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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 $x_s \to F(d)(x_{r})$ 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.)

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  • $\begingroup$ How does having a morphism $x_r \to F(d)(x_s)$ work (typecheck)? Isn't $F(d)$ a functor from $F(r)$ to $F(s)$, while $x_s$ is an object in $F(s)$? $\endgroup$
    – Jason Gross
    Commented Jul 30, 2013 at 18:04
  • $\begingroup$ Thanks, fixed. (Sometimes people talk about contravariant functors instead of covariant ones.) $\endgroup$
    – Mike Shulman
    Commented Jul 31, 2013 at 19:02

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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".

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  • $\begingroup$ I don't think that was the paper I'm thinking of, but thanks for pointing it out; it's certainly closely related! $\endgroup$
    – Mike Shulman
    Commented Jan 31, 2013 at 22:42
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I think the paper I was thinking of when I asked this question was probably

although he only considers the case of inverse categories rather than more general Reedy ones. A more recent reference that does deal with the case of general Reedy categories is

I have collected some more remarks and references on model structures of this sort at the nLab page model structure on sections.

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