[Paper Review] Limits of bimorphic lenses

Jules Hedges|arXiv (Cornell University)|Aug 16, 2018

Artificial Intelligence in Games8 references4 citations

TL;DR

This paper establishes that the category of bimorphic lenses is complete when the underlying category is complete, cocomplete, and cartesian closed, enabling the construction of spans of bimorphic lenses as a foundation for symmetric polymorphic lenses. The key contribution is a formal construction of pullbacks in the category of bimorphic lenses using pullbacks and pushouts in the base category, which supports the development of compact closed categories for open games in compositional game theory.

ABSTRACT

Bimorphic lenses are a simplification of polymorphic lenses that (like polymorphic lenses) have a type defined by 4 parameters, but which are defined in a monomorphic type system (i.e. an ordinary category with finite products). We show that the category of bimorphic lenses is complete when the base category is complete, cocomplete and cartesian closed, and so symmetric bimorphic lenses can be defined as spans of ordinary bimorphic lenses. This is in contrast to monomorphic lenses, which do not have pullbacks, and for which the category of spans can be defined in an ad-hoc way only when the lenses satisfy a certain axiom (the put-get law). This is a step towards a theory of symmetric polymorphic lenses. Bimorphic lenses additionally play an essential role in compositional game theory, and spans of bimorphic lenses are a step towards a compact closed category of open games.

Motivation & Objective

• To establish the completeness of the category of bimorphic lenses under suitable conditions on the base category.
• To provide a categorical foundation for symmetric polymorphic lenses by constructing spans of bimorphic lenses.
• To support the development of compact closed categories for open games in compositional game theory using bimorphic lenses.
• To bridge monomorphic and polymorphic lenses by introducing bimorphic lenses as a middle ground in a monomorphic type system.
• To formalize the role of bimorphic lenses in game-theoretic semantics and future work on polymorphic lens theories.

Proposed method

• Define bimorphic lenses as pairs of morphisms: a view function $ v_\lambda: S \to A $ and an update function $ u_\lambda: S \times B \to T $ in a category $ \mathcal{C} $ with finite products.
• Construct the category $ \mathbf{Bilens}(\mathcal{C}) $ with pairs of objects as types and bimorphic lenses as morphisms.
• Prove that $ \mathbf{Bilens}(\mathcal{C}) $ is complete when $ \mathcal{C} $ is complete, cocomplete, and cartesian closed.
• Construct pullbacks in $ \mathbf{Bilens}(\mathcal{C}) $ using pullbacks in $ \mathcal{C} $ for view components and pushouts for update components.
• Define the span of bimorphic lenses as a construction relying on pullbacks, enabling symmetric composition.
• Use the universal property of pullbacks and pushouts to prove uniqueness and commutativity of the resulting lens structure.

Experimental results

Research questions

• RQ1 Can the category of bimorphic lenses be shown to be complete under reasonable conditions on the base category?
• RQ2 How can spans of bimorphic lenses be constructed in a way that supports symmetric composition and compact closed structure?
• RQ3 What is the role of bimorphic lenses in enabling a theory of symmetric polymorphic lenses?
• RQ4 How do bimorphic lenses support the construction of compact closed categories for open games in compositional game theory?
• RQ5 What categorical constructions (e.g., pullbacks, pushouts) are necessary and sufficient to define spans in the category of bimorphic lenses?

Key findings

• The category $ \mathbf{Bilens}(\mathcal{C}) $ is complete when $ \mathcal{C} $ is complete, cocomplete, and cartesian closed.
• Pullbacks in $ \mathbf{Bilens}(\mathcal{C}) $ are constructed using pullbacks of view morphisms and pushouts of update morphisms in $ \mathcal{C} $.
• The pullback of a cospan $ \lambda: \binom{S}{T} \to \binom{A}{B} \leftarrow \binom{S'}{T'}: \lambda' $ is given by $ \binom{S \times_A S'}{T +_{(S \times_A S') \times B} T'} $, where $ S \times_A S' $ is the pullback of $ v_\lambda $ and $ v_{\lambda'} $, and the second component is a pushout.
• The universal property of the pullback ensures uniqueness of the induced lens, with view and update morphisms constructed via universal morphisms and colimit preservation.
• The construction of spans in $ \mathbf{Bilens}(\mathcal{C}) $ is well-defined and categorical, avoiding ad-hoc definitions used in monomorphic lens categories.
• The result supports the long-term goal of constructing a compact closed category of open games using spans of bimorphic lenses.