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The nominal/FM Yoneda Lemma

Published online by Cambridge University Press:  05 April 2021

R. L. Crole Open the ORCID record for R. L. Crole [Opens in a new window]
R. L. Crole*
Affiliation:
School of Informatics, University of Leicester, University Road, Leicester, LE1 7RH, UK
*
*Corresponding author. Email: rlc3@le.ac.uk
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Abstract

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This paper explores versions of the Yoneda Lemma in settings founded upon FM sets. In particular, we explore the lemma for three base categories: the category of nominal sets and equivariant functions; the category of nominal sets and all finitely supported functions, introduced in this paper; and the category of FM sets and finitely supported functions. We make this exploration in ordinary, enriched and internal settings. We also show that the finite support of Yoneda natural transformations is a theorem for free.

Keywords

Yoneda Lemmanominal setsFM sets
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 30 , Issue 9 , October 2020 , pp. 1011 - 1024
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Borceux, F. (1994a). Handbook of Categorical Algebra: Volume 1, Basic Category Theory, Encyclopedia of Mathematics and its Applications, Cambridge, Cambridge University Press. https://doi.org/10.1017/CBO9780511525858.CrossRefGoogle Scholar
Borceux, F. (1994b). Handbook of Categorical Algebra: Volume 2, Categories and Structures, Encyclopedia of Mathematics and its Applications, Cambridge, Cambridge University Press. https://doi.org/10.1017/CBO9780511525865.CrossRefGoogle Scholar
Carboni, A. and Johnstone, P. (1995). Connected limits, familial representability and artin glueing. Mathematical Structures in Computer Science 5 (4) 441–459.CrossRefGoogle Scholar
Clouston, R. (2011). Nominal Lawvere theories. In: WoLLIC’11, 67–83.CrossRefGoogle Scholar
Clouston, R. (2014). Nominal Lawvere theories: A category theoretic account of equational theories with names. Journal of Computer and System Sciences 80 (6), 1067–1086.CrossRefGoogle Scholar
Clouston, R. and Pitts, A.M. (2007). Nominal equational logic. Electronic Notes in Theoretical Computer Science 172 223–257.CrossRefGoogle Scholar
Crole, R. L. (1993). Categories for Types, Cambridge, Cambridge University Press.CrossRefGoogle Scholar
Crole, R. L. (1996). On fixpoint objects and gluing constructions. Applied Categorical Structures 4 (2 & 3), 251–281. This volume is a Special Edition for the European Colloquium on Category Theory, Tours, France.CrossRefGoogle Scholar
Crole, R. L. and Nebel, F. (2013). Nominal lambda calculus: An internal language for FM-cartesian closed categories. In: Mislove, M. (ed.), Proceedings of Twenty-Ninth Conference on the Mathematical Foundations of Programming Semantics (MFPS XXIX), New Orleans, LA, Electronic Notes in Theoretical Computer Science, vol. 298, 93–117.CrossRefGoogle Scholar
Gabbay, M. and Pitts, A. M. (2002). A new approach to abstract syntax with variable binding. Formal Aspects of Computing 13 (3–5) 341–363.CrossRefGoogle Scholar
Gabbay, M. J. A Theory of Inductive Definitions with alpha-Equivalence.Google Scholar
Gabbay, M. J. and Mathijssen, A. (2009). Nominal universal algebra: Equational logic with names and binding. Journal of Logic and Computation 19 (6) 1455–1508.CrossRefGoogle Scholar
Johnstone, P. (1977). Topos Theory, LMS Monographs, Cambridge, MA, Academic Press.Google Scholar
Johnstone, P. T. (2002). Sketches of an Elephant: A Topos Theory Compendium, Oxford Logic Guides, Oxford, Oxford University Press.Google Scholar
Kelly, G. (1982). Basic Concepts of Enriched Category Theory, London Mathematical Society Lecture Note Series, vol. 64, Cambridge, Cambridge University Press.Google Scholar
Kelly, G. (2005). Basic Concepts of Enriched Category Theory. Reprints in Theory and Applications of Categories, No. 10.Google Scholar
Lambek, J. and Scott, P. (1986). Introduction to Higher Order Categorical Logic, Cambridge, Cambridge University Press.Google Scholar
Lane, S. (1998). Categories for the Working Mathematician, Graduate Texts in Mathematics, Berlin, Heidelberg, Springer.Google Scholar
Moerdijk, I. (1982). Glueing topoi and higher order disjunction and existence. In: Troelstra, A. S. and van Dalen, D. (eds.), The L. E. J. Brouwer Centenary Symposium, Studies in Logic and the Foundation of Mathematics, North-Holland, 359–375.CrossRefGoogle Scholar
Mostowski, A. (1939). Uber die Unabhangigkeit des Wohlordnungssatzes vom Ordnungsprinzip. Fundamenta Mathematicae 32 201–252.CrossRefGoogle Scholar
Pitts, A. M. (2003). Nominal logic, a first order theory of names and binding. Information and Computation 186 165–193.CrossRefGoogle Scholar
Pitts, A. M. (2013). Nominal Sets: Names and Symmetry in Computer Science, Cambridge Tracts in Theoretical Computer Science, vol. 57, Cambridge, Cambridge University Press.CrossRefGoogle Scholar