# Inversion, Iteration, and the Art of Dual Wielding

## TL;DR

This paper explores the interaction of adjoint and fixed point operations in dagger categories enriched in domains, introducing a monotone dagger structure that ensures desirable inversion properties and fixed point adjoints, with implications for reversible programming languages.

## Contribution

It introduces the concept of a monotone dagger structure in enriched dagger categories, linking fixed point adjoints to conjugates and advancing the semantics of reversible computation.

## Key findings

- Monotone dagger structures lead to fixed point inversion properties.
- Existence of fixed point adjoints related to conjugates in involutive monoidal structures.
- Applications to reversible programming language semantics.

## Abstract

The humble $\dagger$ ("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.