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Publication# Homotopical relations between 2-dimensional categories and their infinity-analogues

Abstract

In this thesis, we study the homotopical relations of 2-categories, double categories, and their infinity-analogues. For this, we construct homotopy theories for the objects of interest, and show that there are homotopically full embeddings of 2-categories into (infinity,2)-categories, and of double categories into double (infinity,1)-categories, which are compatible with the inclusions of 2-categories and (infinity,2)-categories into their double categorical analogues.

In the strict setting, we first present two model structures on the category of double categories and double functors, constructed in papers by the author, Sarazola, and Verdugo. Unlike previously defined model structures for double categories, they recover Lack's model structure for 2-categories. More precisely, the horizontal embedding functor from 2-categories to double categories is homotopically well-behaved, and embeds the homotopy theory of 2-categories into that of double categories in a reflective way. While, in the first model structure, all double categories are fibrant, the fibrant objects of the second model structure are the weakly horizontally invariant double categories. We show that both model structures are enriched over 2-categories, and that the model structure for weakly horizontally invariant double categories is further monoidal with respect to the Gray tensor product for double categories.

Going to the infinity-world, we then consider infinity-versions of these 2-dimensional categories. Double (infinity,1)-categories are defined as double Segal objects in spaces which are complete in the horizontal direction, and hence include (infinity,2)-categories in the form of Barwick's 2-fold complete Segal spaces. We then construct a nerve from double categories to double (infinity,1)-categories, and show that it embeds the homotopy theory for weakly horizontally invariant double categories into that of double (infinity,1)-categories in a reflective way. Finally, by restricting the nerve along the horizontal embedding, we obtain a nerve from 2-categories to 2-fold complete Segal spaces, which again embeds the homotopy theory of 2-categories into that of (infinity,2)-categories in a reflective way.

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In mathematics, higher category theory is the part of at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Higher category theory is often applied in algebraic topology (especially in homotopy theory), where one studies algebraic invariants of spaces, such as their fundamental . An ordinary has and morphisms, which are called 1-morphisms in the context of higher category theory.

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In mathematics, particularly in homotopy theory, a model category is a with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstract from the category of topological spaces or of chain complexes ( theory). The concept was introduced by . In recent decades, the language of model categories has been used in some parts of algebraic K-theory and algebraic geometry, where homotopy-theoretic approaches led to deep results.

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In algebraic geometry and algebraic topology, branches of mathematics, A1 homotopy theory or motivic homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties and, more generally, to schemes. The theory is due to Fabien Morel and Vladimir Voevodsky. The underlying idea is that it should be possible to develop a purely algebraic approach to homotopy theory by replacing the unit interval [0, 1], which is not an algebraic variety, with the affine line A1, which is.

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