Higher category theory

Summary

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. A generalizes this by also including 2-morphisms between the 1-morphisms. Continuing this up to n-morphisms between (n − 1)-morphisms gives an n-category. Just as the category known as Cat, which is the and functors is actually a 2-category with natural transformations as its 2-morphisms, the category n-Cat of (small) n-categories is actually an (n + 1)-category. An n-category is defined by induction on n by: A 0-category is a , An (n + 1)-category is a category over the category n-Cat. So a 1-category is just a () category. The structure of Set is the one given by the cartesian product as tensor and a singleton as unit. In fact any category with finite can be given a monoidal structure. The recursive construction of n-Cat works fine because if a category C has finite products, the category of C-enriched categories has finite products too. While this concept is too strict for some purposes in for example, homotopy theory, where "weak" structures arise in the form of higher categories, strict cubical higher homotopy groupoids have also arisen as giving a new foundation for algebraic topology on the border between homology and homotopy theory; see the article Nonabelian algebraic topology, referenced in the book below. Weak n-category In weak n-categories, the associativity and identity conditions are no longer strict (that is, they are not given by equalities), but rather are satisfied up to an isomorphism of the next level. An example in topology is the composition of paths, where the identity and association conditions hold only up to reparameterization, and hence up to homotopy, which is the 2-isomorphism for this 2-category.

Official source

https://en.wikipedia.org/wiki/Higher_category_theory

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Towards an (∞,2)-category of homotopy coherent monads in an ∞-cosmos

Dimitri Zaganidis

This thesis is part of a program initiated by Riehl and Verity to study the category theory of (infinity,1)-categories in a model-independent way. They showed that most models of (infinity,1)-categories form an infinity-cosmos K, which is essentially a category enriched in quasi-categories with some additional structure reminiscent of a category of fibrant objects. Riehl and Verity showed that it is possible to formulate the category theory of (infinity,1)-categories directly with infinity-cosmos axioms. This should also help organize the category theory of (infinity,1)-categories with structure. Given a category K enriched in quasi-categories, we build via a nerve construction a stratified simplicial set N_Mnd(K) whose objects are homotopy coherent monads in K. If two infinity-cosmoi are weakly equivalent, their respective stratified simplicial sets of homotopy coherent monads are also equivalent. We also provide an (infinity,2)-category Adj_r(K) whose objects are homotopy coherent adjunctions in K, that we use to classify the 1-simplices of N_Mnd(K) up to homotopy.

EPFL2017

This thesis, which presents a new approach to the algebraic K-theory, is divided into two parts. The first one is devoted to the category of small simplicial categories. First, we construct a new model structure on sCat = [Δop,Cat] which is called the diagonal model structure, in reference to the diagonal model structure of Moerdijk on bisimplicial sets sSet2. Then we show that the new structure is proper and cellular. Note that this new model structure is not tensored and cotensored over the category of simplicial sets sSet in a manner consistent with the model structure. To remedy this, we use another model structure on sSet2 defined in the article of Cegarra and Remedios [3], which is equivalent to the Moerdijk structure. So we build a second new model structure on [Δop,Cat], which is cofibrantly generated, left proper, cellular and (co)tensored on sSet in a compatible way. Based on the work of [13], we construct the stable category of spectra (not symmetric) SpN(sCat*, Σ). It garantees the existence of Ω-spectra, which allows us to define thenotion of "weak Waldhausen category". The calculation of the simplicial enrichment map of the model category SpN(sCat*, Σ), leads to our new definition of algebraic K-theory of weak Waldhausen categories . The second part of this thesis is an attempt to generalize the previous results for enriched categories. First we begin by recalling the theory of ∞-categories and ∞-groupoids, based on the work of Joyal [14] and Lurie [18]. Then we make comparisons of ∞-categories with the category of simplicial sets equipped with the usual model structure. Our first result is the construction of a model structure on Top – Cat , the category of small categories enriched over the category of topological spaces Top, based on the work of Bergner [1] . The category Top – Cat is Quillen equivalent to sSet – Cat. Note that all objects in Top – Cat are fibrant ; this remark will play an important role in this theory. Our second result is the construction of a new model structure on the category of small simplicial categories enriched over Top, denoted by Top – sCat = [Δop,Top – Cat]. We show that this structure is proper and cellular. The fact that Top – sCat is not (co)tensored over sSet poses a barrier to defining the category of spectra SpN(sCat*, Σ).

EPFL2010