# A categorified excision principle for elliptic symbol families

## TL;DR

This paper introduces a categorical index calculus for elliptic symbol families, providing an excision principle to compare categorified index problems across different manifolds, aiding in orientation problems and related issues.

## Contribution

It develops a novel categorified index calculus and an excision principle for elliptic symbol families, extending traditional index theory to a categorical framework.

## Key findings

- Established a categorified index calculus for elliptic symbol families
- Proved an excision principle for comparing index problems on different manifolds
- Enabled solutions to orientation problems in moduli spaces

## Abstract

We develop a categorical index calculus for elliptic symbol families. The categorified index problems we consider are a secondary version of the traditional problem of expressing the index class in K-theory in terms of differential-topological data. They include orientation problems for moduli spaces as well as similar problems for skew-adjoint and self-adjoint operators. The main result of this paper is an excision principle which allows the comparison of categorified index problems on different manifolds. Excision is a powerful technique for actually solving the orientation problem; applications appear in the companion papers arXiv:1811.01096, arXiv:1811.02405, and arXiv:1811.09658.