nLab generalized Reedy category

Idea

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 $\Lambda$, Segal's category $\Gamma$, the tree category $\Omega$, 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 & Moerdijk 2011 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.

Definitions

Generalized Reedy categories

Definition

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

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

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

3. every isomorphism in $R$ preserves degree,

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

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

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

Definition

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

This appears as (Berger & Moerdijk 2011, def. 1.1).

Remark

Generalized Reedy category structures (as opposed to ordinary structures!) can always be transported along equivalence of categories.

Definition

For a Cisinski generalized Reedy category, the final condition in def. is replaced by

• every morphism in $R_-$ admits a section, and two parallel morphisms in $R_-$ 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 $R_-$ arrows be something less than epimorphic in $R$. By comparison, the Cisinski condition asks that $R_-$ arrows be actually epimorphic in $R$, in fact that they be split epimorphic, and more.

Presheaves on Reedy categories.

For $R$ a generalized Reedy category, and $X$ a presheaf on $R$, there are the evident analogue notions of $n$-cells in $X$, degenerate $n$-cells, faces, boundaries, horns, etc.

(..)

Properties

Normal monomorphisms (Cisinski)

Let $A$ be a Cisinski generalized Reedy category.

Definition

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

See (Cisinski, prop. 8.1.9).

Proposition

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

This is (Cisinski, prop. 8.1.13).

Let $X$ be a presheaf over $A$.

Definition

For $a \in A$, a cell $v \in X(a)$ is called degenerate precisely if there is a morphism $\alpha : a \to a'$ in $A_-$ and a cell $u \in X(a')$

$$X(\alpha) : u \mapsto v$$

See (Cisinski, cor. 8.1.10).

Write $A/X$ for the category of elements of $X$.

Definition

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

An $a$-cell $u \in X(a)$ is called normal if there is a morphism $\alpha : a \to b$ in $A_-$ with $b$ non-degenerate, and a dominant $b$-cell $v \in X(b)$, such that

$$X(\alpha) : v \mapsto u \,.$$

The presheaf $X$ is called normal if all its cells are normal.

See (Cisinski, 8.1.23).

Remark

A non-degenerate cell is normal precisely if it is dominant.

Proposition

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

This is (Cisinski, cor. 8.1.25).

Let $f : X \to Y$ be a morphism of presheaves over $A$.

Definition

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

This is (Cisinski, 8.1.30).

Example

Every monomorphism between normal presheaves is normal.

Proposition

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

• pushouts;

• transfinite composition;

• retracts.

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

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

Model category structure on presheaves

(…) generalized Reedy model structure

Examples

Berger-Moerdijk type

Specific examples

• Any Reedy category is a generalized Reedy category, in particular the simplex category.

• Any (finite) groupoid $G$ is also a generalized Reedy category, with $G_+ = G_- = G$.

• Connes’ category of cycles $\Lambda$.

• Segal's category$\Gamma$.

• The Moerdijk-Weiss tree category $\Omega$ is generalized Reedy. The degree is given by the number of vertices in a tree.

  See also the discussion at dendroidal set and model structure on dendroidal sets.

• Any generalized direct category or generalized inverse category is also a generalized Reedy category, in which either $R_-$ or $R_+$ consists only of the isomorphisms.

The class of crossed group sites

Definition

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

1. for each object $s \in S$ a group structure on $G_s$;

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

such that for all morphisms $\alpha : s \to r$ and $\beta : t \to s$ in $S$ and $g,h \in G_r$ we have

1. $g_*( \alpha \circ \beta) = g_*(\alpha) \circ \alpha^*(g)_* \beta$;

2. $g_* (id_r) = id_r$;

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

4. $\alpha^*(e_r) = e_s$;

where $g_*$, $h_*$ denotes the group action and $\alpha^* : G_r \to G_s$ the presheaf map.

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

$$(\alpha, g) \circ (\beta, h) = (\alpha \cdot g_*(\beta), \beta^*(g) \cdot h) \,.$$

(BM11, def. 2.1).

Definition

If $S$ is equipped with a generalized Reedy structure, then an crossed $S$-group $G$ is called compatible with that generalized Reedy structure if

1. the $G$-action respects $S^+$ and $S^-$;

2. if $\alpha : r \to s$ is in $S^-$ and $g \in G_s$ such that $\alpha^* (g) = e_r$ and $g_*(\alpha) = \alpha$, then $g = e_s$.

Example

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

Proposition

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

(BM11, prop. 2.10).

Related concepts

• A “more generalized” notion is a c-Reedy category.
• A more specialized notion is an Eilenberg-Zilber category.

References

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

• Denis-Charles Cisinski, Les préfaisceaux comme modèles des types d'homotopie, Astérisque, Volume 308, Soc. Math. France (2006), 392 pages (pdf)

The Berger-Moerdijk definition of generalized Reedy category appears in

• Clemens Berger, Ieke Moerdijk, On an extension of the notion of Reedy category, Mathematische Zeitschrift 269 (2011) 977–1004 [arXiv:0809.3341, doi:10.1007/s00209-010-0770-x]