# Reedy categories and their generalizations

@article{Shulman2015ReedyCA, title={Reedy categories and their generalizations}, author={Michael Shulman}, journal={arXiv: Algebraic Topology}, year={2015} }

We observe that the Reedy model structure on a diagram category can be constructed by iterating an operation of "bigluing" model structures along a pair of functors and a natural transformation. This yields a new explanation of the definition of Reedy categories: they are almost exactly those small categories for which the category of diagrams and its model structure can be constructed by iterated bigluing. It also gives a consistent way to produce generalizations of Reedy categories, including… Expand

#### 7 Citations

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This note explores the algebraic perspective on the notion of generalized Reedy category introduced by Berger and Moerdijk [BM08]. The aim is to unify inductive arguments by means of a canonical… Expand

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We construct a new model category presenting the homotopy theory of presheaves on "inverse EI $(\infty,1)$-categories", which contains universe objects that satisfy Voevodsky's univalence axiom. In… Expand

$(\infty,1)$-Categorical Comprehension Schemes.

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In this paper we define and study a notion of comprehension schemes for cartesian fibrations over $(\infty,1)$-categories, generalizing Johnstone's respective notion for ordinary fibered categories.… Expand

On bifibrations of model categories

- Mathematics, Computer Science
- ArXiv
- 2017

Using this Grothendieck construction for model structures, the traditional definition of Reedy model structures is revisited, and possible generalizations are revisit, and exhibit their bifibrational nature. Expand

First-Order Logic with Isomorphism

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The Univalent Foundations requires a logic that allows us to define structures on homotopy types, similar to how first-order logic with equality ($\text{FOL}_=$) allows us to define structures on… Expand

Finite Inverse Categories as Signatures

- Mathematics, Computer Science
- ArXiv
- 2017

A simple dependent type theory is defined and it is proved that its well-formed types correspond exactly to finite inverse categories. Expand

#### References

SHOWING 1-10 OF 44 REFERENCES

On Reedy Model Categories

- Mathematics
- 2007

The sole purpose of this note is to introduce some elementary results on the structure and functoriality of Reedy model categories. In particular, I give a very useful little criterion to determine… Expand

On an extension of the notion of Reedy category

- Mathematics
- 2011

We extend the classical notion of a Reedy category so as to allow non-trivial automorphisms. Our extension includes many important examples occurring in topology such as Segal’s category Γ, or the… Expand

The theory and practice of Reedy categories

- Mathematics
- 2013

The goal of this paper is to demystify the role played by the Reedy category axioms in homotopy theory by reducing the much of the hard work involved to simpler computations involving weighted colimits and Leibniz (pushout-product) constructions. Expand

The Grothendieck construction for model categories

- Mathematics
- 2014

The Grothendieck construction is a classical correspondence between diagrams of categories and coCartesian fibrations over the indexing category. In this paper we consider the analogous… Expand

Homotopy theory for algebras over polynomial monads

- Mathematics
- 2013

We study the existence and left properness of transferred model structures for "monoid-like" objects in monoidal model categories. These include genuine monoids, but also all kinds of operads as for… Expand

Univalence for inverse diagrams and homotopy canonicity

- Computer Science, Mathematics
- Mathematical Structures in Computer Science
- 2014

A homotopical version of the relational and gluing models of type theory that uses the Reedy homotopy theory on inverse diagrams, and relies on the fact that Reedy fibrant diagrams correspond to contexts of a certain shape in type theory. Expand

Enriched categories as a free cocompletion

- Mathematics
- 2013

Abstract This paper has two objectives. The first is to develop the theory of bicategories enriched in a monoidal bicategory—categorifying the classical theory of categories enriched in a monoidal… Expand

Structure of categories

- Mathematics
- 1966

Introduction. This paper sets out to develop a structure theory of categories and carries it, not very far, but far enough for some applications. We need a new definition of complete (coinciding with… Expand

Univalent categories and the Rezk completion

- Mathematics, Computer Science
- Mathematical Structures in Computer Science
- 2015

A definition of ‘category’ for which equality and equivalence of categories agree is proposed, and it is shown that any category is weakly equivalent to a univalent one in a universal way. Expand

Enriched indexed categories

- Mathematics
- 2012

We develop a theory of categories which are simultaneously (1) indexed over a base category S with nite products, and (2) enriched over an S-indexed monoidal category V. This includes classical… Expand