A Higher Structure Identity Principle

We prove a Structure Identity Principle for theories defined on types of h-level 3 by defining a general notion of saturation for a large class of structures definable in the Univalent Foundations. Formalizing mathematics in the framework of the Univalent Foundations [Uni13] presents unique challenges and opportunities. One of the main opportunities is the ability to formalize higher-level mathematics (categories, higher categories etc.) in an invariant way by imposing appropriate saturation conditions. This is done, for example, in [Uni13] for category theory (cf. Definition 9.1.6). An interesting challenge is how to express general saturation conditions that apply to wide classes of structures that can be formalized in UF. The main contribution of this paper is to provide such a general definition of a saturation condition for a wide class of definable structures and to use it to prove a Structure Identity Principle for all “category-level” (or “3-level”) such structures. To illustrate, consider the following theorem from [Uni13]: Theorem 0.1 ([Uni13], Theorem 9.4.16). For any univalent categories C and D, the type of categorical equivalences C ≃cat D is equivalent to C =UniCat D. By regarding univalent categories as a “saturated” version of an unsaturated structure (i.e. of precategories) we can regard the above result as a specific instance of a more general result of the following form: Pre-Theorem 0.2. For any saturated models M and N of an L-theory T, the type of L-equivalences M ≃L N is equivalent to M =SatMod T N . The purpose of this paper is to make precise and prove a result of this form, such that Theorem 9.4.16 follows as a special case (as an assurance of adequacy). This is done as follows. An “L-theory T” will be given by a theory over a FOLDS signature L in the sense of Makkai [Mak95], i.e. a finite inverse category L. A “model M” of T (or just simply an “L-structure”) will be given by an interpretation of FOLDS into HoTT as has been described in [Tse16]. An “L-equivalence” between two such modelsM and N will be given by Makkai’s notion of FOLDS L-equivalence, which in set theory is defined as the existence of a span of fiberwise surjective Lhomomorphisms from M to N . What remains is to define what a “saturated” model is in this setting and indeed this is the main contribution of this paper. Date: February 28, 2017. 2010 Mathematics Subject Classification. 03G99, 03B15, 03B22, 03C99.