A simplicial foundation for differential and sector forms in tangent categories

TL;DR

This paper introduces a new simplicial framework for differential and sector forms within tangent categories, revealing a rich symmetric cosimplicial structure that generalizes classical differential geometry concepts.

Contribution

It establishes that sector forms in tangent categories form a symmetric cosimplicial object, a novel insight even for smooth manifolds, and connects this to the de Rham complex.

Findings

Sector forms form a symmetric cosimplicial object

The complex of sector forms contains a subcomplex isomorphic to the de Rham complex

A new equational presentation of symmetric cosimplicial objects is developed

Abstract

Tangent categories provide an axiomatic framework for understanding various tangent bundles and differential operations that occur in differential geometry, algebraic geometry, abstract homotopy theory, and computer science. Previous work has shown that one can formulate and prove a wide variety of definitions and results from differential geometry in an arbitrary tangent category, including generalizations of vector fields and their Lie bracket, vector bundles, and connections. In this paper we investigate differential and sector forms in tangent categories. We show that sector forms in any tangent category have a rich structure: they form a symmetric cosimplicial object. This appears to be a new result in differential geometry, even for smooth manifolds. In the category of smooth manifolds, the resulting complex of sector forms has a subcomplex isomorphic to the de Rham complex of differential forms, which may be identified with alternating sector forms. Further, the symmetric cosimplicial structure on sector forms arises naturally through a new equational presentation of symmetric cosimplicial objects, which we develop herein.