch(R))$, where there is no ambiguity. Alternately, you could choose the injective model structure on $Fun(P^{op},Ch(R))$ and you know the Reedy is Quillen equivalent to the injective too.

Is one more natural than the other? I guess that's up to you. Ch(R) is a very nice example, even better if R is a field of characteristic zero. So I think pretty much any choice here is going to be convenient and it really doesn't matter which you use. For sure there are plenty of different ways to think about Fun($P^{op},Ch(R))$ but all are Quillen equivalent and it shouldn't matter for whatever application you have in mind. Hope that helps.

• Yes this is helpful, thank you. By the way, any idea where I can find a proof of the local presentability of $Ch(R)$ ? – Vorph Aug 18 '15 at 15:39
• Sure. It's basically there in Lemma 2.3.2 of Hovey's book, but obviously there are more canonical references. The oldest I can find is Grothendieck's Tohoku paper, which proves it for $R$-modules. You can get from there to chain complexes easily via the machinery in Adamek and Rosicky's book, since colimits of chain complexes are computed levelwise. See also the wikipedia article on Grothendieck categories. – David White Aug 18 '15 at 23:19