A Higher Structure Identity Principle

Benedikt Ahrens; Paige Randall North; Michael Shulman; Dimitris Tsementzis

doi:10.1145/3373718.3394755

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arXiv · 2004.06572

A Higher Structure Identity Principle

Benedikt Ahrens·Paige Randall North·Michael Shulman·Dimitris Tsementzis

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Abstract

The ordinary Structure Identity Principle states that any property of set-level structures (e.g., posets, groups, rings, fields) definable in Univalent Foundations is invariant under isomorphism: more specifically, identifications of structures coincide with isomorphisms. We prove a version of this principle for a wide range of higher-categorical structures, adapting FOLDS-signatures to specify a general class of structures, and using two-level type theory to treat all categorical dimensions uniformly. As in the previously known case of 1-categories (which is an instance of our theory), the structures themselves must satisfy a local univalence principle, stating that identifications coincide with "isomorphisms" between elements of the structure. Our main technical achievement is a definition of such isomorphisms, which we call "indiscernibilities", using only the dependency structure rather than any notion of composition.

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BibTeX RIS

Benedikt Ahrens, Paige Randall North, Michael Shulman, Dimitris Tsementzis. 2020-04-14. A Higher Structure Identity Principle. https://doi.org/10.1145/3373718.3394755

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