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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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Related concepts (8)

Infinity

Infinity is something which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes.

Euclidean plane

In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted E2. It is a geometric space in which two real numbers are required to determine the position of each point. It is an affine space, which includes in particular the concept of parallel lines. It has also metrical properties induced by a distance, which allows to define circles, and angle measurement. A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane.

Homotopy

In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from ὁμός "same, similar" and τόπος "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (həˈmɒtəpiː, ; ˈhoʊmoʊˌtoʊpiː, ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces.