The Operadic Spectrum and Obstructions to Spectral Base Change

TL;DR

This paper develops an operadic framework for spectra in algebraic contexts, revealing fundamental obstructions to base change and proposing a universal operadic residue to define a consistent spectral notion.

Contribution

It introduces a new operadic spectrum concept, demonstrates the non-existence of functorial base change, and constructs a universal residue object for spectral invariants.

Findings

Classical spectral invariants lack functorial base change in operadic settings.

A universal operadic residue object induces a well-defined spectrum.

The operadic spectrum reduces to classical spectrum for trivial operads.

Abstract

We introduce an operadic notion of spectrum for algebras over colored operads in a symmetric monoidal category. The construction is defined via a canonical Hochschild-type object together with an operadic residue, which together encode spectral information in a manner compatible with operadic composition. A central result of this work is that classical spectral invariants do not, in general, admit a natural base change in the operadic setting. More precisely, we show that there is no functorial procedure that transports spectra along strong monoidal functors while preserving their expected structural properties. This establishes a fundamental obstruction to spectral base change. To address this issue, we construct a universal operadic residue object and show that it induces a well-defined and functorial notion of operadic spectrum. We further prove that this construction is canonical and reduces to the classical spectrum in the case of the trivial operad. These results provide a conceptual foundation for spectral theory in operadic and higher algebraic contexts, and clarify the limitations of extending classical spectral invariants beyond the linear setting.