# Inversion, Iteration, and the Art of Dual Wielding

arXiv: https://arxiv.org/abs/1904.01679 · HTML: https://mlsanity.com/paper/1904.01679/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Apr 2019.

## Built on by (most cited first, 1 of 1)

- [Categorical Semantics of Reversible Pattern-Matching](https://mlsanity.com/paper/2109.05837/) (2021)

## Abstract (arXiv)

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.