Higher Topos Theory
Book about ∞-categories and ∞-topoi by Jacob Lurie
Higher Topos Theory is a treatise on the theory of ∞-categories written by American mathematician Jacob Lurie. In addition to introducing Lurie's new theory of ∞-topoi, the book is widely considered foundational to higher category theory. Since 2018, Lurie has been transferring the contents of Higher Topos Theory (along with new material) to Kerodon, an "online resource for homotopy-coherent mathematics" inspired by the Stacks Project.
Topics
Higher Topos Theory covers two related topics: ∞-categories and ∞-topoi (which are a special case of the former). The first five of the book's seven chapters comprise a rigorous development of general ∞-category theory in the language of quasicategories, a special class of simplicial set which acts as a model for ∞-categories. The path of this development largely parallels classical category theory, with the notable exception of the ∞-categorical Grothendieck construction; this correspondence, which Lurie refers to as "straightening and unstraightening", gains considerable importance in his treatment.
The last two chapters are devoted to ∞-topoi, Lurie's own invention and the ∞-categorical analogue of topoi in classical category theory. The material of these chapters is original, and is adapted from an earlier preprint of Lurie's. There are also appendices discussing background material on categories, model categories, and simplicial categories.
History
Higher Topos Theory followed an earlier work by Lurie, On Infinity Topoi, uploaded to the arXiv in 2003.
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