Inversion, Iteration, and the Art of Dual Wielding

by   Robin Kaarsgaard, et al.

The humble † ("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.


page 1

page 2

page 3

page 4


A categorical foundation for structured reversible flowchart languages: Soundness and adequacy

Structured reversible flowchart languages is a class of imperative rever...

Parametrized Fixed Points on O-Categories and Applications to Session Types

O-categories generalize categories of domains to provide just the struct...

Fixed-Points for Quantitative Equational Logics

We develop a fixed-point extension of quantitative equational logic and ...

Categorical Semantics of Reversible Pattern-Matching

This paper is concerned with categorical structures for reversible compu...

Truth and Subjunctive Theories of Knowledge: No Luck?

The paper explores applications of Kripke's theory of truth to semantics...

Please sign up or login with your details

Forgot password? Click here to reset