generalized Reedy category


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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 Λ\Lambda, Segal's category Γ\Gamma, the tree category Ω\Omega, 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.


Generalized Reedy categories


A Berger-Moerdijk generalized Reedy category is a category RR together with two wide subcategories R +R_+ and R R_-, and a function d:ob(R)αd\colon ob(R)\to \alpha called degree, for some ordinal α\alpha, such that

  1. every non-isomorphism in R +R_+ raises degree,

  2. every non-isomorphism in R R_- lowers degree,

  3. every isomorphism in RR preserves degree,

  4. R +R R_+\cap R_- is the core of RR (equivalently, every isomorphism is in both R +R_+ and R R_-, i.e. they are not just wide but pseudomonic subcategories),

  5. every morphism ff factors as a map in R R_- followed by a map in R +R_+, uniquely up to isomorphism,

  6. if fR f\in R_- and θ\theta is an isomorphism such that θf=f\theta f = f, then θ=1\theta = 1 (isomorphisms see the maps in R R_- 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.


A morphism of generalized Reedy category SRS \to R is a functor whish takes S +S_+ to R +R_+, takes S S_- to R R_- and preserves the degree.

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

  • every morphism in R R_- admits a section, and two parallel morphisms in R R_- are equal precisely when they have the same sections.

For clarity, in the context of generalized Reedy categories, an ordinary Reedy category may be called a strict Reedy category.

Prima Facie comparison between Cisinski and Berger-Moerdijk.

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 R R_- arrows be something less than epimorphic in RR. By comparison, the Cisinski condition asks that R R_- arrows be actually epimorphic in RR, in fact that they be split epimorphic, and more.

Presheaves on Reedy categories.

For RR a generalized Reedy category, and XX a presheaf on RR, there are the evident analogue notions of nn-cells in XX, degenerate nn-cells, faces, boundaries, horns, etc.



Normal monomorphisms (Cisinski)

Let AA be a Cisinski generalized Reedy category.


An object aAa \in A is called degenerate precisely if there is a non-isomorphism out of aa in A A_-.

See (Cisinski, prop. 8.1.9).


For every object aAa \in A there exists a morphism π:ab\pi : a \to b in A A_- with bb non-degenerate.

This is (Cisinski, prop. 8.1.13).

Let XX be a presheaf over AA.


For aAa \in A, a cell vX(a)v \in X(a) is called degenerate precisely if there is a morphism α:aa\alpha : a \to a' in A A_- and a cell uu \in

X(α):uv X(\alpha) : u \mapsto v

See (Cisinski, cor. 8.1.10).

Write A/XA/X for the category of elements of XX.


For aAa \in A an aa-cell vX(a)v \in X(a) is called dominant if (a,v)A/X(a,v) \in A/X has trivial automorphism group.

An aa-cell uX(a)u \in X(a) is called normal if there is a morphism α:ab\alpha : a \to b in A A_- with bb non-degenerate, and a dominant bb-cell vX(b)v \in X(b), such that

X(α):vu. X(\alpha) : v \mapsto u \,.

The presheaf XX 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.


XX is normal precisely if all its non-degenerate cells are dominant.

This is (Cisinski, cor. 8.1.25).

Let f:XYf : X \to Y be a morphism of presheaves over AA.


The morphism f:XYf : X \to Y is called normal if every cell of YY not in the image of ff is dominant.

This is (Cisinski, 8.1.30).


Every monomorphism between normal presheaves is normal.


The class of normal monomorphisms in PSh(A)PSh(A) is closed under

In fact, the class of normal monomorphisms is that generated under these operations from the boundary inclusions I:={aa}I := \{\partial a \hookrightarrow a\}.

This is (Cisinski, prop. 8.1.31, 8.1.35).

Model category structure on presheaves

(…) generalized Reedy model structure


Berger-Moerdijk type

Specific examples

The class of crossed group sites


For SS a small category, a crossed SS-group is a presheaf G:S opSetG : S^{op} \to Set equipped with

  1. for each object sSs \in S a group structure on G sG_s;

  2. for all s,rSs, r\in S a left G rG_r-action on the hom-set S(s,r)S(s,r) ;

such that for all morphisms α:sr\alpha : s \to r and β:ts\beta : t \to s in SS and g,hG rg,h \in G_r we have

  1. g *(αβ)=g *(α)α *(g) *βg_*( \alpha \circ \beta) = g_*(\alpha) \circ \alpha^*(g)_* \beta;

  2. g *(id r)=id rg_* (id_r) = id_r;

  3. α *(gh)=h *(α) *(g)α *(h)\alpha^* (g \cdot h) = h_*(\alpha)^*(g)\cdot \alpha^*(h);

  4. α *(e r)=e s\alpha^*(e_r) = e_s;

where g *g_*, h *h_* denotes the group action and α *:G rG s\alpha^* : G_r \to G_s the presheaf map.

The total category SGS G of an crossed SS-group GG is the category with the same objects as SS, and with morphisms rsr \to s being pairs (α,g)S(s,r)×G r(\alpha, g) \in S(s,r)\times G_r and with composition defined by

(α,g)(β,h)=(αg *(β),β *(g)h). (\alpha, g) \circ (\beta, h) = (\alpha \cdot g_*(\beta), \beta^*(g) \cdot h) \,.

(Ber-Moe, def. 2.1).


If SS is equipped with a generalized Reedy structure, then an SS-crossed group GG is called compatible with that generalized Reedy structure if

  1. the GG-action respects S +S^+ and S S^-;

  2. if α:rs\alpha : r \to s is in S S^- and gG sg \in G_s such that α *(g)=e r\alpha^* (g) = e_r and g *(α)=αg_*(\alpha) = \alpha, then g=e sg = e_s.


The category Ω pl\Omega_{pl} of planar finite rooted trees is a strict Reedy category. The category Ω\Omega of non-planar finite rooted trees is the total category of an Ω pl\Omega_{pl}-crossed group which to a planar tree TT assigns its group of non-planar automorphisms.


Let SS be a strict Reedy category and let GG be a compatible SS-crossed group. Then there exists a unique dualizabe generalized Reedy structure on SGS G for which the embedding SSGS \hookrightarrow S G is a morphism of generalized Reedy categories.

(Ber-Moe, prop. 2.10).


Cisinski’s notion of generalized Reedy category appears as def 8.1.1 in

The Berger-Moerdijk definition of generalized Reedy category appears in

Revised on October 5, 2015 14:56:00 by Mike Shulman (