# zbMATH — the first resource for mathematics

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)
Full Text:
##### References:
 [1] Angeltveit, V, Enriched reedy categories, Proc. Am. Math. Soc., 136, 2323-2332, (2008) · Zbl 1153.18012 [2] Baues, HJ, Geometry of loop spaces and the cobar construction, Mem. Amer. Math. Soc., 25, 230, (1980) · Zbl 0473.55009 [3] Berger, C, A cellular nerve for higher categories, Adv. Math., 169, 118-175, (2002) · Zbl 1024.18004 [4] Berger, Clemens, Ieke, Moerdijk: On an extension of the notion of Reedy category. Math. Z. (3-4), 977-1004 (2011) · Zbl 1244.18017 [5] Gabriel, P., Zisman, M.: Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35. Springer, New York (1967) · Zbl 0186.56802 [6] Hischhorn, P.S.: Model Categories and Their Localizations, Mathematical Surveys and Monographs 99. AMS, Providence (2003) [7] Isaacson, SB, Symmetric cubical sets, J. Pure Appl. Algebra, 215, 1146-1173, (2011) · Zbl 1228.18014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.