homotopy-theory

Formal -category theory is to -category theory what formal category theory is to 1-category theory, that is: a synthetic approach to -categories standing in relation to synthetic homotopy theory as -category theory stands to homotopy theory. Consequently, formal -category theory has also been called synthetic -category theory. There have been two main styles of approaches to formal -category theo…

This page is to record the reference: Local homotopy theory Springer Monographs in Mathematics (2015) on the abstract homotopy theory (model category theory) of presheaves of homotopy types (presenting -topoi) and stable homotopy types (model structures on simplicial presheaves, model structures on sheaves of spectra, Brown-Gersten property, etc.) with an eye towards motivic homotopy theory. Brie…

homotopy theory, (∞,1)-category theory, homotopy type theory flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed… models: topological, simplicial, localic, … see also algebraic topology Introductions Definitions Paths and cylinders Homotopy groups Basic facts Theorems Special and general types group cohomology, nonabelian group cohomology, Lie group cohomology co…

The James construction type is an axiomatization of the James construction in the context of homotopy type theory. As a higher inductive type, the James construction type of a pointed type is given by the following constructors In Rocq pseudocode this becomes Inductive JamesConstruction (A : Type) (a : A) : Type | epsilon : JamesConstruction A a | alpha : A -> JamesConstruction A a -> JamesConstr…

homotopy theory, (∞,1)-category theory, homotopy type theory flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed… models: topological, simplicial, localic, … see also algebraic topology Introductions Definitions Paths and cylinders Homotopy groups Basic facts Theorems Homotopy products are Cartesian products in homotopy theory, hence are a special case of homotop…

Homotopy coproducts are a special case of homotopy colimits, when the indexing diagram is a discrete category. Homotopy coproducts can be defined in any relative category, just like homotopy colimits, but practical computations are typically carried out in presence of additional structures such as model structures. In any model category, the homotopy coproduct of a family of objects can be comput…

Dominic Verity is a British higher category theorist, based in Australia. He is an Emeritus Professor at Macquarie University. Verity has worked on the theory of complicial sets and their weak analogues, which followed up on ideas of John Roberts on cohomology and, effectively, omega-category theory. More recently he has worked with Emily Riehl on foundations of -category theory seen through thei…

Emily Riehl is Professor in the Department of Mathematics at Johns Hopkins University. On rigidification of quasi-categories: On topos levels in simplicial sets and cubical sets: On Reedy model structures via weighted colimits: An introductory category theory textbook for beginning graduate students or advanced undergraduates with an emphasis on applications of categorical concepts to a variety o…