The theory and practice of Reedy categories

By Emily Riehl, Dominic R. Verity. Theory and Applications of Categories, 2013

97 citations recorded by the source corpus.

The goal of this paper is to demystify the role played by the Reedy category axioms in homotopy theory. With no assumed prerequisites beyond a healthy appetite for category theoretic arguments, we present streamlined proofs of a number of useful technical results, which are well known to folklore but difficult to find in the literature. While the results presented here are not new, our approach to their proofs is somewhat novel. Specifically, we reduce the much of the hard work involved to simpler computations involving weighted colimits and Leibniz (pushout-product) constructions. The general theory is developed in parallel with examples, which we use to prove that familiar formulae for homotopy limits and colimits indeed have the desired properties.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f4d588ddf6a7e60cb7bd9addbc4ce56339701e3f

Citations in the corpus

• The 2-category theory of quasi-categories (2013, unknown): “A review of the Reedy category theory necessary to understand its statement and proof, written in part for this purpose, can be found in [23].”
• Homotopy coherent adjunctions and the formal theory of monads (2013, unknown): “Our free homotopy coherent adjunction is exactly the Schanuel-Street free adjunction, regarded as a simplicial category via the inclusion 2-Cat ↪ sSet-Cat; this explains the common notation Adj.”
• Categorical Homotopy Theory (2014, unknown): “There has been a cross-pollination between this chapter and the expository article [RV13c], which gives a much more comprehensive development of Reedy category theory from the perspective of weighted limits and colimits.”
• The Frobenius condition, right properness, and uniform fibrations (2015, unknown): “we encapsulate in the notion of a suitable wfs, given in Definition 2.1 below. In order to state that definition, we need some notation. For k∈ {0,1} and a map i: A→ Bin E, we define the Leibniz product [36] of the endpoint inclusion δk⊗(−) and ito be the map δk⊗ˆ i: (I⊗A) +AB→ I⊗B (2.1) determined uni”
• Magnitude homology of enriched categories and metric spaces (2017, unknown): “⊙∂∆n LnB ⊙∆n Cn Bn ⊙∆n |B|n−1 |B|n p p MAGNITUDE HOMOLOGY OF ENRICHED CATEGORIES AND METRIC SPACES 15 Proof. This is well-known in homotopy theory; we sketch a modern proof based on the technology of [RV14]. In the terminology and notation of that paper, we have (Cn →Bn ⊙∆n) = (LnB →Bn)⊙b(∂∆n →∆n) ∼”