related by the Dold-Kan correspondence
The notion of Reedy category, though useful, is in some contexts inconveniently restrictive, since no Reedy category can contain any nonidentity isomorphisms. This is problematic for many “shape categories” such as Connes’ category of cycles , Segal's category , the tree category , and so on. The notion of generalized Reedy category lifts this restriction, while maintaining the truth of the main theorem about Reedy categories: the existence of the Reedy model structure.
In fact, there are two notions of generalized Reedy category in the literature. Cisinski’s “catégories squelletiques” (Ch. 8 in PCMH) provide a natural generalization of the simplex category, so that diagrams based on them behave much like simplicial objects. They were introduced primarily for the purposes of modeling homotopy types. The “generalized Reedy categories” of Berger and Moerdijk are a strictly broader generalization suitable e.g. for describing dendroidal sets. They were introduced for the purposes of modelling more general classes of structure, particularly operads.
every non-isomorphism in raises degree,
every non-isomorphism in lowers degree,
every isomorphism in preserves degree,
every morphism factors as a map in followed by a map in , uniquely up to isomorphism,
if and is an isomorphism such that , then (isomorphisms see the maps in as epis).
The last condition implies that the isomorphism in the penultimate condition must be unique. It is not self-dual, but has an obvious dual version. A BM generalized Reedy category is said to be dualizable if it satisfies both this condition and its dual.
This appears as (Berger-Moerdijk, def. 1.1).
Generalized Reedy category structures (as opposed to ordinary structures!) can always be transported along equivalence of categories.
For a Cisinski generalized Reedy category, the final condition in def. 1 is replaced by
For clarity, in the context of generalized Reedy categories, an ordinary Reedy category may be called a strict Reedy category.
The only difference between the Cisinski notion and the Berger-Moerdijk notion is in the final condition – let’s say, between the Cisinski condition and the Berger-Moerdijk condition.
It’s easy to see that the Cisinski condition is strictly stronger than the Berger-Moerdijk condition. The Berger-Moerdijk condition asks that arrows be something less than epimorphic in . By comparison, the Cisinski condition asks that arrows be actually epimorphic in , in fact that they be split epimorphic, and more.
For a generalized Reedy category, and a presheaf on , there are the evident analogue notions of -cells in , degenerate -cells, faces, boundaries, horns, etc.
Let be a Cisinski generalized Reedy category.
An object is called degenerate precisely if there is a non-isomorphism out of in .
See (Cisinski, prop. 8.1.9).
For every object there exists a morphism in with non-degenerate.
This is (Cisinski, prop. 8.1.13).
Let be a presheaf over .
For , a cell is called degenerate precisely if there is a morphism in and a cell
See (Cisinski, cor. 8.1.10).
Write for the category of elements of .
For an -cell is called dominant if has trivial automorphism group.
An -cell is called normal if there is a morphism in with non-degenerate, and a dominant -cell , such that
The presheaf is called normal if all its cells are normal.
See (Cisinski, 8.1.23).
A non-degenerate cell is normal precisely if it is dominant.
is normal precisely if all its non-degenerate cells are dominant.
This is (Cisinski, cor. 8.1.25).
Let be a morphism of presheaves over .
The morphism is called normal if every cell of not in the image of is dominant.
This is (Cisinski, 8.1.30).
Every monomorphism between normal presheaves is normal.
The class of normal monomorphisms in is closed under
In fact, the class of normal monomorphisms is that generated under these operations from the boundary inclusions .
This is (Cisinski, prop. 8.1.31, 8.1.35).
Any (finite) groupoid is also a generalized Reedy category, with .
Connes’ category of cycles .
The Moerdijk-Weiss tree category is generalized Reedy. The degree is given by the number of vertices in a tree.
such that for all morphisms and in and we have
where , denotes the group action and the presheaf map.
The total category of an crossed -group is the category with the same objects as , and with morphisms being pairs and with composition defined by
If is equipped with a generalized Reedy structure, then an -crossed group is called compatible with that generalized Reedy structure if
the -action respects and ;
if is in and such that and , then .
The category of planar finite rooted trees is a strict Reedy category. The category of non-planar finite rooted trees is the total category of an -crossed group which to a planar tree assigns its group of non-planar automorphisms.
Let be a strict Reedy category and let be a compatible -crossed group. Then there exists a unique dualizabe generalized Reedy structure on for which the embedding is a morphism of generalized Reedy categories.
Cisinski’s notion of generalized Reedy category appears as def 8.1.1 in
The Berger-Moerdijk definition of generalized Reedy category appears in