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Reedy categories and the \(\varTheta\)-construction. (English) Zbl 1270.55014
The authors use the notion of multi-Reedy category to prove that if \(\mathcal{C}\) is a Reedy category, then \(\Theta\mathcal{C}\) is a Reedy category as well. This result gives a new proof that the categories \(\Theta_n\) are Reedy categories. Then the authors introduce the notion of elegant Reedy category, for which it is proved that the Reedy and injective model structures coincide. The authors conclude the paper with the notion of \(EZ\)-Reedy category (for Eilenberg Zilberg). It is proved that any \(EZ\)-Reedy category is elegant, and therefore Reedy and injective model structures coincide. And that the categories \(\Theta_k\) are \(EZ\)-Reedy for all \(k\geq 0\).

MSC:
55U35 Abstract and axiomatic homotopy theory in algebraic topology
55U10 Simplicial sets and complexes in algebraic topology
18G55 Nonabelian homotopical algebra (MSC2010)
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