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

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.