TY - GEN
T1 - Inversion, Iteration, and the Art of Dual Wielding
AU - Kaarsgaard, Robin
PY - 2019
Y1 - 2019
N2 - The humble (Formula presented) (“dagger”) is used to denote two different operations in category theory: Taking the adjoint of a morphism (in dagger categories) and finding the least fixed point of a functional (in categories enriched in domains). While these two operations are usually considered separately from one another, the emergence of reversible notions of computation shows the need to consider how the two ought to interact. In the present paper, we wield both of these daggers at once and consider dagger categories enriched in domains. We develop a notion of a monotone dagger structure as a dagger structure that is well behaved with respect to the enrichment, and show that such a structure leads to pleasant inversion properties of the fixed points that arise as a result. Notably, such a structure guarantees the existence of fixed point adjoints, which we show are intimately related to the conjugates arising from a canonical involutive monoidal structure in the enrichment. Finally, we relate the results to applications in the design and semantics of reversible programming languages.
AB - The humble (Formula presented) (“dagger”) is used to denote two different operations in category theory: Taking the adjoint of a morphism (in dagger categories) and finding the least fixed point of a functional (in categories enriched in domains). While these two operations are usually considered separately from one another, the emergence of reversible notions of computation shows the need to consider how the two ought to interact. In the present paper, we wield both of these daggers at once and consider dagger categories enriched in domains. We develop a notion of a monotone dagger structure as a dagger structure that is well behaved with respect to the enrichment, and show that such a structure leads to pleasant inversion properties of the fixed points that arise as a result. Notably, such a structure guarantees the existence of fixed point adjoints, which we show are intimately related to the conjugates arising from a canonical involutive monoidal structure in the enrichment. Finally, we relate the results to applications in the design and semantics of reversible programming languages.
KW - Dagger categories
KW - Domain theory
KW - Enriched categories
KW - Iteration categories
KW - Reversible computing
UR - http://www.scopus.com/inward/record.url?scp=85068232293&partnerID=8YFLogxK
U2 - 10.1007/978-3-030-21500-2_3
DO - 10.1007/978-3-030-21500-2_3
M3 - Article in proceedings
AN - SCOPUS:85068232293
SN - 9783030214999
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 34
EP - 50
BT - Reversible Computation - 11th International Conference, RC 2019, Proceedings
A2 - Soeken, Mathias
A2 - Thomsen, Michael Kirkedal
PB - Springer
T2 - 11th International Conference on Reversible Computation, RC 2019
Y2 - 24 June 2019 through 25 June 2019
ER -