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.