Adjoining adjoints
type: Item
instance of: scholarly article
title: Adjoining adjoints
zbMATH Open document ID: 1030.18001
DOI: 10.1016/S0001-8708(02)00068-3
author: Robert Paré
author: Dorette Pronk
author: Robert Dawson
published in: Advances in Mathematics
publication date: 2003-08-25T00:00:00Z
review text: If \(\mathcal{C}\) is a category with pullbacks, the bicategory, \(\text{Span}(\mathcal{C})\), has the universal property that it is the result of freely adjoining right adjoints to the arrows in \(\mathcal{C}\), subject just to the Beck condition. Here the authors describe a 2-category \(\Pi_2\mathcal{C}\), which is similarly the result of freely adjoining right adjoints to the arrows in \(\mathcal{C}\) for any category, \(\mathcal{C}\). The construction uses zig-zag diagrams reminiscent of the Dwyer-Kan hammock localisation from homotopical algebra. The resulting 2-cell diagrams, here called `fences', are very richly structured objects. The 2-cells in \(\Pi_2\mathcal{C}\) are equivalence classes of fences. Of particular interest is \(\Pi_2\mathcal{C}\), when \(\mathcal{C}\) is a free category on a directed graph. In this case the authors explore a beautiful link with planar \((p,q)\)-diagrams (Kauffman diagrams) and thus the tantalising possibility of links with areas of \(C^*\)-algebra theory.
Mathematics Subject Classification ID: 18A40
Mathematics Subject Classification ID: 18D05
Mathematics Subject Classification ID: 18G55
zbMATH DE Number: 1970567
zbMATH Keywords: free adjoint construction
zbMATH Keywords: Beck condiction
zbMATH Keywords: Dwyer-Kan hammock localisation
zbMATH Keywords: zig-zag diagrams
zbMATH Keywords: Kauffman diagrams
zbMATH Keywords: \(C^*\)-algebra
reviewed by: Timothy Porter
MaRDI profile type: Publication
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full work available at URL: s0001-8708(02)00068-3
OpenAlex ID: W4205338989
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description: scientific article; zbMATH DE number 1970567