Lifting twisted coreflections against delta lenses
type: Item
instance of: scholarly article
title: Lifting twisted coreflections against delta lenses
zbMATH Open document ID: 1545.18006
author: Bryce Clarke
published in: Theory and Applications of Categories
publication date: 2024-08-19T00:00:00Z
review text: Delta lenses were introduced in 2011 in [\url{https://link.springer.com/chapter/10.1007/978-3-642-24485-8_22}] as a framework for bidirectional transformations [\textit{F. Abou-Saleh} et al., Lect. Notes Comput. Sci. 9715, 1--28 (2018; Zbl 1476.68034)]. Johnson and Rosenbrugh [\url{https://eceasst.org/index.php/eceasst/article/view/2090}] initiated the study of delta lenses using category theory.\N\NThis paper considers a way of comparing delta lenses and split opfibrations: the class of functors that they lift against. These are the twisted coreflections and split coreflections, respectively.\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 1] recalls the concepts required to state the definition of an algebraic weak factorization system (AWFS).\N\N\item[\S 2] recalls the notion of delta lense [\url{https://link.springer.com/chapter/10.1007/978-3-642-24485-8_22}; \textit{P. J. Freyd} and \textit{G. M. Kelly}, J. Pure Appl. Algebra 2, 169--191 (1972; Zbl 0257.18005)], constructing a thin double category \(\mathbb{L}\mathrm{ens}\)\ of categories, functors and delta lenses [\url{https://bryceclarke.github.io/other/the-double-category-of-lenses.pdf}]. It is shown (Theorem 2.6) that the double category of delta lenses is isomorphic to the double category \(\mathbb{R}\mathrm{LP}\left( \mathbb{J}\right) \)\ of right lifts for a small double category \(\mathbb{J}\). It is also shown that \ has tabulators, which is exploited to show that a delta lens is equivalent to a commutative diagram of functors\N\[\N\begin{array} [c]{ccc} X & \overset{\varphi}{\longrightarrow} & A\\\N& \underset{f\varphi}{\searrow} & \downarrow_{f}\\\N& & B \end{array}\N\]\Nwhere \(\varphi\)\ is bijective-on-objects and \(f\varphi\)\ is a discrete opfibration.\N\N\item[\S 3] introduces a special kind of split coreflection, called a twisted coreflection, constructing the double category \(\mathbb{T}\mathrm{wCoref}\)\ of categories, functors and twisted coreflections. It is sho…
reviewed by: Hirokazu Nishimura
Mathematics Subject Classification ID: 18A32
Mathematics Subject Classification ID: 18A40
Mathematics Subject Classification ID: 18C15
Mathematics Subject Classification ID: 18D30
Mathematics Subject Classification ID: 18N10
zbMATH DE Number: 7898685
zbMATH Keywords: algebraic weak factorisation system
zbMATH Keywords: coreflection
zbMATH Keywords: double category
zbMATH Keywords: lens
zbMATH Keywords: lifting
zbMATH Keywords: opfibration
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7898685