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The Duskin nerve of 2-categories in Joyal’s cell category \(\Theta_2\). (English) Zbl 1442.18047

A \(2\)-categorical analog of the nerve of ordinary categories goes by the name of Duskin nerve. It first occurred as an instance of Street’s nerve of \(\omega\)-categories from [R. Street, J. Pure Appl. Algebra 49, 283–335 (1987; Zbl 0661.18005)], and was then studied in detail by J. W. Duskin [Theory Appl. Categ. 9, 198–308 (2001; Zbl 1046.18009)].
After having understood the Duskin nerve of suspension \(2\)-categories \(\Sigma\mathcal{D}\), the authors give an explicit and purely combinatorial description of the Duskin nerve of any \((r+1)\)-point suspension \(2\)-category \(\Sigma[\mathcal{D}_1,\ldots, \mathcal{D}_r]\). In particular, of \(2\)-categories belonging to Joyal’s cell category \(\Theta_2\) and being motivating examples of \((r+1)\)-point suspension \(2\)-categories.

MSC:

18N10 2-categories, bicategories, double categories
18N50 Simplicial sets, simplicial objects
55U35 Abstract and axiomatic homotopy theory in algebraic topology
55U10 Simplicial sets and complexes in algebraic topology
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References:

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