A Categorified Excision Principle for Elliptic Symbol Families

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 that allows the comparison of categorified index problems on different manifolds. Excision is a powerful technique for actually solving the orientation problem; applications will appear in companion papers Joyce–Tanaka–Upmeier [16], Joyce–Upmeier [17] and Cao–Gross–Joyce [8].