2cellsforms

2cellsforms is a term used in some discussions of higher category theory to describe a framework in which two-dimensional morphisms (2-cells) are enriched by differential-form data. In this usage, a bicategory or 2-category is equipped not only with objects, 1-cells, and 2-cells, but also with a graded algebra of differential forms attached to the hom-categories. A 2-cell form can be viewed as a 2-cell paired with a differential form label, and the compositions of 2-cells are required to satisfy compatibility rules that mirror the familiar Leibniz rule from differential geometry.

The formal setup typically includes a bicategory B together with a mechanism that assigns to each hom-category a de Rham-like complex of differential forms and a differential operator d of degree one. The forms interact with the categorical compositions through rules that resemble, in spirit, the Leibniz rule: the differential of a composite 2-cell should relate to the composites of the differentials of its factors, using a wedge-like product on forms. This leads to a structure where the algebra of forms and the 2-cells influence each other, producing a richer invariant than either structure alone.

Examples and applications of the idea are explored in more speculative or exploratory work. In some proposed instances, objects are smooth manifolds, 1-cells are smooth maps, and 2-cells are homotopies; the attached forms may encode curvature, obstruction, or connection data associated with the homotopy. The concept also relates to differential graded or enriched higher categories and to the study of 2-bundles and higher gauge theories.

Overall, 2cellsforms represents an area at the intersection of higher category theory and differential geometry, intended to bridge transformations between morphisms with geometric form data. See also 2-categories, bicategories, differential forms, and higher gauge theory.