This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in
homotopical and higher categorical contexts. In this first part we investigate a notion of …

We prove a rectification theorem for enriched∞–categories: if V is a nice monoidal model
category, we show that the homotopy theory of∞–categories enriched in V is equivalent to …

Relative categories: Another model for the homotopy theory of homotopy theories -
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Category theory provides structure for the mathematical world and is seen everywhere in
modern mathematics. With this book, the author bridges the gap between pure category …

We propose a notion of weak (n+ k, n)–category, which we call (n+ k, n)–Θ–spaces. The (n+
k, n)–Θ–spaces are precisely the fibrant objects of a certain model category structure on the …

The study of higher categories is attracting growing interest for its many applications in
topology, algebraic geometry, mathematical physics and category theory. In this highly …

We use the terms∞-categories and∞-functors to mean the objects and morphisms in an∞-
cosmos: a simplicially enriched category satisfying a few axioms, reminiscent of an enriched …

This paper is an introduction to the theory of “model categories”, which was devel-oped by
Quillen in [22] and [23]. By definition a model category is just an ordinary category with three …

The aim of this book is to explain modern homotopy theory in a manner accessible to
graduate students yet structured so that experts can skip over numerous linear …

The purpose of this monograph, which is aimed at the graduate level and beyond, is to
obtain a deeper understanding of Quillen's model categories. A model category is a …