Macquarie University
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The category of asymmetric lenses and its proxy pullbacks

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posted on 2022-07-06, 04:44 authored by Matthew Di Meglio

We study the categorical properties of the category of small categories and asymmetric delta lenses, continuing the work begun by Chollet et al. at the Applied Category Theory Adjoint School 2020. We give complete elementary characterisations of the monic and epic lenses, confirming several of Chollet et al.’s conjectures. We also initiate the study of lens co-equalisers, ultimately showing that every epic lens is regular, and that discrete opfibrations have pushouts along monic lenses. An important construction for proving many of these results is Johnson and Rosebrugh’s “pullback” of lenses, which we call the proxy pullback of lenses. We give a new treatment of the proxy pullback in terms of compatibility—a stronger notion of commutativity for squares of lenses. We also prove that the proxy pullback has several pullback-like properties, including an analogue of the well-known pullback pasting lemma. The proxy pullback is sometimes, but not always, a real pullback. Using new notions of sync-minimal and independent lens spans, we characterise when a lens span that forms a commuting square with a lens cospan has a comparison lens to a proxy pullback of the cospan.


Table of Contents

Chapter 1. Introduction -- Chapter 2. Background -- Chapter 3. Proxy pullbacks -- Chapter 4. Universal properties of the proxy pullback -- Chapter 5. Monic and epic lenses -- Chapter 6. Coequalisers of lenses -- Chapter 7. Conclusion


Bibliography: pages 50-51

Awarding Institution

Macquarie University

Degree Type

Thesis MRes


MRes, Macquarie University, Faculty of Science and Engineering, Department of Mathematics and Statistics

Department, Centre or School

Department of Mathematics and Statistics

Year of Award


Principal Supervisor

Michael Johnson

Additional Supervisor 1

Richard Garner

Additional Supervisor 2

Samuel Muller

Rights Copyright Matthew Di Meglio 2021




v, 51 pages