Talk given on Thursday October 10, 2019 at The Graduate Center. Higher-dimensional categories play an increasing role in many areas of Mathematics, such as topology, geometry, number theory, and logic. Higher categories are often defined analytically, i.e. entirely based on sets and ordinary categories (endowed with some additional structure), even though these objects themselves do not carry any higher-dimensional information. Riehl and Shulman have introduced a synthetic framework for (∞,1)-category theory where, in contrast, the basic objects intrinsically are of homotopical flavor. One motivation in developing such a synthetic theory is to find a language that is closer to ordinary 1-category theory, and hide much of the (mathematical) implementation of higher-dimensional and homotopical objects. We present a variation of this synthetic theory that allows for additional constructions such as opposite categories and twisted arrow spaces. Our approach is crucially based on Licata--Riley--Shulman's fibrational framework for modal type theories. Furthermore, we suggest a notion of synthetic co-/cartesian fibrations, based on Riehl--Shulman's work on synthetic discrete co-/contravariant fibrations. The topics presented are joint work in progress with Ulrik Buchholtz.
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