model category, model -category
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homotopy theory, (∞,1)-category theory, homotopy type theory
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A Reedy category is a category equipped with a structure enabling the inductive construction of diagrams and natural transformations of shape . It is named after Christopher Reedy.
The most important consequence of a Reedy structure on is the existence of a certain model structure on the functor category whenever is a model category (no extra hypotheses on are required): the Reedy model structure.
A Reedy category is a category equipped with two wide subcategories and and a total ordering on the objects of , defined by a degree function , where is an ordinal number, such that:
The prototypical examples of Reedy categories are the simplex category and its opposite . (More generally, for any simplicial set , its category of simplices is a Reedy category.)
(Reedy structure on the simplex category)
The Reedy category structure on is defined by:
The degree function is defined by .
a map is in precisely if it is injective;
a map is in precisely if it is surjective.
And the Reedy category structure on is defined by switching and .
Any ordinal , considered as a poset and hence a category, is a Reedy category with , the discrete category on , and the identity.
The opposite of any Reedy category is a Reedy category: use the same degree function, and exchange and .
The integers regarded as a poset is NOT a Reedy category, since it is not well-founded in either direction.
Joyal's category is also a Reedy category.
Many very small categories of diagram shapes are Reedy categories, such as , or , or . This is of importance for the construction of homotopy limits and colimits over such diagram shapes.
A Reedy category in which contains only identities is called a direct category; the factorization axiom then says simply that . Similarly, if contains only identities it is said to be an inverse category.
Any ordinal is of course a direct category, and so is the subcategory 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 of any Reedy category considered as a category in its own right. This amounts to “discarding the faces” in a shape category.
One problem with the notion of Reedy category is that it violates the principle of equivalence: 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 , let and be the unique factorizations. Then , so and , whence and since . Thus . The same argument applied to shows that preserves the degree, hence .
This is problematic for many -like categories such as the category of cycles, Segal’s category , the tree category , 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
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 and .
Eilenberg-Zilber categories are a special sort of generalized Reedy category that behave rather like an elegant strict Reedy category.
There is also a generalization of the notion of Reedy category to the context of enriched category theory: this is an enriched Reedy category.
If is a direct category, then for any model category the colimit functor is a left Quillen functor. However, there are non-direct Reedy categories with the same property, they are called Reedy categories with fibrant constants.
See the references at Reedy model structure
Last revised on June 29, 2023 at 17:27:39. See the history of this page for a list of all contributions to it.