Introduction

Topos theory grew out of Grothendieck's work on étale cohomology. Its basic objects are categories of sheaves, which encode how local data can be glued. Higher topos theory replaces set-valued sheaves by sheaves of animae, so that both the objects and their descent data retain their full homotopy-coherent structure. Consequently, the resulting notion of topos carries an internal homotopy theory, including truncations and connected covers, loop and delooping constructions, homotopy group objects, and Eilenberg–MacLane objects. In this way, the higher-categorical notion of a topos directly captures the cohomology theory, without a need to talk about derived functors.

In these notes, we take this higher-categorical notion as the primary one. Classical topos theory appears as the \(0\)-truncated part of the higher theory: Given a topos \(T\), the full subcategory \(T_{\leq 0}\) of \(0\)-truncated objects is a classical topos. We accordingly use “topos” for the higher notion and “classical topos” when we specifically mean the classical one.

As a rough guide, higher topos theory may be viewed as classical topos theory with sets replaced by animae:

As will become clear during the course, this analogy does not capture the entire theory, and the left-hand side is a lot richer than the right-hand side suggests. For example, higher topoi support approximation procedures, such as Postnikov and Goodwillie towers, which cannot be seen within the classical topos \(T_{\leq 0}\) alone.

Descent

The organizing idea in topos theory is descent: reconstructing global objects from local data together with gluing information.

Consider first the familiar setting of vector bundles on a manifold \(M\). Given an open cover \(\{U_i\}\) of \(M\), a vector bundle on \(M\) is completely determined by its restrictions to each open set \(U_i\), together with transition functions on the overlaps \(U_i \cap U_j\) satisfying the usual cocycle condition on triple overlaps. In categorical terms, the category \(\Vect(M)\) of vector bundles on \(M\) satisfies descent for open covers.

This same pattern appears in algebraic geometry. A scheme is built by gluing affine pieces \(\Spec(R)\) along Zariski open subschemes. For an affine scheme \(\Spec(R)\), a quasi-coherent sheaf is simply an \(R\)-module. For a general scheme \(X\) obtained by gluing affines, the category \(\QCoh(X)\) of quasi-coherent sheaves on \(X\) can be defined via descent: we specify compatible \(R\)-modules for all affine maps \(\Spec(R) \to X\), where these modules must satisfy gluing conditions on overlaps. The assignment \(X \mapsto \QCoh(X)\) thus satisfies descent for affine covers, by construction.

These examples motivate the classical notion of a Grothendieck topos. One starts with a small category \(C\) of “local models” (such as open subsets of a topological space, or affine schemes), equips it with a notion of “coverings” generalizing open covers, and forms the category of sheaves on \(C\) with respect to these coverings. This sheaf category is a left exact localization of the presheaf category of \(C\). The presheaf category freely adds colimits to the local models, while the localization imposes the relations expressing that covering families glue effectively.

But the presentation in terms of a category \(C\) with coverings is not intrinsic to the resulting topos: different choices of \((C, \text{coverings})\) can give rise to equivalent sheaf categories. The invariant content is the descent property itself, and this can be recognized directly on the category. An object parametrized by \(X \in T\) is an object of the slice category \(T_{/X}\). If \(X\) is exhibited as the colimit of a diagram \(X_{\bullet}\), such an object restricts to a compatible family of objects parametrized by the \(X_i\); descent says that every compatible family arises uniquely in this way. Equivalently, the assignment \(X \mapsto T_{/X}\) takes colimits in \(T\) to limits of categories. This leads to the definition of a topos as a presentable category satisfying descent for all colimits.

The geometric viewpoint

Even though a topos is defined as a category satisfying descent, we often think of it as a geometric object in its own right. This viewpoint is justified already by the basic example coming from topology.

Indeed, every topological space \(X\) gives rise to a topos \(\Shv(X)\), its category of sheaves of animae. This assignment determines a functor \(\Top \to \Topos\), which is fully faithful when restricted to sober spaces, including all Hausdorff spaces. Thus we may “do topology” by studying topoi rather than topological spaces. In fact, the terminology “topos” itself was chosen by the Grothendieck school to reflect the view that topology is fundamentally the study of topoi [McLarty 1990].

Classical geometric structures generalize naturally to this setting. For instance, the notion of a locally ringed space, a topological space equipped with a structure sheaf of rings, extends to that of a locally ringed topos. One can develop a theory of schemes as certain locally ringed topoi, recovering classical algebraic geometry while allowing for higher categorical phenomena.

Taking the geometric viewpoint seriously leads to a natural convention regarding morphisms. Every continuous map \(f\colon X \to Y\) of topological spaces induces a pullback functor \(f^*\colon \Shv(Y) \to \Shv(X)\). This functor is left exact and preserves all colimits; in other words, it preserves the descent structure on the sheaf categories. Given topoi \(S\) and \(T\), a natural class of functors \(f^*\colon S \to T\) between them is the class of left exact colimit-preserving functors. Following the geometric viewpoint, we treat such a functor as a morphism of topoi from \(T\) to \(S\), rather than the other way around.

Content

The notes follow three intertwined developments. The first concerns the intrinsic structure of a single topos. In Chapter 2, we formulate descent as an exactness property of colimits, identify effective epimorphisms as the intrinsic covers, and prove the Giraud and presheaf-localization characterizations of topoi. Chapter 3 then develops the internal homotopy theory made possible by this exactness: connected–truncated factorization systems, homotopy group objects, hypercompletion, and gerbes.

The second development concerns morphisms, quotients, and approximation procedures. In Chapter 4, the algebraic perspective of logoi leads to presentations by generators and relations and hence to limits and colimits of topoi, while the geometric perspective identifies étale topoi over \(T\) with objects of \(T\). Chapter 5 studies fiberwise factorization systems and their relation to left exact localizations. The generalized Blakers–Massey theorem, the description of generated congruences by iterated diagonals, and products of modalities culminate in the Goodwillie tower of a topos.

The third development applies the general theory and explores further structural and geometric topics. Chapter 6 treats sites, groupoid actions and principal bundles, hypercovers, convergence and dimension, shape, coherence and pretopoi, exponentiable topoi, and parametrized objects. Finally, Chapter 7 explains how schemes and other locally defined geometric objects are encoded by structured and fractured topoi, how six-functor formalisms produce such structures, and how these ideas appear in analytic stacks and Gestalten.

Appendices. Appendix A collects background on factorization systems and saturated classes used throughout the text. Appendix B records basic facts about limits and colimits in categories of sections of cartesian and cocartesian fibrations.