A Synthetic Perspective on (infinity,1)-Category Theory: Fibrational and Semantic Aspects

Jonathan Weinberger

The field of category theory seeks to unify and generalize constructions from mathematical areas such as algebra, topology, and geometry. E.g. in modern homotopy theory and algebraic geometry, higher-dimensional versions of ordinary categories play an important role. These go by the name of infinity-categories. Since these structures morally are less of set-theoretic but rather of homotopy-theoretic nature one can seek to develop their theory in a different foundational language than set theory, more faithful to their homotopical flavor.

In our thesis, we develop parts of infinity-category theory in such a synthetic setting, namely simplicial homotopy type theory due to Riehl–Shulman. In this theory, the basic objects are not sets but a kind of generalized space. The condition of being an infinity-category can then be stated within the theory in relatively simple terms. This enables a synthetic and intrinsic development of infinity-category theory which is invariant under homotopy equivalence. Our main developments include synthetic versions of fibered infinity-categories, i.e., families of infinity-categories parametrized by another infinity-category. Our work is informed in an essential way by Riehl--Verity's far-reaching program of infinity-cosmos theory which develops higher category theory in a model-independent way. Building on previous joint work with Buchholtz, we obtain type-theoretic versions of results known from infinity-cosmos theory. Importantly, this includes the fibered Yoneda Lemma and various closure properties, valid for different notions of fibrations. We also obtain synthetic and higher versions of results about extensive fibrations supporting a well-behaved notion of sum, classically due to Bénabou, Jibladze, Moens, and Streicher. We conclude by discussing the semantic meaning of our synthetic treatment which shows that it can even be interpreted as a theory about infinity-categories internal to a given an ambient infinity-topos, due to results by Shulman, Riehl–Verity, and Rasekh.

https://tuprints.ulb.tu-darmstadt.de/20716/