Doctor of Philosophy
Kapulkin, Krzysztof R.
This thesis concerns model structures on presheaf categories, modeling the theory of infinity-categories. We introduce the categories of simplicial and cubical sets, and review established examples of model structures on these categories for infinity-groupoids and (infinity, 1)-categories, including the Quillen and Joyal model structures on simplicial sets, and the Grothendieck model structure on cubical sets. We also review the complicial model structure on marked simplicial sets, which presents the theory of (infinity, n)-categories. We then construct a model structure on the category of cubical sets whose cofibrations are the monomor- phisms and whose fibrant objects are defined by the right lifting property with respect to inner open boxes, the cubical analogue of inner horns. We show that this model structure is Quillen equivalent to the Joyal model structure on simplicial sets via the triangulation functor. To do this, we develop a theory of cones in cubical sets. As an application, we show that cubical quasicategories admit a convenient notion of a mapping space, which we use to characterize the weak equivalences between fibrant objects in our model structure. We also develop model structures for (infinity,n)-categories on marked cubical sets, and show that these are equivalent to the complicial model structures on marked simplicial sets.
Summary for Lay Audience
The field of higher category theory, which studies abstract mathematical objects known as higher categories or infinity-categories, has applications to a wide range of mathematical disciplines. While many of its key motivating examples come from the study of topological spaces, higher category theory has also found applications to areas of pure mathematics such as formal logic, algebra, and geometry, as well as to theoretical physics and computer science. Many of the most successful frameworks for the study of higher category theory make use of simplices, higher-dimensional shapes analogous to the triangle and the tetrahedron. In recent years, there has been significant interest in developing a cubical framework for higher category theory – one which would make use of higher-dimensional analogues of the square and the familiar three-dimensional cube. It is expected that such a framework will have many useful applications to the above-mentioned scientific areas. In this thesis, we begin by reviewing some of the basic theory of higher categories, and the established simplicial models for two specific types of higher categories, known as ∞-groupoids and (infinity, 1)-categories, as well as an established cubical model for the theory of infinity-groupoids. Building on this previous work, we then construct and study a cubical model for the theory of (infinity, 1)- categories. We show that this model is equivalent, in a suitable sense, to the previously-established simplicial model of (infinity, 1)-categories, thereby showing that they do indeed model the same kinds of higher categories. To prove this equivalence, we develop a theory of cubical cones, shapes which are intermediate, in a suitable sense, between simplices and cubes. As an application of our work, we use our cubical framework to construct certain infinity-groupoids known as mapping spaces from a given (infinity, 1)-category, and show how this construction is simplified compared to its traditional simplicial analogue. We also adapt the theory of cubical cones to more general objects, called (infinity,n)-categories.
Doherty, Brandon, "Cubical Models of Higher Categories" (2021). Electronic Thesis and Dissertation Repository. 7930.
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