Higher topos theory.

*(English)*Zbl 1175.18001
Annals of Mathematics Studies 170. Princeton, NJ: Princeton University Press (ISBN 978-0-691-14049-0/pbk; 978-0-691-14048-3/hbk). xv, 925 p. (2009).

Although cohomology groups \(\text{H}^{n}(X;G)\) of a topological space \(X\) in an abelian group \(G\) are extremely useful, the usual definition of \(\text{H}^{n}(X;G)\) in terms of singular \(G\)-valued cochains on \(X\) is somewhat unenlightening. The motivation behind this book is the question: can we understand the cohomology group \(\text{H}^{n}(X;G)\) in more conceptual terms? It is well known that each element of \(\text{H}^{1}(X;G)\) is conceptually a \(G\)-torsor on \(X\), which is well defined up to isomorphism.

Grothendieck proposed in his infamous letter to Quillen that there should be a theory of \(n\)-stacks on \(X\) for every integer \(n\geq0\). Moreover, for every sheaf of abelian groups \({\mathcal G}\) on \(X\), the cohomology group \(\text{H}_{\text{sheaf}}^{n+1}(X;{\mathcal G})\) should have an interpretation as classifying a special type of \(n\)-stack, namely, the class of \(n\)-gerbes banded by \({\mathcal G}\). When the space \(X\) is a point, the theory of \(n\)-stacks on \(X\) should recover the classical homotopy theory of \(n\)-types. Therefore we should think of an \(n\)-stack in groupoids on a general space \(X\) as a sheaf of \(n\)-types on \(X\).

A \(0\)-stack on a topological space \(X\) is simply a sheaf of sets on \(X\). The totality of such sheaves can be organized into a category \({\mathcal S}hv _{{\mathcal S}et}(X)\), which is the prototype of a Grothendieck topos. The principal objective in this book is to obtain an analogous understanding of the situation for \(n\)-stacks on \(X\). To complete this understanding, the language of higher-order category theory is indispensible. It is usually regarded as a forbidding subject, but a well-behaved fragment of it suffices for the author’s purpose, namely, \((\infty,1)\)-categories.

The book consists of seven chapters, the first four of which should be regarded as genuinely formal. Chapter 5 introduces \(\infty\)-categorical analogues of classical category theory such as presheaves, Pro-categories, Ind-categories, accessible categories, presentable categories and localization. The main theme is that most of the \(\infty\)-categories which appear in nature are determined by small subcategories. Taking advantage of this fact, the author can deduce a number of pleasant results such as an \(\infty\)-categorical version of the adjoint functor theorem. Chapter 6 is the heart of the book, dealing with \(\infty \)-topoi. The main result is an \(\infty\)-analogue of Giraud’s theorem, which claims the equivalence of extrinsic and intrinsic approaches to the subject. Roughly speaking, an \(\infty\)-topos is an \(\infty\)-category which looks like the \(\infty\)-category of all homotopy types. In other words, an \(\infty\)-topos is a world in which one can do homotopy theory. Chapter 7 is devoted to the relationship between the theory of \(\infty\)-topoi and ideas from classical topology. It is shown that, if \(X\) is a paracompact space, then the \(\infty \)-topos of sheaves of homotopy types on \(X\) can be interpreted in terms of the classical homotopy theory of spaces over \(X\). Another main theme is that various ideas from geometric topology such as dimension theory and shape theory can be described naturally in terms of \(\infty\)-topoi. Nonabelian generalizations of classical cohomological results such as Grothendieck’s vanishing theorem for the cohomology of Noetherian topological spaces and the proper base change theorem are also discussed.

The book is kept as self-contained as possible, assuming familiarity with the classical homotopy theory of simplicial sets. The book is accompanied by appendices on category theory, model categories and simplicial categories. It is nicely written, and it should be useful to both experts and novices.

Grothendieck proposed in his infamous letter to Quillen that there should be a theory of \(n\)-stacks on \(X\) for every integer \(n\geq0\). Moreover, for every sheaf of abelian groups \({\mathcal G}\) on \(X\), the cohomology group \(\text{H}_{\text{sheaf}}^{n+1}(X;{\mathcal G})\) should have an interpretation as classifying a special type of \(n\)-stack, namely, the class of \(n\)-gerbes banded by \({\mathcal G}\). When the space \(X\) is a point, the theory of \(n\)-stacks on \(X\) should recover the classical homotopy theory of \(n\)-types. Therefore we should think of an \(n\)-stack in groupoids on a general space \(X\) as a sheaf of \(n\)-types on \(X\).

A \(0\)-stack on a topological space \(X\) is simply a sheaf of sets on \(X\). The totality of such sheaves can be organized into a category \({\mathcal S}hv _{{\mathcal S}et}(X)\), which is the prototype of a Grothendieck topos. The principal objective in this book is to obtain an analogous understanding of the situation for \(n\)-stacks on \(X\). To complete this understanding, the language of higher-order category theory is indispensible. It is usually regarded as a forbidding subject, but a well-behaved fragment of it suffices for the author’s purpose, namely, \((\infty,1)\)-categories.

The book consists of seven chapters, the first four of which should be regarded as genuinely formal. Chapter 5 introduces \(\infty\)-categorical analogues of classical category theory such as presheaves, Pro-categories, Ind-categories, accessible categories, presentable categories and localization. The main theme is that most of the \(\infty\)-categories which appear in nature are determined by small subcategories. Taking advantage of this fact, the author can deduce a number of pleasant results such as an \(\infty\)-categorical version of the adjoint functor theorem. Chapter 6 is the heart of the book, dealing with \(\infty \)-topoi. The main result is an \(\infty\)-analogue of Giraud’s theorem, which claims the equivalence of extrinsic and intrinsic approaches to the subject. Roughly speaking, an \(\infty\)-topos is an \(\infty\)-category which looks like the \(\infty\)-category of all homotopy types. In other words, an \(\infty\)-topos is a world in which one can do homotopy theory. Chapter 7 is devoted to the relationship between the theory of \(\infty\)-topoi and ideas from classical topology. It is shown that, if \(X\) is a paracompact space, then the \(\infty \)-topos of sheaves of homotopy types on \(X\) can be interpreted in terms of the classical homotopy theory of spaces over \(X\). Another main theme is that various ideas from geometric topology such as dimension theory and shape theory can be described naturally in terms of \(\infty\)-topoi. Nonabelian generalizations of classical cohomological results such as Grothendieck’s vanishing theorem for the cohomology of Noetherian topological spaces and the proper base change theorem are also discussed.

The book is kept as self-contained as possible, assuming familiarity with the classical homotopy theory of simplicial sets. The book is accompanied by appendices on category theory, model categories and simplicial categories. It is nicely written, and it should be useful to both experts and novices.

Reviewer: Hirokazu Nishimura (Tsukuba)

##### MSC:

18-02 | Research exposition (monographs, survey articles) pertaining to category theory |

18B25 | Topoi |

18G30 | Simplicial sets; simplicial objects in a category (MSC2010) |

18G55 | Nonabelian homotopical algebra (MSC2010) |

18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |