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Reedy category

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Idea

A Reedy category is a category RR equipped with a structure enabling the inductive construction of diagrams and natural transformations of shape RR. It is named after Christopher Reedy.

The most important consequence of a Reedy structure on RR is the existence of a certain model structure on the functor category M RM^R whenever MM is a model category (no extra hypotheses on MM are required): the Reedy model structure.

Definition

A Reedy category is a category RR equipped with two wide subcategories R +R_+ and R R_- and a total ordering on the objects of RR, defined by a degree function d:Ob(R)αd:Ob(R) \to \alpha, where α\alpha is an ordinal number, such that:

  • Every nonidentity morphism in R +R_+ raises degree,
  • Every nonidentity morphism in R R_- lowers degree, and
  • Every morphism ff in RR factors uniquely as a map in R R_- followed by a map in R +R_+.

That is, we have two subcategories R +R_+ and R R_-, such that all arrows in R +R_+ are degree-raising (more precisely, non-degree-lowering) and all arrows in R +R_+ are degree-lowering, and every arrow in RR is uniquely decomposible as a degree-lowering followed by a degree-raising.

Examples

  • Any ordinal α\alpha, considered as a poset and hence a category, is a Reedy category with α +=α\alpha_+=\alpha, α \alpha_- the discrete category on Ob(α)Ob(\alpha), and dd the identity.

  • The opposite of any Reedy category is a Reedy category: use the same degree function, and exchange R +R_+ and R R_-.

  • The integers regarded as a poset is NOT a Reedy category, since it is not well-founded in either direction.

  • Joyal's category, Θ\Theta, is also a Reedy category.

  • Many very small categories of diagram shapes are Reedy categories, such as ()(\cdot\to\cdot\to \dots), or ()(\cdot\leftarrow \cdot\rightarrow\cdot), or ()(\cdot\rightrightarrows\cdot). This is of importance for the construction of homotopy limits and colimits over such diagram shapes.

The simplex category

The prototypical examples of Reedy categories are the simplex category Δ\Delta and its opposite Δ op\Delta^{op}. More generally, for any simplicial set XX, its category of simplices Δ/X\Delta/X is a Reedy category.

The Reedy category structure on Δ\Delta is defined by:

  • The degree function d:Ob(Δ)d: Ob(\Delta) \to \mathbb{N} is defined by [k]k[k] \mapsto k.

  • a map [k][n][k] \to [n] is in Δ +\Delta_+ precisely if it is injective;

  • a map [n][k][n] \to [k] is in Δ \Delta_- precisely if it is surjective.

And the Reedy category structure on Δ op\Delta^{op} is defined by switching Δ +\Delta_+ and Δ \Delta_-.

Direct and inverse categories

A Reedy category in which R R_- contains only identities is called a direct category; the factorization axiom then says simply that R=R +R=R_+. Similarly, if R +R_+ contains only identities it is said to be an inverse category.

Any ordinal is of course a direct category, and so is the subcategory R +R_+ of any Reedy category considered as a category in its own right. This amounts to “discarding the degeneracies” in a shape category. In some examples there are no degeneracies to begin with, such as the category of opetopes; thus these are naturally direct categories.

Dually, so is the subcategory R R_- of any Reedy category considered as a category in its own right. This amounts to “discarding the faces” in a shape category.

Generalized Reedy categories

One problem with the notion of Reedy category is that it is evil: it is not invariant under equivalence of categories. It’s not hard to see that any Reedy category is necessarily skeletal. In fact, it’s even worse: no Reedy category can have any nonidentity isomorphisms!

Proof: Take any isomorphism ff, let f=ghf = g h and hf 1=ghh f^{-1} = g' h' be the unique factorizations. Then id=ghf 1=(gg)hid = g h f^{-1} = (g g') h', so h=idh' = id and gg=idg g' = id, whence g=idg = id and g=idg' = id since g,gR +g, g' \in R_+. Thus f=hR f = h \in R_-. The same argument applied to f 1f^{-1} shows that ff preserves the degree, hence f=idf = id. \qed

This is problematic for many Δ\Delta-like categories such as the category of cycles, Segal’s category Γ\Gamma, the tree category Ω\Omega, and so on. The concept of

due to Clemens Berger and Ieke Moerdijk, avoids these problems. There is a similar notion (which however does not comprise all Reedy categories) due to Denis-Charles Cisinski. A further generalization which allows noninvertible level morphisms is a

Elegant Reedy categories

The notion of elegant Reedy category, introduced by Julie Bergner and Charles Rezk, is a restriction of the notion which captures the property that the Reedy model structure and injective model structure coincide. Several important Reedy categories are elegant, such as the Δ\Delta and Θ\Theta.

Eilenberg-Zilber categories

Eilenberg-Zilber categories are a special sort of generalized Reedy category that behave rather like an elegant strict Reedy category.

Enriched Reedy categories

There is also a generalization of the notion of Reedy category to the context of enriched category theory: this is an enriched Reedy category.

Reedy categories with fibrant constants.

If RR is a direct category, then for any model category MM the colimit functor colim R:M RM\colim_R \colon M^R \to M is a left Quillen functor. However, there are non-direct Reedy categories with the same property, they are called Reedy categories with fibrant constants.

References

See the references at Reedy model structure

Last revised on July 12, 2020 at 12:50:09. See the history of this page for a list of all contributions to it.