Starting from a generalized Reedy category $R$ satisfying a simple condition, we construct an absolutely dense functor $\mathbf{D}_R \to R$ with domain a strict Reedy category. In the case of a generalized inverse category $R$, and given any tribe $\mathcal{T}$, we leverage this construction to provide a tribe structure on a subcategory of fibrant diagrams in $\mathcal{T}^R$.
Given a combinatorial model category M and any small category C, both the projective and injective model structure on the category of diagrams M C exist. If C is a Reedy category, a third model structure, the Reedy model structure, which is in general different from the other two, always exists even when dropping the combinatoriality assumption on M.
For fibration categories, the situation is a bit different. Indeed, given a fibration category F, the “natural” fibration category structure on a category of C-shaped diagrams in F is arguably the Reedy model structure, assuming that C is an inverse category. By this, we mean that the notion of Reedy model structure adapts in a straightforward manner to yield a definition of Reedy fibrant diagram, forming a category where it is possible to factor any map suitably as required from the axioms of a fibration category. If C is not assumed to be a direct category, it is still possible to endow the category F C with the structure of a fibration category, where both the fibrations and the weak equivalences are the pointwise ones: this is established in [Rad06, Theorem 9.5.5], relying on the Reedy fibration category structure on F ∆ ′ ↓C , where ∆ ′ is the direct category spanned by the injective maps in the simplex category ∆. In [Cis10, Théorème 6.17], another fibration category structure is established on a subcategory of “τ -fibrant” diagrams in F C , working in a slightly different manner with the usual functor τ : ∆ ′ ↓ C → C.
Finally, in the case of tribes, it seems that a tribe structure on a category of diagram T C , where T is a tribe and C a small category, has only been considered in the literature when C is an inverse category, in which case the notion of Reedy fibrancy can be used (see [KS19,Lemma 2.22]).
In the present document, we will be interested in the notion of generalized Reedy category, which aims at relaxing the definition of a Reedy category in order to allow non-identity isomorphisms. It has been introduced by Cisinski in [Cis+06], then slightly generalized and further discussed by Berger and Moerdijk in [BM11].
Our objective is to provide a tribe structure on a subcategory of fibrant diagrams T C , assuming C to be a generalized inverse category.
In this section, we consider a generalized Reedy category R satisfying the following condition:
• There exists a functor c : R 0 → R from a (strict) Reedy category, such that every arrow f : a → b in R lifts to an arrow k : x → y in R 0 up to isomorphism, i.e., such that there is an isomorphism w : a ≃ c(x) and w ′ : b ≃ c(y) fitting in a commutative square:
Example 1.1. Our main source of examples is that of categories of cubes. In [Cam23], several variations on the category of cubes are considered. For a given category of cubes □ without isomorphisms, there is usually a counterpart with symmetries □ s . The inclusion
then satisfies the condition above. When the categories of cubes do not involve diagonals, the author establishes in Theorem 7.9 that □ s is a generalized Reedy category (more precisely an EZ-category), in addition to □ being a strict Reedy category. Hence, concrete examples include the standard category of cubes □, which is the free monoidal category generated by two projections and one degeneracy. The corresponding category of cubes with symmetries □ s is the free symmetric monoidal category generated by □.
Definition 1.1. Write F for the free category comonad on Cat, then form the pushout as in the diagram below:
Finally, write D R for the full subcategory of the twisted arrow category Tw(F ≃ (R)) spanned by the objects consisting of an arrow x → y of the form x → z → y where x → z comes from a map in R 0 , and z → y corresponds to a “free” isomorphism in R or is an identity arrow (that is, an arrow obtained by taking the image under F(R) → F ≃ (R) of a path of length one in R whose underlying arrow is an isomorphism). It comes with a projection p : D R → R.
Lemma 1.1. D R inherits a (strict) Reedy category structure.
Proof. Consider an object of D R given by a map f : x → z → y. We take the degree of such an object to be deg(y) -deg(x) + k, where k = 0 if z → y is an identity arrow, and k = 1 otherwise. We define the subcategory D R,+ as the one spanned by the arrows F : X → Y of the form
where f comes from a map in R 0,-and f ′ arise from a path, of length at most 2, [f ′ 0 , f ′ 1 ] of arrows in R such that each f ′ 0 comes from an arrow in R 0,+ and f ′ 1 from an isomorphism in R (either, or both, being omitted if the path is of length strictly less than 2).
Dually, we define a subcategory D R,-similarly, but where the role of f and f ′ have been swapped. Now consider an arrow F : X → Y , given by a diagram as the one above. Then, f is of the form w • f s where f s comes from an arrow in R 0 and w is a “free” isomorphism in R or an identity arrow. Similarly, f ′ is of the form w ′ • f ′ s , where w ′ is a “free” isomorphism or an identity, and f ′ s comes from an a
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