Multicategories of multiary lenses

Recently lens-like definitions of multidirectional transformations have appeared [2], encompassing both a "propagation" form (in the style of [5]) and a "wide span" form (in the style of [6]). These "multiary lenses" raise a number of new challenges that are addressed in this paper. First, in common with classical symmetric lenses, they need to be studied modulo equivalence relations that factor out differences in hidden data and declare two lenses to be equivalent if their observable behaviours correspond. Then, modulo those equivalence relations the precise relationship between the propagation style lenses and the wide span lenses needs to be established. All of this is carried through here. But more significantly, the compositions of multiary lenses proposed to date have suffered various limitations: well-behaved amendment lenses don't necessarily compose to give well-behaved amendment lenses; technical conditions on model spaces can interfere with composition; junction conditions needed to be introduced because certain kinds of compositions of multiary lenses might not always be defined; and in any case, composition of multiary lenses comes with unusual challenges and can't be expected to form a category structure since when there are more than two data sources there are not even single notions of domain or codomain for multiary lenses. In this paper we introduce a class of asymmetric amendment lenses called spg-lenses (stable putget lenses) which is more general than well-behaved amendment lenses, but is closed under composition, and we show how to use spg-lenses to capture, via wide spans, a wide class of mutidirectional transformations which compose well and form a well-known and long-standing structure, a multicategory — a multicategory of multiary lenses.