Concept# Category theory

Summary

Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in almost all areas of mathematics. In particular, numerous constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality.
Many areas of computer science also rely on category theory, such as functional programming and semantics.
A is formed by two sorts of objects: the s of the category, and the morphisms, which relate two objects called the source and the target of the morphism. One often says that a morphism is an arrow that maps its source to its target. Morphisms can be composed if the target of the first morphism equals the source of the second one,

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Kan spectra provide a combinatorial model for the stable homotopy category. They were introduced by Dan Kan in 1963 under the name semisimplicial spectra. A Kan spectrum is similar to a pointed simplicial set, but it has simplices in negative degrees as well and all its simplices have infinitely many faces and degeneracies. In the first part of this thesis, we define a functor from the category of Gamma-spaces to the category of Kan spectra without passing through any other category of spectra. We show that the resulting Kan spectrum of a Gamma-space A agrees with the usual object associated to A in the stable homotopy category by comparing it to Bousfield-Friedlander's spectrum construction. In particular, applying our construction to the Gamma-space associated to a symmetric monoidal category provides a combinatorial model of its algebraic K-theory spectrum. For the Gamma-space associated to an abelian group, this yields via the stable Dold-Kan correspondence the unbounded chain complex with the abelian group concentrated in degree zero. The second part of this work concerns group spectra and twisting structures. Group spectra are the group objects in the category of Kan spectra. They provide an algebraic, combinatorial model for the stable homotopy category. We transfer Brown's model structure from the category of Kan spectra to a Quillen equivalent model structure on the category of group spectra. We then construct the analogues of Kan's loop group functor and its right adjoint Wbar together with corresponding classifying bundles, so that the category of Kan spectra becomes a twisted homotopical category in the sense of Farjoun and Hess.

In previous work, we defined the category of functors F-quad, associated to F-2-vector spaces equipped with a nondegenerate quadratic form. In this paper, we define a special family of objects in the category F-quad, named the mixed functors. We give the complete decompositions of two elements of this family that give rise to two new infinite families of simple objects in the category F-quad.

2007This 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.