Pursuing Stacks

The manuscript began in correspondence before it became an object with a title. In 1982 Ronald Brown, a topologist at Bangor, encountered a reference to Grothendieck’s interest in higher categories. Brown and Philip Higgins had been developing higher-dimensional forms of the van Kampen theorem, using groupoids and related structures. Before traveling to a conference in Marseille-Luminy, Brown wrote to Grothendieck, sent papers, and asked whether a visit might be possible. The answer did not open a seminar. It offered copies of earlier letters written to Larry Breen about $n$-categories. 11Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”.

The letters were in French, and one was handwritten. Brown prepared an English draft, took it to the conference, and received corrections from Jean-Louis Loday and Breen. He then sent the translation to Grothendieck with comments about the reach and limits of the higher groupoids Brown’s circle had studied. What followed was not a simple acknowledgment. Grothendieck began to reconsider foundations for homotopy types, higher categories, and nonabelian cohomology in sustained English correspondence with Brown and Tim Porter. The language of that exchange helped determine the language of the manuscript. 22Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”; Grothendieck and Brown, Correspondance Grothendieck–Brown.

The open manuscript circulated in installments during 1983 and was last dated November 12, 1983; Chapter 54 follows its composition phases.33Grothendieck, À la Poursuite des Champs (Pursuing Stacks); Scharlau, Spirituality, ch. 22, pp. 2–3.

The title has two forms: Pursuing Stacks and À la poursuite des champs. In French mathematical usage, champ is “stack,” though the ordinary word also means “field.” “Pursuit” accurately captures an investigation that advances by following whatever becomes urgent. But the title is historically misleading if used as a table of contents. As the 2022 editor’s description emphasizes, only the early sections and some later passages directly treat stacks; most of the 1983 typescript concerns homotopy theory, models for homotopy types, test categories, localizers, and related structures. 44Alexandre Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1, ed. Georges Maltsiniotis, Documents Mathématiques 20 (Société Mathématique de France, 2022), https://smf.emath.fr/sites/default/files/2022-06/DM20-version-def__sample.pdf.

By the printed January 1984 terminal date of Esquisse—which remains in unresolved tension with a 1983 catalog label—the pursuit had changed direction. In Esquisse, Grothendieck described the “Modelizing Story” as an unforeseen digression and joked that the famous stacks might not return until around page 1,000. The remark should not be read as confession that the intervening mathematics was irrelevant. The digression became the work. It reveals a method in which foundations are not erected before exploration; they emerge because an intended theory demands objects the existing foundations cannot supply.55Grothendieck, À la Poursuite des Champs (Pursuing Stacks); Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1; Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”.

The need for a stack

Begin with local data. A presheaf assigns information to every open set and restriction maps when one open set lies inside another. A sheaf adds a gluing principle: compatible local sections join to a unique global section. The standard example is a function. If functions on overlapping open sets agree where the opens meet, they glue to one function on their union.

Many geometric problems do not classify functions. They classify objects with symmetries: vector bundles, principal bundles, curves, torsors, or families of such objects. Local objects may be isomorphic on overlaps without being literally equal. The isomorphisms must agree on triple overlaps. Moreover, an object can possess nontrivial automorphisms. A set of isomorphism classes loses this structure.

A stack is designed for this situation. In a useful shorthand, it is a category-valued sheaf satisfying descent: objects and arrows can be glued from compatible local data, and compatibility is expressed up to coherent isomorphism. The shorthand is incomplete because one must specify the site, fibered category or equivalent framework, and descent conditions. It identifies the human problem of assembling local objects when equality is too rigid and symmetry is part of the object.

Suppose a line bundle is trivial on each member of an open cover. Choosing a local trivialization does not make the bundle globally trivial. On overlaps, trivializations differ by invertible functions. Those transition functions satisfy a cocycle relation on triple overlaps. Changing trivializations changes the cocycle by a coboundary. The bundle is not captured by a list of local copies of a line; it is captured by copies plus gluing isomorphisms plus relations. A stack generalizes this pattern when the local objects themselves form categories.66Grothendieck, À la Poursuite des Champs (Pursuing Stacks); Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1; Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”.

The projective line gives a true example. Over a field $k$, cover $\mathbb P^1_k$ by the standard affine charts $U_0$ and $U_\infty$. Their overlap is $\mathbb G_m$, with coordinate $z$. Start with a trivial rank-one bundle on each chart and identify the two copies over the overlap by multiplication with $z^n$. The resulting line bundle is $\mathcal O(n)$. Replacing either trivialization multiplies the transition function by invertible functions from the two charts but does not change the isomorphism class of the glued bundle.

The example separates three levels. The local bundles are objects. Multiplication by $z^n$ is a gluing isomorphism. Changes of trivialization are arrows between two presentations of the same global object. A set containing only isomorphism classes can remember the integer $n$, but it does not retain the automorphisms of each bundle or the arrows needed for descent in families. The stack of line bundles retains that groupoid-valued information.

The boundary is equally important. This two-chart example has no nonempty triple overlap, so the cocycle condition is invisible. A cover with three or more members makes the triple-overlap equation explicit. Higher stacks add further levels of coherent comparison; $\mathcal O(n)$ by itself does not explain those levels.

This anchor belongs here because it connects the late pursuit to Grothendieck’s earlier practice. Descent for bundles and sheaves was established mathematical terrain. The 1983 problem was how to extend such gluing when coefficients themselves carried higher homotopy. The concrete bundle does not prove the higher theory, but it identifies the operation that the theory had to preserve.77Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1; Georges Maltsiniotis, La théorie de l’homotopie de Grothendieck, Astérisque 301 (Société Mathématique de France, 2005).

The phrase “stack of curves” thus does not mean a pile of curves. It means a moduli object that remembers families and automorphisms and satisfies descent. This is why Chapter 55’s moduli of pointed curves naturally require stacks. A coarse moduli space can identify isomorphism classes, but it may forget stabilizer groups. Those groups affect geometry, intersection theory, and gluing. The stack retains them rather than treating symmetry as an inconvenience to be divided out.88Grothendieck, À la Poursuite des Champs (Pursuing Stacks); Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1; Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”.

The idea was not new in 1983. Grothendieck’s earlier work on descent, fibered categories, and nonabelian cohomology had prepared it; Giraud and others had developed stacks. The late manuscript sought a higher extension: $n$-stacks and ultimately $\infty$-stacks that could serve as coefficients for nonabelian cohomology. If ordinary stacks glue objects with isomorphisms, higher stacks must also manage isomorphisms between isomorphisms and coherence at successive levels.99Grothendieck, À la Poursuite des Champs (Pursuing Stacks); Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1; Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”.

From paths to higher groupoids

A groupoid is a category in which every arrow is invertible. A space supplies a fundamental groupoid: points are objects, homotopy classes of paths are arrows, and reversing a path gives an inverse. Choosing one base point reduces this to a fundamental group, but the reduction can hide symmetries among several natural points. Brown’s work stressed the value of keeping a set of base points, especially in van Kampen theorems.

Now refuse to collapse paths to homotopy classes. Keep paths as 1-arrows and homotopies between paths as 2-arrows. Homotopies between homotopies become 3-arrows, and so on. Every arrow is invertible in the appropriate higher sense. The result one seeks is a weak $\infty$-groupoid encoding the space’s entire homotopy type.1010Grothendieck, À la Poursuite des Champs (Pursuing Stacks); Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1; Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”.

“Weak” is indispensable. Path composition is not literally associative in every convenient model; it is associative up to a specified homotopy. That homotopy must satisfy its own coherence, perhaps up to a higher homotopy. A strict higher category demands equations on the nose. A weak one permits coherent higher comparison. The apparent mess is not defective bookkeeping. It reflects the geometry of deformation.

The homotopy hypothesis says, roughly, that homotopy types and weak $\infty$-groupoids are two ways of presenting the same information. A topological space has paths and higher homotopies; a sufficiently good weak $\infty$-groupoid can be realized as a homotopy type. In 1983 this was a guiding hypothesis and a demand for adequate models, not one completed theorem in Grothendieck’s preferred formalism.

The hypothesis connects the title’s two worlds. An $\infty$-stack on a point is an $\infty$-groupoid. Before developing higher stacks on arbitrary topoi, one needs a trustworthy language for the coefficients themselves. The long “digression” into models was therefore foundational preparation. If the model of higher groupoids is too strict, too large, or badly behaved under localization, the intended nonabelian cohomology will inherit the defect.1111Grothendieck, À la Poursuite des Champs (Pursuing Stacks); Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1; Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”.

What “same homotopy type” means

Two spaces can be different as sets and manifolds yet equivalent from the perspective of homotopy. A weak homotopy equivalence induces isomorphisms on all homotopy groups, with appropriate base-point conditions. Model categories and related frameworks formally invert chosen weak equivalences, producing a homotopy category. But the homotopy category alone forgets higher information about maps and diagrams. Grothendieck’s search concerns models in which weak equivalences, higher arrows, and constructions such as homotopy limits remain controlled.1212Grothendieck, À la Poursuite des Champs (Pursuing Stacks); Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1; Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”.

No one formula defines every weak $\infty$-groupoid model. Simplicial sets satisfying Kan conditions, topological spaces, various higher categories, and other structures can present homotopy types. Equivalence among models is itself mathematical work. The safe statement is not “Grothendieck defined the modern $\infty$-groupoid.” It is that he articulated the homotopy hypothesis and explored foundational strategies—including coherators, test categories, and localizers—that strongly influenced later theories.

Shapes on which to test spaces

Classical algebraic topology often models a space using simplices. A simplicial set records vertices, edges, triangles, tetrahedra, and higher simplices with face and degeneracy maps. Cubical sets use cubes. Each family of shapes supports combinatorial constructions. Grothendieck asked more generally which small categories of shapes can model all homotopy types through their presheaves.1313Grothendieck, À la Poursuite des Champs (Pursuing Stacks); Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1; Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”.

Let $A$ be a small category. A presheaf on $A$ assigns data contravariantly to each object of $A$. If $A$ is the simplex category, these presheaves are simplicial sets. With an appropriate class of weak equivalences, their homotopy theory models spaces. A test category is, roughly, an indexing category whose presheaves can serve this universal modeling role.

The change from the observation that simplices work to the classification of categories of shapes is characteristic. The problem concerns which properties of the simplex category are essential, whether cubes or geometry-specific cells work, and how two models can be compared. The goal is not abstraction for its own sake. Different shapes can preserve different combinatorial advantages.

Grothendieck used terms translated as modelizers, contractibility structures, and asphericity structures. He sought axioms that identify when an object behaves like a contractible shape and when a functor should count as a weak equivalence. A basic localizer is a class of functors in the category of small categories satisfying closure properties and playing the role of weak equivalences. Inverting them should produce the desired homotopy theory. 1414Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1.

The word “localization” can sound spatial, but here it is categorical. To localize a category at a class $W$ is to formally make every arrow in $W$ invertible. Fractions provide an analogy: passing from integers to rational numbers makes nonzero integers invertible. The analogy illuminates the formal move and fails if it suggests that categorical localization is always a simple set of fractions. Size, coherence, and existence issues can be substantial.

The manuscript’s ambition was comparative. Simplicial, cubical, hemispherical, and other structures should not remain isolated technologies. A general theory would say when they present the same homotopy types and how constructions transfer. Later work by Maltsiniotis and Cisinski developed this line with proofs and modern formalism. Their books are not polished summaries of a finished 1983 theory; they are major mathematical continuations. 1515Maltsiniotis, La théorie de l’homotopie de Grothendieck; Denis-Charles Cisinski, Les préfaisceaux comme modèles des types d’homotopie, Astérisque 308 (Société Mathématique de France, 2006).

Coherence and the temptation of strictness

Higher mathematics repeatedly faces a bargain. Strict structures are easier to define and manipulate: associativity is an equation, units behave exactly, diagrams commute. Geometric constructions often supply these properties only up to homotopy. One can try to replace every weak structure by an equivalent strict one, but such strictification works only under specific conditions and may fail in high dimensions.

Grothendieck’s proposed “coherators” were intended to generate the operations and coherence needed for weak higher groupoids. One begins with a category of formal pasting shapes and adds fillers or lifting data so that compositions exist with coherent higher relations. The details are difficult and evolved after him. The historical point is that he recognized coherence as the central object, not debris to be swept away after defining composition.1616Grothendieck, À la Poursuite des Champs (Pursuing Stacks); Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1; Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”.

This recognition joined older and newer concerns. Descent already required cocycle compatibility. Stacks replaced equality with isomorphism. Higher stacks replace single isomorphisms with towers of comparison. Homotopy theory treats deformation as structure. The same movement appears at increasing height: local objects, arrows between them, arrows between arrows, and rules ensuring that many routes through the data agree appropriately.

A manuscript in the mail

The mathematics traveled in installments. Brown had asked whether he could duplicate material Grothendieck sent him, and Grothendieck agreed. Brown forwarded pages to a small number of interested people; Larry Breen also received material. Others entered correspondence as the manuscript became known. Bangor was not a publishing house. It was a node in a permission-based copying network. 1717Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”.

This mode of circulation shaped reception. A recipient encountered a recent typescript, perhaps before later sections existed. Questions could return to the author. Brown and Porter supplied references and objections from traditions Grothendieck did not follow closely. The manuscript’s English made the dialogue visible. It also contains turns toward family events and daily life, reminders that the pages remained letters even as their mathematics expanded.1818Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”; Grothendieck and Brown, Correspondance Grothendieck–Brown.

The epistolary style differs from a textbook. Definitions may be provisional. A theme disappears and returns. Grothendieck comments on his own ignorance of literature. He promises future volumes. Errors remain. The openness can make the text unusually close to discovery and unusually hard to cite as authority. A later reader must distinguish a speculation on one page from a stable definition adopted hundreds of pages later.1919Grothendieck, À la Poursuite des Champs (Pursuing Stacks); Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1; Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”.

The style also complicates editing. Correcting an error makes the mathematics easier to use but may falsify the documentary record if the correction is silent. Preserving every error respects the manuscript but burdens students. An edition can solve the dilemma through apparatus: retain authorial text, mark corrections, supply commentary, and index concepts. That work is slow because the editor must understand both mathematics and sequence.2020Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1.

In September 1983, Grothendieck referred to a promise to publisher Pierre Berès and hoped a first volume might be ready by year’s end. It was not. The statement records publication intention, not a publication. The proposed series Réflexions Mathématiques would have included Pursuing Stacks volumes and other late projects. The enormous 1983 typescript remained outside formal print for decades. 2121Scharlau, Spirituality, ch. 22, p. 3.

Correspondence continued after the main dated sequence. Hans-Joachim Baues invited Grothendieck to a small Bonn meeting; he declined, explaining his aversion to learned gatherings. Brown later visited. In 1986 Grothendieck expressed appreciation of higher van Kampen questions. These contacts disprove a picture of mathematical silence while confirming that he avoided ordinary conference life.2222Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”; Grothendieck and Brown, Correspondance Grothendieck–Brown.

The network was selective and fragile. If one recipient lacked time, a line of development could stop. If a photocopy was poor, notation became uncertain. If no publisher fixed pagination, references used manuscript sections or local copies. Informal circulation can move faster than publication, but preservation and discoverability remain uneven. Pursuing Stacks demonstrates both advantages.2323Grothendieck, À la Poursuite des Champs (Pursuing Stacks); Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1; Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”.

What the title does not contain

The Société Mathématique de France’s description of the edited Volume I offers an important corrective. Only the first 13 sections and parts of sections 15–21 and 27 concern stacks in a direct sense. Much of the 140-section project deals with homotopy theory: models of homotopy types, presheaf categories, test functors, asphericity, weak equivalences, model structures, derivators, and integration or cointegration. 2424Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1.

The publisher’s summary also notes that reflections on $\infty$-stacks over arbitrary topoi remain largely heuristic. Grothendieck planned one or two further volumes returning to them and incorporating the earlier Breen letters. He did not write that completed sequel. The phrase “Pursuing Stacks founded higher stack theory” is therefore too blunt. The manuscript supplied powerful proposals and vocabulary, especially around weak $\infty$-groupoids and homotopy models; other mathematicians built modern higher-stack theories through many frameworks.2525Grothendieck, À la Poursuite des Champs (Pursuing Stacks); Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1; Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”.

This limit does not diminish the work. A pursuit can matter because it discovers that the intended object requires a new continent of foundations. But historical credit must attach to what is present. The text does not contain every later definition of an $(\infty,1)$-category, model stack, derived stack, or higher topos. Similarity of ambition is not identity of theorem.2626Grothendieck, À la Poursuite des Champs (Pursuing Stacks); Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1; Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”.

Descent before height

The higher project becomes clearer by returning to ordinary descent. Let a space or scheme $X$ be covered by pieces $U_i$. To construct a vector bundle globally, choose bundles $E_i$ on each piece and isomorphisms

$$\varphi_{ij}:E_j|_{U_i\cap U_j}\longrightarrow E_i|_{U_i\cap U_j}.$$

On triple overlaps, the isomorphisms must satisfy

$$\varphi_{ij}\circ\varphi_{jk}=\varphi_{ik}.$$

This cocycle equation is not decorative. Without it, pairwise agreements may fail to produce one global object.

Now allow morphisms between two such descent data. They too must agree on overlaps. The collection of bundles is therefore categorical: objects have automorphisms, and arrows between objects matter. The stack condition says, in an appropriate language, that this descent data is effective and that arrows descend as well.2727Grothendieck, À la Poursuite des Champs (Pursuing Stacks); Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1; Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”.

For gerbes and higher stacks, one encounters data coherent only up to a specified 2-isomorphism. The triple-overlap equation may be replaced by a comparison arrow, and quadruple overlaps impose a relation among comparisons. At the next height, that relation may itself hold up to higher data. This iterative pattern explains why higher coherence cannot be added as one final correction.

Nonabelian cohomology motivates the complexity. Ordinary cohomology with abelian coefficients benefits from addition and commutativity. Torsors under a nonabelian group already form a pointed set rather than a group in low degree. Stacks and higher stacks offer coefficients rich enough to classify bundles, gerbes, and more elaborate objects while respecting automorphisms.2828Grothendieck, À la Poursuite des Champs (Pursuing Stacks); Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1; Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”.

Grothendieck’s earlier geometry had normalized descent as a method: construct an object after a covering, then determine whether it descends to the base. Pursuing Stacks extends the question vertically to the coefficients needed for higher transformations, the homotopy type represented by a stack, and the integration of cohomology from local higher data.2929Grothendieck, À la Poursuite des Champs (Pursuing Stacks); Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1; Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”.

The manuscript did not finish this theory on arbitrary topoi. That negative fact is mathematically informative. Before the theory could be written, Grothendieck judged that the available account of homotopy types and higher groupoids was inadequate. The foundation swallowed the application.3030Grothendieck, À la Poursuite des Champs (Pursuing Stacks); Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1; Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”.

Strict, weak, and what can be modeled

Brown’s initial comments emphasized that some strict higher groupoids then under discussion modeled restricted homotopy types, not all of them in the form Grothendieck wanted. The correspondence later refined the issue as Brown and Loday developed results for multiple groupoids. This is a case where “strict versus weak” cannot be summarized by saying strict models never work.

Different strict structures model different ranges. Crossed modules model connected homotopy 2-types. Crossed complexes and cubical higher groupoids with connections can encode broad homotopical information. Strict globular $\infty$-groupoids face other limitations. The shape, composition laws, and extra operations matter. 3131Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”.

Grothendieck’s broad response was to seek weak structures capable of all homotopy types. Weakness does not mean that axioms are optional. It means equations such as associativity are replaced by coherent higher equivalences. The coherence laws become more numerous and, paradoxically, can require a more exact formalism than strict equations.

At dimension two, a bicategory illustrates the pattern. Given composable 1-arrows $f,g,h$, the composites $(h\circ g)\circ f$ and $h\circ(g\circ f)$ need not be equal, but an associator 2-isomorphism relates them. For four arrows, different sequences of associators must agree through the pentagon identity. In higher dimensions, coherence continues.

A weak $\infty$-groupoid adds the condition that arrows at every positive level are invertible in the relevant weak sense. An $(\infty,1)$-category, by contrast, allows noninvertible 1-arrows while requiring higher arrows to be invertible. Modern writing often uses the latter for homotopy-coherent category theory. Grothendieck’s homotopy hypothesis concerns the groupoidal case.

The distinction prevents anachronism. Saying “he invented $\infty$-categories” ignores the variety of modern definitions and earlier contributors. Saying “he proposed weak $\infty$-groupoids as models of homotopy types and higher stacks as coefficients” identifies a documented claim.

Coherators as a strategy

One approach in the manuscript begins with formal shapes for globes and pasting. A coherator is intended to be sufficiently free and sufficiently contractible that models of it carry all required compositions and higher coherences without arbitrary extra equations.

“Free” and “contractible” pull in different directions. Freeness avoids imposing accidental relations; contractibility ensures that parallel operations have higher comparison data. The construction proceeds by adjoining lifts for admissible pairs. Models then interpret formal operations in sets or another category.

The intuition is like designing a grammar that contains every legitimate way to compose expressions and explicit transformations among equivalent parses. The analogy fails because a coherator is categorical machinery with lifting properties, not a natural language grammar, and because proving that its models represent all homotopy types is a major theorem.

Later authors developed coherator-based and other definitions of weak $\infty$-groupoids. Equivalence among definitions is difficult. A historical survey must not select one later formalization and write as if every technical detail were fixed in 1983. The manuscript contains a strategy, terminology, examples, and conjectural equivalences.3232Grothendieck, À la Poursuite des Champs (Pursuing Stacks); Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1; Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”.

Model categories and their comparison

Quillen’s model categories supplied one major language for homotopy theory. A model category has three distinguished classes of maps—weak equivalences, fibrations, and cofibrations—satisfying axioms that permit factorizations and lifting. It allows spaces, chain complexes, simplicial sets, and other settings to be handled by a common calculus.

The letter to Quillen therefore addressed someone whose machinery exemplified the kind of general foundation at issue. Grothendieck was extending stacks in algebraic geometry while testing how model structures, presheaf categories, and higher groupoids might fit a more global synthesis.

A model category is a presentation of homotopy theory, not the homotopy theory itself in an absolute sense. Different model categories can be Quillen equivalent and present the same underlying homotopy type theory. This distinction supports the search for modelizers: identify combinatorial presentations and criteria for when they encode the desired weak equivalences.

Presheaf categories are attractive because they are flexible and generated from simple representables. Given a small category of shapes $A$, the Yoneda embedding places each shape inside $\widehat A=\mathrm{Set}^{A^{op}}$. One then seeks a localizer $W$ so that $\widehat A[W^{-1}]$ models spaces up to homotopy.

For the simplex category $\Delta$, the presheaf category is simplicial sets. Geometric realization and singular complex relate simplicial sets to topological spaces. The success of this case raises Grothendieck’s characteristic classification problem: which properties of $\Delta$ make it work and which other categories share them.

A test functor can transport homotopical information from one category of shapes to another. An aspherical functor behaves, roughly, as if its fibers have contractible homotopy type under the chosen localizer. Smooth and proper functors in this categorical homotopy setting mimic stability properties familiar from geometry. The vocabulary shows old geometric instincts entering abstract homotopy.

Cisinski’s later work makes many of these ideas precise in a developed theory of presheaves as models for homotopy types. Citing the later theorem is not evidence that every proof lies hidden in Pursuing Stacks. It is evidence that the manuscript set a rigorous research direction capable of sustained development.

Integration, realization, and the return to spaces

A combinatorial model must connect back to homotopy types. A realization functor sends formal cells or presheaves to spaces. A nerve sends categories to simplicial sets. Grothendieck used language of integration and cointegration for passages among categorical and homotopical structures.

The nerve of a small category $C$ records strings of composable arrows as simplices. Its geometric realization $BC$ is the classifying space. A functor $C\to D$ can be declared a weak equivalence when it induces a weak homotopy equivalence $BC\to BD$. This gives a basic localizer in the category of small categories.

But the manuscript asks how much of this construction can be characterized intrinsically. If weak equivalence is always defined by first realizing into a trusted category of spaces, the new foundation remains dependent on the old one. An intrinsic set of axioms would identify the correct maps inside categories themselves.

This is another instance of reconstruction. Chapter 55 sought arithmetic geometry inside profinite group actions. Here the aim is to recover homotopy theory inside categories of shapes and presheaves. The objects differ, but the method asks what structural relations characterize the intended world without constant return to an external model.

Intrinsic characterization is difficult because many classes of maps satisfy plausible closure properties. One may obtain different localizers and different homotopy theories. Minimality, accessibility, and compatibility with products or intervals become substantive requirements.

Error, speculation, and editorial voice

The typescript’s openness includes mistakes. Grothendieck sometimes lacked references, proposed statements too broadly, or changed notation. A student using the manuscript needs warnings. A historian needs the original sequence. A critical edition must serve both.

One solution is layered typography: reproduce authorial text, number manuscript pages, and mark editorial corrections in brackets. The SMF edition and related Maltsiniotis portals describe such practices. Silent correction would make the author appear more prescient and the editors less visible.

Speculation has its own status. Grothendieck defended it as essential to creative mathematics. A speculative paragraph can be valuable even when false because it identifies a test or analogy. But later citation must say “conjectured” or “suggested.” The prestige of the author cannot promote it to theorem.

The manuscript’s conversations with Brown are particularly useful because objections remain visible. Brown could say that a structure modeled only a certain class of types; Grothendieck could react; later correspondence could revise the assessment. The record counters the myth that ideas emerged complete.

English phrasing sometimes carries signs of a French writer thinking across languages. An editor can smooth grammar but risks removing voice and technical nuance. Quotation should therefore use the manuscript or established edition, not a silent stylistic rewrite. This chapter paraphrases almost everything and preserves only a five-word epigraph.

Personal material inside mathematical pages

The typescript includes brief records of family events, travel, and daily interruption. Their presence reflects the letter form and the belief that discovery unfolds inside life. They are not invitations to mine private relatives for narrative color.

One dated passage follows news of the death of a young granddaughter; another marks the birth of a grandson. The source account identifies them, but this chapter does not reproduce intimate detail. The mathematical history needs only the formal observation that the manuscript did not maintain a wall between technical reflection and life events.

This integration of life and mathematics anticipates Récoltes et Semailles, where the act of writing becomes part of the subject. Pursuing Stacks is not autobiography, but it preserves its own conditions of production more visibly than a journal article.

A map of influence

Influence radiated through several channels. Brown and Porter connected the manuscript to higher-dimensional group theory. Maltsiniotis developed localizers, test categories, and Grothendieck homotopy theory. Cisinski built a broad presheaf theory. Other readers found the homotopy hypothesis and higher-stack vision relevant to emerging higher categories.

Some influence was direct: a person read a circulating copy or correspondence. Some was mediated by talks and later papers. Some was convergence: mathematicians developed similar structures independently. The word “influenced” should be reserved for a documented path rather than inferred from resemblance.

Mikhail Kapranov later recalled that Gelfand advocated reading Esquisse and Pursuing Stacks, and that Drinfeld and Voevodsky were interested in related directions. This is valuable testimony about reception in Soviet mathematics, recorded years later on Brown’s page. It is not a substitute for citations in Drinfeld’s or Voevodsky’s papers.

The manuscript’s absence from standard publication both limited and magnified its aura. Readers with copies belonged to an informal circle; others encountered secondhand claims about hundreds of pages. Scarcity can make a text legendary before its assertions are checked. Formal editing reduces scarcity and permits criticism.

The 2022 edition arrived after many ideas had independent literatures. Its value is therefore historical and mathematical. It lets readers see exact formulations, abandoned paths, and connections later summaries omit. It also lets them identify errors rather than treating “Grothendieck’s manuscript” as a single oracle.3333Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1.

The intended return to stacks

The early sections and Breen letters indicate the purpose of the deferred volumes: a program of higher stacks over a topos, serving as coefficients for nonabelian cohomology. Ordinary sheaves take values in sets or groups; higher stacks would take values in higher groupoids. Cohomology would classify torsors, gerbes, and an ascending range of geometric structures.

On a site $C$, one can imagine assigning to every object $U$ an $\infty$-groupoid $\mathcal F(U)$, with restriction along maps and descent for coverings. Compatible local objects should glue, compatible equivalences should glue, and higher coherences should glue. In modern language, this is a homotopy-coherent sheaf condition.

There are several ways to express that condition: hypercover descent, model structures on simplicial presheaves, $\infty$-topoi, and other frameworks. These mature formulations postdate or develop independently from the manuscript. They illuminate the target without being silently attributed to it.

The intended nonabelian cohomology would then be geometric. A class need not be an element of an abelian group; it may be a torsor, gerbe, or higher object with automorphisms. “Coefficients” become a stack of possible local structures. This shift was already implicit in descent and gerbes; the infinite height enlarged it.

Applications could reach deformation, obstruction theory, bundles, and moduli. But Pursuing Stacks does not supply a completed catalog of such applications. The manuscript spends its energy ensuring that the coefficient objects model homotopy correctly.

The deferred return therefore marks both coherence and incompletion. It was coherent to settle coefficients first. It was incomplete because the promised general theory on topoi remained heuristic. A history should say both in one sentence.

A correspondence network with uneven knowledge

Brown knew higher groupoids and van Kampen methods; Grothendieck brought stacks, topoi, and broad homotopical aims; Porter and other correspondents supplied further expertise. No participant possessed every relevant literature. The network worked by exchanging partial knowledge.

Grothendieck openly described gaps in his culture of homotopy theory. This admission matters. It counters a portrait of universal mastery and explains why letters could redirect him. A conceptual imagination of unusual range still depended on bibliographic help.

Brown’s duplication practice created several audiences. Some recipients received pages directly from Grothendieck, some through Bangor, and some heard about the text. Their versions and dates may differ. Citation to “the 1983 manuscript” should not assume every person saw the final foliation.3434Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”; Grothendieck and Brown, Correspondance Grothendieck–Brown.

Permission was relational: Brown asked and Grothendieck agreed. That clear exchange is a positive model for informal circulation. It did not necessarily authorize every later medium, commercial edition, or unrestricted online copy. Scope should be documented.

The network also demonstrates how a peripheral institution can become central to an idea’s route. Bangor was not IHÉS, yet Brown’s initiative and copying made it a distribution center. Intellectual geography is made by correspondence as well as prestigious addresses.

Computation after concepts

Grothendieck’s stated priority of conceptual understanding can be tested against the manuscript’s subjects. Test categories turn conceptual criteria into comparisons among simplicial, cubical, and other models. Localizers turn a philosophy of weak equivalence into closure axioms. Derivators turn lost diagram data into a system of functors. These are operational consequences.

Still, many explicit calculations remain for others. Determining homotopy groups, constructing concrete fibrant replacements, or computing nonabelian cohomology can require machinery not supplied by a conceptual slogan. The manuscript’s influence often lies in problem formation rather than ready algorithms.

Brown’s own higher van Kampen theorems illustrate a complementary path. They provide calculations of higher homotopy information through algebraic models and decompositions. Grothendieck recognized their foundational significance while pursuing a broader synthesis. Neither “concept” nor “computation” belongs to one side alone.

The most faithful conclusion is conditional: conceptual understanding can reveal what should be computed and why invariants behave; explicit work tests whether the concept truly controls examples. A field needs both.

Citing an unfinished source

A claim drawn from Pursuing Stacks should include section or manuscript page, not only title. It should say whether the source is the original typescript, Brown scan, Maltsiniotis transcription, or SMF Volume I. Different witnesses can correct or reorder.

If a later theorem proves a conjecture in the manuscript, cite both. The manuscript receives conjectural priority; the later paper receives proof credit. If the later definition differs, describe the difference rather than use one name for both.

If an idea appears in correspondence before the typescript, the letter date may be the earliest witness. Public availability may begin later. Priority, circulation, and publication are three separate dates.

This protocol is demanding because the text is difficult. It is also the only alternative to aura. Exact citation lets readers find where the pursuit actually went.

Difficulty creates a final editorial temptation: quote a striking general sentence and skip the technical neighborhood. That practice makes Grothendieck sound prophetic while hiding whether a definition follows. This chapter pairs every broad claim with a mathematical card.

Conversely, extracting only definitions makes the manuscript look like a modern reference work. Its hesitations, letters, and promised returns are part of the source. A critical history needs both conceptual statement and compositional form.

The 2022 edition makes such balanced reading possible for Volume I. Future review should record which chapter claims have been checked directly against it and which still rely on scan or editorial description. Source quality can improve without rewriting composition history.3535Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1.

The pursuit remains open in a precise sense: later theories answer some questions in languages Grothendieck did not fix, while other conjectural identifications require comparison. “Open” does not mean untouched. It means the history cannot be closed by one edition or one framework.

That openness should not excuse vagueness. A review can still decide whether a particular definition is coherent, a proof correct, or a priority claim supported. “Programmatic” labels the scale and status of a passage; it is not immunity from mathematical criticism.

Nor should the manuscript’s difficulty become a gatekeeping badge. Expository maps, glossaries, and examples can widen access without pretending the source is easy. The long editorial labor exists so readers need not prove seriousness by deciphering every rough page alone.

The deepest fidelity to a foundational manuscript may therefore be a clear account of where it succeeds and where it stops. Pursuit is not worship of unfinishedness. It is the shared work of carrying a question farther.

That shared work includes deciding when not to use the manuscript. For a standard definition now available in a reviewed text, students may cite the modern source and consult Pursuing Stacks for history. Historical priority does not require an unfinished typescript to remain the most practical reference.

Conversely, modern convenience should not erase the source. When a concept’s motivation or early range matters, the manuscript and correspondence can reveal alternatives polished accounts suppress. Use depends on the question.

The editorial achievement is to make both choices possible: reliable access for historians and clear guidance for mathematicians. A facsimile alone supplies the first incompletely; a rewritten textbook alone supplies the second at the cost of history.

Derivators in historical perspective

Section 69 contains the term that a later editorial record reports as central to Les Dérivateurs. This chapter records that documentary bridge but does not define the modern formalism. Chapter 59 explains the mathematics in full; Chapter 60 treats the editor-dated late manuscript as an object of retirement-era work.

The bounded point here is chronological. An idea appears inside the open 1983 typescript. The editorial account associated with Matthias Künzer’s transcription, Jean Malgoire’s manuscript consultation, and Georges Maltsiniotis’s edition assigns a much larger handwritten treatment from October 1990 into the first half of 1991. Neither the publisher’s webpage nor the original colophon could be consulted for this account; the precise span therefore remains editorial testimony rather than an independently established date. Alex Heller’s independent 1988 work and later derivator formalisms prevent exclusive priority claims. The exact relation among those lines remains a specialist question rather than a conclusion drawn from resemblance. 3636Georges Maltsiniotis, “Les Dérivateurs (The Derivators),” https://webusers.imj-prg.fr/~georges.maltsiniotis/groth/Derivateursengl.html.

This allocation also protects the shape of Pursuing Stacks. Its main mathematical burden in Part X is higher groupoids, homotopy models, test categories, and the deferred return to stacks. A complete derivator lesson at this point would duplicate the declared Chapter 59 owner and distort the manuscript’s already complex itinerary. 3737Heller, Homotopy Theories.

One method across two late programs

Esquisse and Pursuing Stacks address different mathematical objects, yet a common method appears. In Esquisse, Grothendieck seeks arithmetic geometry through all finite coverings and the relations of a moduli tower. In Pursuing Stacks, he seeks homotopy types through all higher arrows and categories of combinatorial models.

Both resist premature reduction. A single fundamental group at one base point may hide groupoid symmetry. A coarse moduli space may hide automorphisms. A homotopy category may hide coherent diagrams. An abelian invariant may hide noncommuting structure. The response is to retain relations at the next level.

Both also seek reconstruction: recognition of the absolute Galois group through its tower action, representation of a homotopy type by a weak $\infty$-groupoid, and characterization of a category of shapes as a test category. The word “recognized” means characterized by structure rather than embedded in a larger known world.

The parallel has limits. Dessins and Belyi maps involve established algebraic curves and a theorem. Weak $\infty$-groupoids in the manuscript remain a foundational program with competing models. Similar method does not imply equal maturity.

The comparison explains why late work should not be divided into an arithmetic phase and a categorical digression with no connection. Moduli, descent, groupoids, and gluing pass between them. The same insistence on preserving symmetry drives both.

It also explains the cost. Every preserved relation generates coherence conditions. The richer object is more faithful and harder to construct. Grothendieck repeatedly accepted this cost in exchange for a language he regarded as natural.

Editing after decades

Around 2002, the Société Mathématique de France asked Georges Maltsiniotis to prepare Pursuing Stacks for publication. Matthias Künzer contributed extensive transcription and checking. The scale was formidable. The original typescript mixed mathematical development, correspondence, and local revision. Editors had to establish text, notation, sections, corrections, and relationships to letters.3838Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1; Maltsiniotis, La théorie de l’homotopie de Grothendieck; Cisinski, Les préfaisceaux comme modèles des types d’homotopie.

The process encountered another difficulty. In 2010, Grothendieck communicated a wish that his works not be published or reproduced. The history of earlier permissions, later restrictions, estate authority, and public-interest arguments is legally and ethically complex. This account does not adjudicate it. It records only editions and publisher statements and uses short quotations. Formal publication by SMF in 2022 is a fact; the normative dispute about publication belongs to the legacy chapters.3939Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1.

The 2022 book is À la poursuite des champs (volume I) / Pursuing Stacks (volume I), Documents Mathématiques 20. It contains a long editorial front matter and 446 main pages, covering sections 1–91 and 95–98. The publisher explicitly anticipates a second volume for remaining sections, Breen letters, and related correspondence. 4040Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1.

The gap in section ranges is not a typographical curiosity. Volume I’s selected order reflects editorial decisions about coherent chapters. A citation to a section absent from Volume I must return to the typescript or another edited file. “SMF edition” is not yet a universal locator for the whole work.

The publisher’s bilingual title also resolves one translation question while leaving others. À la poursuite des champs and Pursuing Stacks are presented as parallel official titles. Internal subtitles such as The Modelizing Story retain their historical spellings and should not be modernized silently.

The cxxi pages of front matter are evidence of editorial scale. Introductions, textual notes, correspondence context, and indexes are not padding; they make an unfinished manuscript usable. Their authorship belongs to Maltsiniotis and collaborators, not Grothendieck.

If Volume II appears, its publication date will be later than Volume I and must be recorded separately. Until then, calling the project “published in 2022” requires the qualifier “Volume I.” 4141Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1.

Formal publication also changes quotation rights. A scan once circulated with personal permission may now have a publisher’s typesetting and apparatus under separate rights. An image of the work must distinguish authorial text, facsimile, and publisher design.

This means three statements must not be confused:

1. The open manuscript circulated in installments during 1983.
2. Editors transcribed and prepared portions over many later years.
3. SMF formally published Volume I in 2022, covering sections 1–91 and 95–98 rather than the whole typescript.

Saying “Pursuing Stacks was published in 1983” is false. Saying “it first existed in 2022” is equally false. Saying “the complete work was published by SMF in two volumes in 2022” exceeds the publisher record. Precise verbs preserve the history.4242Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1.

The edition’s delay changed mathematical context. By 2022, higher categories, $\infty$-categories, model categories, derivators, and higher stacks had large literatures. Readers could recognize anticipations and divergences invisible in 1983. That advantage creates hindsight bias. A phrase resembling a modern definition may have had a different role; an idea now standard may have been one speculative branch among many. Editors and historians must resist translating the manuscript entirely into current vocabulary.4343Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1.

Influence without automatic inheritance

Maltsiniotis’s La théorie de l’homotopie de Grothendieck and Cisinski’s Les préfaisceaux comme modèles des types d’homotopie are major examples of rigorous continuation. They develop localizers, test categories, presheaf models, and homotopical structures with proofs that cannot be credited to the manuscript’s questions. Their work demonstrates influence because it is new work.

Brown’s own account emphasizes another influence: encouragement to take higher groupoids seriously as models and to seek many-base-point formulations. Correspondence can influence by changing which problems seem legitimate. It can also reveal disagreement. Grothendieck initially reacted to limitations Brown described, and later revised his view of what strict multiple groupoids could model. The exchange was not transmission from master to disciple in one direction.

Later higher category theory developed through many lines: Boardman–Vogt, Bénabou, Street, Joyal, Quillen, Segal, Dwyer–Kan, Lurie, and others. Listing names does not settle conceptual ancestry. The safe historical claim is local: Pursuing Stacks contains a broad 1983 program for weak higher groupoids, the homotopy hypothesis, higher stacks, and models of homotopy types; circulated copies affected identifiable mathematicians; later theories share and transform parts of that program.4444Grothendieck, À la Poursuite des Champs (Pursuing Stacks); Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1; Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”.

The manuscript also matters as evidence of Grothendieck’s mathematical practice after 1970. It is difficult to reconcile with the phrase “he abandoned mathematics” unless abandonment means leaving institutions and standard publication. In fewer than nine months, he produced hundreds of pages, maintained correspondence, and pursued foundational questions. The pages may be unfinished; they are not absence.4545Grothendieck, À la Poursuite des Champs (Pursuing Stacks); Grothendieck, Pursuing Stacks (À La Poursuite Des Champs), vol. 1; Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”.

Three ways into the manuscript

A general reader can enter through descent: local objects, overlap isomorphisms, and the need to preserve symmetry. That route explains why stacks exist without requiring the full manuscript.

A topology reader can enter through paths and higher homotopies. Fundamental groupoids lead to weak $\infty$-groupoids and the homotopy hypothesis. Brown’s correspondence supplies historical motivation.

A category theorist can enter through test categories, localizers, and derivators. That route follows the manuscript’s actual expansion and connects to Maltsiniotis and Cisinski.

No route should be mistaken for a summary of every section. The work’s scale defeats linear exposition in one chapter. Multiple entrances are more faithful than a false single thesis.

The original diagram for this chapter uses layered arrows rather than a pyramid. A pyramid would imply that higher cells are superior to lower ones. The layers instead show dependence: objects, arrows, 2-arrows, and continuing coherence.

The diagram also separates “stack on a site” from “$\infty$-groupoid at a point.” Collapsing them visually would repeat the title’s misleading suggestion that all 600 pages directly construct higher stacks over spaces.

Labels identify “program/hypothesis” where appropriate. The homotopy hypothesis is not drawn as an equality already proved in the manuscript’s formalism. A dashed equivalence and caption state the intended correspondence.

This visual honesty matters because diagrams look authoritative. A clean arrow can conceal decades of proof. The caption names which part is definition, which is intuition, and which is later development.

Understanding and computation

Brown asked how a conceptual formulation could lead to computations. Grothendieck replied that the question did not trouble him and described explicit calculations as by-products of conceptual understanding across his mathematical work. The epigraph condenses that priority; its wording has not been checked against the original letter. 4646Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”.

The claim is both powerful and contestable. New concepts can make calculations possible by exposing structure. Homological algebra, schemes, and cohomology supply examples. But computation does not always emerge “by itself.” Algorithms, estimates, examples, and implementation require specialized labor. A conceptual framework can hide computational difficulty as easily as dissolve it.

In Pursuing Stacks, the priority of understanding licensed long foundational detours. Before computing nonabelian cohomology with higher stacks, one should know what the coefficients are and how they model homotopy. That patience is coherent. It also risks an infinite regress: every foundation calls for another. The unfinished state of the project records both the fertility and cost of the method.

The best assessment avoids verdict by outcome alone. The manuscript did not become the standard textbook from which higher stacks were constructed. It did articulate questions and structures that later research made central. Its concepts were generative even where its exact formalism was not adopted. A program can succeed partially, diffusely, and through transformation.

Pursuit as a form of writing

The word “pursuing” describes more than topic. The manuscript follows thought in real time. It retains wrong turns, personal asides, changes of terminology, and promises. In a polished article, these would be removed. Here they show how an apparently small response becomes a continent of questions.

That record is valuable because foundational mathematics is often narrated backward. Once a definition stabilizes, earlier uncertainty looks like confusion rather than necessary search. Pursuing Stacks restores the multiplicity of possible models. It shows an expert learning literature he did not command, testing analogies with older work, and asking correspondents for correction.

It also complicates the cult of solitary genius. The pages were written by Grothendieck, but their genesis includes Duskin’s reference, Breen’s letters, Brown’s inquiry, Loday’s corrections, Higgins’s and Brown’s theorems, Porter’s participation, Quillen as addressee, and the postal labor of circulation. Later transcription required Maltsiniotis and Künzer. The work is a powerful individual text with a collective documentary frame.

This frame returns us to the ethics of Chapter 57. Brown’s permission question was explicit. Grothendieck agreed that material sent to him could be duplicated. That permission enabled influence. Later restrictions complicated publication. Neither moment should be erased. Consent has a history, and the scope of a 1982 permission is not automatically the scope of a 2022 edition.4747Brown, “The Origins of Alexander Grothendieck’s *Pursuing Stacks*”; Grothendieck and Brown, Correspondance Grothendieck–Brown.

The open manuscript stopped without a definitive conclusion. Grothendieck moved among Récoltes et Semailles, the CNRS report, correspondence, topology, and later spiritual writings. Yet section 69’s derivator would return near the end of the decade. The pursuit had not reached its promised stacks; one of its digressions would become the last great mathematical manuscript of his life.4848Maltsiniotis, “Les Dérivateurs (The Derivators)”; Alexandre Grothendieck, Lettre d’Alexandre Grothendieck à Robert Thomason, ed. Matthias Künzer, 1991, https://webusers.imj-prg.fr/~georges.maltsiniotis/groth/ps/lettreder.pdf; Heller, Homotopy Theories.

The chronology then turns to writing whose claims cannot be assessed by the methods used for Belyi maps or homotopy models. In La Clef des Songes, Grothendieck treated dreams as messages, named a Dreamer, and identified that Dreamer with God. The biography must neither translate those statements into theorems nor explain them by diagnosis. It must read them as dated claims in texts and preserve the simultaneous dated evidence of abstract mathematical writing.