Title: Segal-type models of higher categories,
Abstract: Higher category theory is an exceedingly active area of research, whose rapid
growth has been driven by its penetration into a diverse range of scientific
fields. Its influence extends through key mathematical disciplines, notably
homotopy theory, algebraic geometry and algebra, mathematical physics, to
encompass important applications in logic, computer science and beyond. Higher
categories provide a unifying language whose greatest strength lies in its
ability to bridge between diverse areas and uncover novel applications.
In this foundational work we introduce a new approach to higher categories.
It builds upon the theory of iterated internal categories, one of the simplest
possible higher categorical structures available, by adopting a novel and
remarkably simple "weak globularity" postulate and demonstrating that the
resulting model provides a fully general theory of weak n-categories. The
latter are among the most complex of the higher structures, and are crucial for
applications. We show that this new model of "weakly globular n-fold
categories" is suitably equivalent to the well studied model of weak
n-categories due to Tamsamani and Simpson.