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

A categorified excision principle for elliptic symbol families