The Dialectics Archetypes/Types (Universal Categorical Constructions/Concrete Models) in the Work of Alexander Grothendieck

Abstract

We present two basic directions that Grothendieck explores in the transit between archetypes (universal categorical constructions) and types (concrete models): (1) projecting archetypes to types in the 1950s (around the Tôhoku and Riemann-Roch), (2) embedding types into archetypes in the 1980s (around Pursuing Stacks and Dérivateurs). We also discuss (3) his general remarks about “models” in Récoltes et semailles.

Along with David Hilbert, Alexander Grothendieck (1928–2014) is certainly one of the two greatest mathematicians of the twentieth century. The range (functional analysis, complex variables, algebraic geometry, category theory, number theory, topological algebra) and the depth (nuclear spaces, sheaves, schemes, topos, motives, stacks) of his contributions are simply outstanding. In his work, Grothendieck systematically explored the pendulum of the abstract and the concrete, the universal and the particular, a back-and-forth that we can subsume in a dialectics between archetypes and types. Archetypes are mathematical constructions which act as universals in certain categories, and which are projected into the many types living in those categories. On the other hand, types are often embedded into global archetypes, which govern the local behavior of the types. We will thus understand ‘archetypes’ as universal categorical constructions, and ‘types’ as concrete models, and study the correlations between them.

The article is divided in three parts: (1) an exploration of the notion of ‘archetype,’ and its projectivity to ‘types,’ at the beginning of Grothendieck's algebraic geometry period (the Tôhoku paper, published in 1957 and the Rapport on classes on sheaves, also published in 1957),¹ (2) a study of the embedding of types into archetypes, at the end of Grothendieck's topological algebra period (Pursuing Stacks (1983)² and Dérivateurs (1991)³), (3) an account of the appearances of the notion of “modèle” and its variants in Récoltes et semailles (1986).⁴

Archetypes and Types in the Tôhoku and the Rapport

The Tôhoku introduced sheaves and category theory into the landscape of ‘real’ mathematics. Along with David Corfield in his 2003 Towards a Philosophy of Real Mathematics,⁵ we understand by ‘real’ mathematics, the corpus of the hard-working mathematicians: combinatorics, number theory, abstract algebra, algebraic geometry, topology, complex variables, functional analysis, differential geometry, etc., well beyond sets and classical logic. Jean Leray invented sheaves in 1942. Saunders MacLane and Samuel Eilenberg invented categories in 1945.⁶ Nevertheless, their actual use in real mathematics, as signaled by the same MacLane is due to Grothendieck.⁷ Grothendieck's contributions to category theory are truly impressive on their own (introduction of sub-objects, adjoint functors, equivalences, representable functors, additive and abelian categories, generators, infinitary axioms), but the central points of the Tôhoku paper deal with two kinds of archetypes related to mathematical practice: (1) to prove that in an abelian category with a generator and a suitable infinitary axiom, any object (‘type’) can be embedded into an injective object (‘archetype’),⁸ (2) to prove that the cohomology of a space, with coefficients in a sheaf (‘types’), can be reconstructed as a projection of a formalism of derived functors (‘archetypes’).⁹ In that way, (co)homology, one of the main themes of mathematical development in the period 1895–1950, can be seen as a projection of an even more fundamental theme: a theory of abstract derivators in suitable categories.¹⁰

Grothendieck's Tôhoku strategy is very interesting in terms of ‘models.’¹¹ First, categories, abstract universal properties, are, through a series of axioms, held in the diversity of mathematical regions: associativity and identities (all categories), existence of an abelian group structure in the Hom-sets (additive categories, e.g. category of holomorphic fibered spaces over a Riemann surface),¹² existence of kernels, cokernels, and adequate factorizations through them (abelian categories, e.g. category of sheaves with fibers abelian groups),¹³ etc. In this approach, the axiomatic abstract definitions have their expected concrete models. Second, in the abstract setting, that is in general categories which satisfy the axioms, not yet ‘reified’ in concrete models, archetypes occur in a natural way (definitions and properties related to the quantifier ‘exists only’ ∃!), and only afterwards are they projected into existential objects of concrete categories. Thus, abstract categories are ‘doubly archetyped’ (inside the categories free objects appear, outside the categories free objects incarnate in apparently very different disguises), while concrete categories become just partial modelizations of general categorical transits. For example, an initial object 0 in an abstract category is defined through the property that there exists only a single morphism from 0 to any other object of the category. If the abstract category is modelized in diverse concrete categories, the initial object ‘incarnates’ in apparently very different constructions: 0 is the empty set in the category of sets, 0 is a group with one element in the category of groups, 0 is the ring of integers in the category of rings, 0 is the discrete topology in the category of topological spaces, etc. Third, in the abstract settings, the general theorems (e.g. (1) and (2) above) occur thanks to free machinery that has escaped from the particularity of the concrete models.

For Grothendieck, abstraction is never artificial,¹⁴ but, on the contrary, has the very precise purpose of smoothing many of the obstructions that live along the diverse particular regions of mathematics. Modelizing through axioms and categories allows for the integration of many possible differences. Archetypes (universal categorical constructions) allow to unify smoothly a diversity of types (concrete models). It is a simple matter of elevation and perspective: while, in the maze of the concrete, walls surround us, when we elevate ourselves to the general, we are afforded the possibility of viewing from a wider panorama. Orientation is obtained at the top of the mountain, if we can escape the underlying jungle.

Grothendieck's extension of the Riemann-Roch theorem is a good example of the abstract orientation advantages obtained through categorical modelization. To recall, the Riemann-Roch theorem unifies two very different approaches to obtain a natural invariant for a complex surface. On one hand, the genus g of the surface is obtained by counting the number of holes of the surface, or, equivalently, the number of cuts (minus one) with which the surface becomes disconnected (e.g., the genus of the sphere is 0, the genus of the torus is 1, etc.) On the other hand, one can think about the ‘good’ functions (holomorphic) and the ‘bad’ functions (meromorphic) definable on the surface. If we fix n points on the surface, consider the (vector) space Hol of holomorphic functions with zeroes on those points, and the (vector) space Mer of holomorphic functions with poles on those points. The Riemann-Roch theorem says that

$$ n \, {-} \, g + { 1 } = {\text{ dim }}\left( {Mer} \right) \, {-}{\text{ dim }}\left( {Hol} \right). $$

In this way, an intrinsic geometric invariant (the genus) is related to an extrinsic complex variable invariant (harmonic difference between Mer and Hol). The Riemann-Roch theorem marks the beginning of modern algebraic geometry at the very heart of profound connections between geometry, topology, complex variables, differential geometry and algebra.

To examine Grothendieck’s approach in more detail, recall that coherent sheaves may be understood as the simplest, well-behaved sheaves, emerging from ring representations.¹⁵ For a variety X with enough smooth conditions, Grothendieck imagined the free group K(X) generated by all coherent sheaves on X. Coherence, freeness and totality—diverse forms of simplicity—help to understand why K(X) may behave like an archetype. The main theorem in Grothendieck's K-theory assures that it is the case. K becomes a functor related to the homology functor H, through a natural transformation C given by Chern's classes.¹⁶ Then, an obstruction to commutativity (CK does not equal KC) occurs, but it can be factored through Todd's classes,¹⁷ producing an extended Riemann-Roch-Serre-Hirzebruch formula. The particular case of the Riemann-Roch formula is obtained from the general case, specifying a variety in one point. Thus, the general archetype (universal categorical construction K(X)), while projected on a trivial homology, captures the specific type (concrete equation on the surface

$$ X, n - g +1 = {\text{ dim }}\left( {Mer} \right) \, {-}{\text{ dim }}\left( {Hol} \right). $$

Here, the modelization does not occur though axioms, but it uses instead a very soft categorical environment, where little is specified. Well-behaved coherence, freeness, and universality, are the only ingredients used in the construction of the K-theory group K(X). Categorical abstraction provides an archetypal guide, and concrete modeling provides its associated types. The fact that Riemann-Roch's wonderful connection between magnitude (genus) and number (harmonic difference) can be seen as a projection from an abstract sheaf’s behavior shows the depth of Grothendieck's approximation. By way of sheaves, geometry and algebra become unified, something, which further developments, will also underline forcefully.¹⁸

Types and Archetypes in Pursuing Stacks and Dérivateurs

Both in the Tôhoku paper and the Rapport, both from 1957, precedence is given to archetypes (derived functors, group of the K-theory), which then become projected into types (homology constructions). In Grothendieck's work in the 1980s, the method is somewhat inverted: a study of many types (e.g., moduli spaces, that is, isomorphism classes of Riemann surfaces, and the absolute Galois group $Gal (\overline{\mathbb{Q} },{\mathbb{Q}}$)) gives rise to an emergence of archetypes (e.g., anabelian conjectures, stacks, derivators). In this section I will explore some of these situations in Pursuing Stacks (1983) and Dérivateurs (1991).¹⁹

Grothendieck's La longue marche à travers la théorie de Galois (1981)²⁰ establishes a new research program, after his works of the first (1949–1957) and second (1958–1970) decades. His attention is now turned to ‘low-level’ complexity objects, that is, surfaces in general and Riemann surfaces in particular, modular curves, complex towers, and combinatorial approximations to the absolute Galois group Gal($\overline{\mathbb{Q} },{\mathbb{Q}}$). Given an algebraic extension X of ${\mathbb{Q}},$, one can ask if his algebraic group π₁^alg(X) (profinite completion of the fundamental group π₁^top(X)) fully characterizes X: if the answer is positive, Grothendieck says that we have an anabelian variety (‘an’ stands for ‘non,’ since the homotopy groups involved are strongly non-commutative). One of his main conjectures (still open) states that the anabelian varieties over ${\mathbb{Q}}$ are essentially the isomorphism classes of Riemann surfaces. In this way, an archetypal property (anabelianity) is expected to capture some of the main types of modern mathematical thought, the Riemann surfaces.

Pursuing stacks (1983) opens the stage for a profound connection between homotopy and homology. A hierarchy of concepts, objects and techniques is fundamental to be able to delve into the many levels of the homological and homotopical groups. In that sense, an n-stack can be understood as an n-truncated homotopy type, which in turn can be imagined as an (n + 1)-sheaf. The iteration of types gives rise to ∞-categories, where all information on n-morphisms (morphisms, functors, natural transformations, etc.) is piled-up. ∞-categories act as very large archetypes, which cover huge varieties of models. Pursuing stacks, whose first episode is called “histoire de modèles” (modelizing story),²¹ studies the many ways in which adequate functors in Cat (the category of all categories) can model fragments of homotopical and homological constructions. In particular, the localizers of a category through weak equivalences cover a lot of concrete realizations. We can observe here the action of both grothendieckian strategies (i) bottom-up, and (ii) top-down. In fact, with the first strategy (i) motivated by the visualization of concrete objects (Riemann surfaces, homotopy groups), the abstract anabelian conjectures emerge, while, on the other hand, with the second (ii) motivated by the desire to unify homotopy and homology, the concrete modelizers in Cat appear.

Dérivateurs (1991) proposes several axiomatic treatments of the constructions first imagined in Pursuing stacks. Axioms are initially imposed on classes W of morphisms in Cat, to try to model some properties of weak equivalences in the category Hot of CW-complexes with homotopies between them. The program consists of attempting to elevate types in Hot to archetypes in Cat. Axioms, as we have mentioned, free the lower level constructions (say, in Hot), in order to get smoothness properties at higher levels (say, in Cat). Other axioms are afterwards imposed on functors Cat → Cat: C → C W⁻¹, in order to capture the good structural properties of localizations. Such well-behaved functors are called derivators, since they generalize, in a canonical way, the nice properties of derived functors (Tôhoku). In this way, in an abstract, free, archetypal environment, derivators capture some of the main constructions of homotopy (coming from topoi and homotopy groups) and homology (coming from abelian categories and homology groups).

Models in Récoltes et Semailles

Récoltes et semailles (1986) is considered without doubt as the most profound text ever written²² on the emergence of mathematical creativity. Beyond unnecessary (but comprehensible) quarrels with the mathematical community, Récoltes et semailles provides, through the lenses of a mathematician of the highest caliber, an extremely detailed analysis of his mathematical career. The result illuminates the many polarities, forces and layers that govern mathematical thought. If we follow in Récoltes et semailles the term “modèle” and some related concepts (“unité,” “universel”), we obtain another complementary perspective on the archetypes/types dialectics that we have quickly perused in Grothendieck's mathematical work.

The term “modèle” appears in approximately sixty pages of the manuscript. Four sorts of uses of “modèle” are on display: (i) “modèle” as a concrete mathematical model, that is, a structure with some well determined properties (e.g., modelizers, homotopical constructions, motivic Galois groups, étale topos, set-theoretic models, Chern classes),²³ (ii) “modèle” as an abstract mathematical framework of ideas (e.g., algebraic geometry, Euclidean geometry, Newtonian mechanics, Einsteinian relativity, coherent duality),²⁴ (iii) “modèle” as a philosophical trend of thought (e.g., mixturing continuity/discreteness, foundations, yin/yang),²⁵ (iv) “modèle” as a human model of conduct (e.g., simplicity, perfection, Guru and Krishnamurti behavior, maternity).²⁶ One can see how Grothendieck explores, as usual, many layers and many different perspectives, which, afterwards, become unified. In fact, unity and universality (circa 50 appearances each in Récoltes et semailles) become closely connected with modelization, since any of the four reference “modèle” categories (i)-(iv) provides a common ground, in order to understand, in correlative, reticular ways, either mathematical structures, mathematical thought, philosophy, or ethics.

Conclusion

The archetype/type dialectics between categorical constructions and structured models enriches considerably the panorama of mathematical practice. Both levels—the abstract and the concrete, the universal and the particular—are necessary to nurture mathematical imagination. Mathematics has to explore all possible worlds, beyond mere actuality, but, on the other hand, many of its greatest achievements occur when a general architecture illuminates an actual conjecture.²⁷ The back-and-forth between types and archetypes, elevated to a method in Grothendieck's work, helps to understand the importance of a hierarchy of models in mathematical thought.