A diagrammatic approach to symmetric lenses

Abstract

Lenses are a mathematical structure for maintaining consistency between a pair of systems. In their ongoing research program, Johnson and Rosebrugh have sought to unify the treatment of symmetric lenses with spans of asymmetric lenses. This paper presents a diagrammatic approach to symmetric lenses between categories, through representing the propagation operations with Mealy morphisms. The central result of this paper is to demonstrate that the bicategory of symmetric lenses is locally adjoint to the bicategory of spans of asymmetric lenses, through constructing an explicit adjoint triple between the hom-categories.

Delta lenses as coalgebras for a comonad

“…In recent work, it has been demonstrated that delta lenses arise as algebras for a monad on Cat/B, providing a dual to the main result of this paper and strengthening the previous work of Johnson and Rosebrugh [12]. Finally, given the importance of the category Lens(B) in the study of symmetric lenses [13,6], it is also hoped that the coalgebraic perspective provides new insights into this area, and this will be the subject of further investigation.…”

“…Therefore this paper introduces a new category Cof(B), whose objects are cofunctors into a fixed category B (Definition 5). The category Lens(B), whose objects are delta lenses into a fixed category B, was previously studied in [13,6]. Surprisingly, we show that the category Lens(B) can be defined (Definition 7) as the slice category Cof(B)/1 B .…”

An orthogonal approach to algebraic weak factorisation systems

“…Cofunctors 1 are a natural kind of morphism between categories [2,22] which fundamentally involve a lifting operation and admit a straightforward sequential composition. The characterisation of delta lenses as a compatible functor and cofunctor [3,7], together with related characterisations of state-based lenses and split opfibrations [8], provides a clear understanding of their composition and lifting, and has led to several fruitful developments in the study of lenses in applied category theory [6,10,14]. However the question remains: why do lenses frequently arise as algebras for a monad?…”

The more legs the merrier: A new composition for symmetric (multi-)lenses

“…Definition 2 This formulation is due to Clarke [4]: A (representative for a) symmetric lens is a span of asymmetric lenses as shown,…”