Question

Suppose $R$ is a category and $F:R\to Cat$ is a functor (or pseudofunctor). The oplax limit of $F$ is the category whose objects consist of an object $x_r \in F(r)$ for all $r$ together with a morphism $x_r \to F(d)(x_{s})$ for all morphisms $d:r\to s$ in $R$, satisfying obvious compatibility conditions.

I have some vague memory of reading a paper in which, given a functor as above in which $R$ is a Reedy category, each category $F(r)$ is a model category, and probably some other conditions, a "Reedy-type" model structure on the oplax limit (or perhaps some related category) was constructed. However, I have been totally unable to find this paper again; the closest I can find is this paper which considers "injective-type" model structures on lax limits. Can anyone point me to the paper I am thinking of?

(I am not interested in seeing proofs or "it seems like this should work" arguments written out in the answers. I only want the reference.)

Answer 1

You may be thinking of

Johnson, Mark W. On modified Reedy and modified projective model structures. Theory Appl. Categ. 24 (2010), No. 8, 179–208.

but his constructions (Definitions 3.3 and 5.2) have a fixed category at each object of $R$ and only the model structures are allowed to vary.

As a side note, a particular instance of this construction is used in these new notes about global homotopy theory, but here the indexing category is not even a Reedy category, but some sort of "enriched generalized Reedy category".