# A Sheaf of Modules is a Geometric Generalization of a Module over a Ring – A Case Derivative of Abelian Closure

A coherent sheaf is a generalization of, on the one hand, a module over a ring, and on the other hand, a vector bundle over a manifold. Indeed, the category of coherent sheaves is the “abelian closure” of the category of vector bundles on a variety.

Given a field which we always take to be the field of complex numbers C, an affine algebraic variety X is the vanishing locus

X = 􏰐(x1,…, xn) : fi(x1,…, xn) = 0􏰑 ⊂ An

of a set of polynomials fi(x1,…, xn) in affine space An with coordinates x1,…, xn. Associated to an affine variety is the ring A = C[X] of its regular functions, which is simply the ring C[x1,…, xn] modulo the ideal ⟨fi⟩ of the defining polynomials. Closed subvarieties Z of X are defined by the vanishing of further polynomials and open subvarieties U = X \ Z are the complements of closed ones; this defines the Zariski topology on X. The Zariski topology is not to be confused with the complex topology, which comes from the classical (Euclidean) topology of Cn defined using complex balls; every Zariski open set is also open in the complex topology, but the converse is very far from being true. For example, the complex topology of A1 is simply that of C, whereas in the Zariski topology, the only closed sets are A1 itself and finite point sets.

Projective varieties X ⊂ Pn are defined similarly. Projective space Pn is the set of lines in An+1 through the origin; an explicit coordinatization is by (n + 1)-tuples

(x0,…, xn) ∈ Cn+1 \ {0,…,0}

identified under the equivalence relation

(x0,…, xn) ∼ (λx0,…, λxn) for λ ∈ C

Projective space can be decomposed into a union of (n + 1) affine pieces (An)i = 􏰐[x0,…, xn] : xi ≠ 0􏰑 with n affine coordinates yj = xj/xi. A projective variety X is the locus of common zeros of a set {fi(x1,…, xn)} of homogeneous polynomials. The Zariski topology is again defined by choosing for closed sets the loci of vanishing of further homogeneous polynomials in the coordinates {xi}. The variety X is covered by the standard open sets Xi = X ∩ (An)i ⊂ X, which are themselves affine varieties. A variety􏰭 X is understood as a topological space with a finite open covering X = ∪i Ui, where every open piece Ui ⊂ An is an affine variety with ring of global functions Ai = C[Ui]; further, the pieces Ui are glued together by regular functions defined on open subsets. The topology on X is still referred to as the Zariski topology. X also carries the complex topology, which again has many more open sets.

Given affine varieties X ⊂ An, Y ⊂ Am, a morphism f : X → Y is given by an m-tuple of polynomials {f1(x1, . . . , xn), . . . , fm(x1, . . . , xn)} satisfying the defining relations of Y. Morphisms on projective varieties are defined similarly, using homogeneous polynomials of the same degree. Morphisms on general varieties are defined as morphisms on their affine pieces, which glue together in a compatible way.

If X is a variety, points P ∈ X are either singular or nonsingular. This is a local notion, and hence, it suffices to define a nonsingular point on an affine piece Ui ⊂ An. A point P ∈ Ui is nonsingular if, locally in the complex topology, a neighbourhood of P ∈ Ui is a complex submanifold of Cn.

The motivating example of a coherent sheaf of modules on an algebraic variety X is the structure sheaf or sheaf of regular functions OX. This is a gadget with the following properties:

1. On every open set U ⊂ X, we are given an abelian group (or even a commutative ring) denoted OX(U), also written Γ(U, OX), the ring of regular functions on U.
2. Restriction: if V ⊂ U is an open subset, a restriction map resUV : OX(U) → OX(V) is defined, which simply associates to every regular function f defined over U, the restriction of this function to V. If W ⊂ V ⊂ U are open sets, then the restriction maps clearly satisfy resUW = resVW ◦ resUV.
3. Sheaf Property: suppose that an open subset U ⊂ X is covered by a collection of open subsets {Ui}, and suppose that a set of regular functions fi ∈ OX(Ui) is given such that whenever Ui and Uj intersect, then the restrictions of fi and fj to Ui ∩ Uj agree. Then there is a unique function f ∈ OX(U) whose restriction to Ui is fi.

In other words, the sheaf of regular functions consists of the collection of regular functions on open sets, together with the obvious restriction maps for open subsets; moreover, this data satisfies the Sheaf Property, which says that local functions, agreeing on overlaps, glue in a unique way to a global function on U.

A sheaf F on the algebraic variety X is a gadget satisfying the same formal properties; namely, it is defined by a collection {F(U)} of abelian groups on open sets, called sections of F over U, together with a compatible system of restriction maps on sections resUV : F(U) → F(V) for V ⊂ U, so that the Sheaf Property is satisfied: sections are locally defined just as regular functions are. But, what of sheaves of OX-modules? The extra requirement is that the sections F(U) over an open set U form a module over the ring of regular functions OX(U), and all restriction maps are compatible with the module structures. In other words, we multiply local sections by local functions, so that multiplication respects restriction. A sheaf of OX-modules is defined by the data of an A-module for every open subset U ⊂ X with ring of functions A = OX(U), so that these modules are glued together compatibly with the way the open sets glue. Hence, a sheaf of modules is indeed a geometric generalization of a module over a ring.

# Philosophical Equivariance – Sewing Holonomies Towards Equal Trace Endomorphisms.

In d-dimensional topological field theory one begins with a category S whose objects are oriented (d − 1)-manifolds and whose morphisms are oriented cobordisms. Physicists say that a theory admits a group G as a global symmetry group if G acts on the vector space associated to each (d−1)-manifold, and the linear operator associated to each cobordism is a G-equivariant map. When we have such a “global” symmetry group G we can ask whether the symmetry can be “gauged”, i.e., whether elements of G can be applied “independently” – in some sense – at each point of space-time. Mathematically the process of “gauging” has a very elegant description: it amounts to extending the field theory functor from the category S to the category SG whose objects are (d − 1)-manifolds equipped with a principal G-bundle, and whose morphisms are cobordisms with a G-bundle. We regard S as a subcategory of SG by equipping each (d − 1)-manifold S with the trivial G-bundle S × G. In SG the group of automorphisms of the trivial bundle S × G contains G, and so in a gauged theory G acts on the state space H(S): this should be the original “global” action of G. But the gauged theory has a state space H(S,P) for each G-bundle P on S: if P is non-trivial one calls H(S,P) a “twisted sector” of the theory. In the case d = 2, when S = S1 we have the bundle Pg → S1 obtained by attaching the ends of [0,2π] × G via multiplication by g. Any bundle is isomorphic to one of these, and Pg is isomorphic to Pg iff g′ is conjugate to g. But note that the state space depends on the bundle and not just its isomorphism class, so we have a twisted sector state space Cg = H(S,Pg) labelled by a group element g rather than by a conjugacy class.

We shall call a theory defined on the category SG a G-equivariant Topological Field Theory (TFT). It is important to distinguish the equivariant theory from the corresponding “gauged theory”. In physics, the equivariant theory is obtained by coupling to nondynamical background gauge fields, while the gauged theory is obtained by “summing” over those gauge fields in the path integral.

An alternative and equivalent viewpoint which is especially useful in the two-dimensional case is that SG is the category whose objects are oriented (d − 1)-manifolds S equipped with a map p : S → BG, where BG is the classifying space of G. In this viewpoint we have a bundle over the space Map(S,BG) whose fibre at p is Hp. To say that Hp depends only on the G-bundle pEG on S pulled back from the universal G-bundle EG on BG by p is the same as to say that the bundle on Map(S,BG) is equipped with a flat connection allowing us to identify the fibres at points in the same connected component by parallel transport; for the set of bundle isomorphisms p0EG → p1EG is the same as the set of homotopy classes of paths from p0 to p1. When S = S1 the connected components of the space of maps correspond to the conjugacy classes in G: each bundle Pg corresponds to a specific point pg in the mapping space, and a group element h defines a specific path from pg to phgh−1 .

G-equivariant topological field theories are examples of “homotopy topological field theories”. Using Vladimir Turaev‘s two main results: first, an attractive generalization of the theorem that a two-dimensional TFT “is” a commutative Frobenius algebra, and, secondly, a classification of the ways of gauging a given global G-symmetry of a semisimple TFT.

Definition of the product in the G-equivariant closed theory. The heavy dot is the basepoint on S1. To specify the morphism unambiguously we must indicate consistent holonomies along a set of curves whose complement consists of simply connected pieces. These holonomies are always along paths between points where by definition the fibre is G. This means that the product is not commutative. We need to fix a convention for holonomies of a composition of curves, i.e., whether we are using left or right path-ordering. We will take h(γ1 ◦ γ2) = h(γ1) · h(γ2).

A G-equivariant TFT gives us for each element g ∈ G a vector space Cg, associated to the circle equipped with the bundle pg whose holonomy is g. The usual pair-of-pants cobordism, equipped with the evident G-bundle which restricts to pg1 and pg2 on the two incoming circles, and to pg1g2 on the outgoing circle, induces a product

Cg1 ⊗ Cg2 → Cg1g2 —– (1)

making C := ⊕g∈GCg into a G-graded algebra. Also there is a trace θ: C1  → C defined by the disk diagram with one ingoing circle. The holonomy around the boundary of the disk must be 1. Making the standard assumption that the cylinder corresponds to the unit operator we obtain a non-degenerate pairing

Cg ⊗ Cg−1 → C

A new element in the equivariant theory is that G acts as an automorphism group on C. That is, there is a homomorphism α : G → Aut(C) such that

αh : Cg → Chgh−1 —– (2)

Diagramatically, αh is defined by the surface in the immediately above figure. Now let us note some properties of α. First, if φ ∈ Ch then αh(φ) = φ. The reason for this is diagrammatically in the below figure.

If the holonomy along path P2 is h then the holonomy along path P1 is 1. However, a Dehn twist around the inner circle maps P1 into P2. Therefore, αh(φ) = α1(φ) = φ, if φ ∈ Ch.

Next, while C is not commutative, it is “twisted-commutative” in the following sense. If φ1 ∈ Cg1 and φ2 ∈ Cg2 then

αg212 = φ2φ1 —– (3)

The necessity of this condition is illustrated in the figure below.

The trace of the identity map of Cg is the partition function of the theory on a torus with the bundle with holonomy (g,1). Cutting the torus the other way, we see that this is the trace of αg on C1. Similarly, by considering the torus with a bundle with holonomy (g,h), where g and h are two commuting elements of G, we see that the trace of αg on Ch is the trace of αh on Cg−1. But we need a strengthening of this property. Even when g and h do not commute we can form a bundle with holonomy (g,h) on a torus with one hole, around which the holonomy will be c = hgh−1g−1. We can cut this torus along either of its generating circles to get a cobordism operator from Cc ⊗ Ch to Ch or from Cg−1 ⊗ Cc to Cg−1. If ψ ∈ Chgh−1g−1. Let us introduce two linear transformations Lψ, Rψ associated to left- and right-multiplication by ψ. On the one hand, Lψαg : φ􏰀 ↦ ψαg(φ) is a map Ch → Ch. On the other hand Rψαh : φ ↦ αh(φ)ψ is a map Cg−1 → Cg−1. The last sewing condition states that these two endomorphisms must have equal traces:

TrCh 􏰌Lψαg􏰍 = TrCg−1 􏰌Rψαh􏰍 —– (4)

(4) was taken by Turaev as one of his axioms. It can, however, be reexpressed in a way that we shall find more convenient. Let ∆g ∈ Cg ⊗ Cg−1 be the “duality” element corresponding to the identity cobordism of (S1,Pg) with both ends regarded as outgoing. We have ∆g = ∑ξi ⊗ ξi, where ξi and ξi ru􏰟n through dual bases of Cg and Cg−1. Let us also write

h = ∑ηi ⊗ ηi ∈ Ch ⊗ Ch−1. Then (4) is easily seen to be equivalent to

∑αhii = 􏰟 ∑ηiαgi) —– (5)

in which both sides are elements of Chgh−1g−1.

# The Case of Morphisms of Representation Corresponding to A-Module Holomorphisms. Part 2

Representations of a quiver can be interpreted as modules over a non-commutative algebra A(Q) whose elements are linear combinations of paths in Q.

Let Q be a quiver. A non-trivial path in Q is a sequence of arrows am…a0 such that h(ai−1) = t(ai) for i = 1,…, m:

The path is p = am…a0. Writing t(p) = t(a0) and saying that p starts at t(a0) and, similarly, writing h(p) = h(am) and saying that p finishes at h(am). For each vertex i ∈ Q0, we denote by ei the trivial path which starts and finishes at i. Two paths p and q are compatible if t(p) = h(q) and, in this case, the composition pq can defined by juxtaposition of p and q. The length l(p) of a path is the number of arrows it contains; in particular, a trivial path has length zero.

The path algebra A(Q) of a quiver Q is the complex vector space with basis consisting of all paths in Q, equipped with the multiplication in which the product pq of paths p and q is defined to be the composition pq if t(p) = h(q), and 0 otherwise. Composition of paths is non-commutative; in most cases, if p and q can be composed one way, then they cannot be composed the other way, and even if they can, usually pq ≠ qp. Hence the path algebra is indeed non-commutative.

Let us define Al ⊂ A to be the subspace spanned by paths of length l. Then A = ⊕l≥0Al is a graded C-algebra. The subring A0 ⊂ A spanned by the trivial paths ei is a semisimple ring in which the elements ei are orthogonal idempotents, in other words eiej = ei when i = j, and 0 otherwise. The algebra A is finite-dimensional precisely if Q has no directed cycles.

The category of finite-dimensional representations of a quiver Q is isomorphic to the category of finitely generated left A(Q)-modules. Let (V, φ) be a representation of Q. We can then define a left module V over the algebra A = A(Q) as follows: as a vector space it is

V = ⊕i∈Q0 Vi

and the A-module structure is extended linearly from

eiv = v, v ∈ Mi

= 0, v ∈ Mj for j ≠ i

for i ∈ Qand

av = φa(vt(a)), v ∈ Vt(a)

= 0, v ∈ Vj for j ≠ t(a)

for a ∈ Q1. This construction can be inverted as follows: given a left A-module V, we set Vi = eiV for i ∈ Q0 and define the map φa: Vt(a) → Vh(a) by v ↦ a(v). Morphisms of representations of (Q, V) correspond to A-module homomorphisms.

# Morphism of Complexes Induces Corresponding Morphisms on Cohomology Objects – Thought of the Day 146.0

Let A = Mod(R) be an abelian category. A complex in A is a sequence of objects and morphisms in A

… → Mi-1 →di-1 Mi →di → Mi+1 → …

such that di ◦ di-1 = 0 ∀ i. We denote such a complex by M.

A morphism of complexes f : M → N is a sequence of morphisms fi : Mi → Ni in A, making the following diagram commute, where diM, diN denote the respective differentials:

We let C(A) denote the category whose objects are complexes in A and whose morphisms are morphisms of complexes.

Given a complex M of objects of A, the ith cohomology object is the quotient

Hi(M) = ker(di)/im(di−1)

This operation of taking cohomology at the ith place defines a functor

Hi(−) : C(A) → A,

since a morphism of complexes induces corresponding morphisms on cohomology objects.

Put another way, an object of C(A) is a Z-graded object

M = ⊕i Mi

of A, equipped with a differential, in other words an endomorphism d: M → M satisfying d2 = 0. The occurrence of differential graded objects in physics is well-known. In mathematics they are also extremely common. In topology one associates to a space X a complex of free abelian groups whose cohomology objects are the cohomology groups of X. In algebra it is often convenient to replace a module over a ring by resolutions of various kinds.

A topological space X may have many triangulations and these lead to different chain complexes. Associating to X a unique equivalence class of complexes, resolutions of a fixed module of a given type will not usually be unique and one would like to consider all these resolutions on an equal footing.

A morphism of complexes f: M → N is a quasi-isomorphism if the induced morphisms on cohomology

Hi(f): Hi(M) → Hi(N) are isomorphisms ∀ i.

Two complexes M and N are said to be quasi-isomorphic if they are related by a chain of quasi-isomorphisms. In fact, it is sufficient to consider chains of length one, so that two complexes M and N are quasi-isomorphic iff there are quasi-isomorphisms

M ← P → N

For example, the chain complex of a topological space is well-defined up to quasi-isomorphism because any two triangulations have a common resolution. Similarly, all possible resolutions of a given module are quasi-isomorphic. Indeed, if

0 → S →f M0 →d0 M1 →d1 M2 → …

is a resolution of a module S, then by definition the morphism of complexes

is a quasi-isomorphism.

The objects of the derived category D(A) of our abelian category A will just be complexes of objects of A, but morphisms will be such that quasi-isomorphic complexes become isomorphic in D(A). In fact we can formally invert the quasi-isomorphisms in C(A) as follows:

There is a category D(A) and a functor Q: C(A) → D(A)

with the following two properties:

(a) Q inverts quasi-isomorphisms: if s: a → b is a quasi-isomorphism, then Q(s): Q(a) → Q(b) is an isomorphism.

(b) Q is universal with this property: if Q′ : C(A) → D′ is another functor which inverts quasi-isomorphisms, then there is a functor F : D(A) → D′ and an isomorphism of functors Q′ ≅ F ◦ Q.

First, consider the category C(A) as an oriented graph Γ, with the objects lying at the vertices and the morphisms being directed edges. Let Γ∗ be the graph obtained from Γ by adding in one extra edge s−1: b → a for each quasi-isomorphism s: a → b. Thus a finite path in Γ∗ is a sequence of the form f1 · f2 ·· · ·· fr−1 · fr where each fi is either a morphism of C(A), or is of the form s−1 for some quasi-isomorphism s of C(A). There is a unique minimal equivalence relation ∼ on the set of finite paths in Γ∗ generated by the following relations:

(a) s · s−1 ∼ idb and s−1 · s ∼ ida for each quasi-isomorphism s: a → b in C(A).

(b) g · f ∼ g ◦ f for composable morphisms f: a → b and g: b → c of C(A).

Define D(A) to be the category whose objects are the vertices of Γ∗ (these are the same as the objects of C(A)) and whose morphisms are given by equivalence classes of finite paths in Γ∗. Define a functor Q: C(A) → D(A) by using the identity morphism on objects, and by sending a morphism f of C(A) to the length one path in Γ∗ defined by f. The resulting functor Q satisfies the conditions of the above lemma.

The second property ensures that the category D(A) of the Lemma is unique up to equivalence of categories. We define the derived category of A to be any of these equivalent categories. The functor Q: C(A) → D(A) is called the localisation functor. Observe that there is a fully faithful functor

J: A → C(A)

which sends an object M to the trivial complex with M in the zeroth position, and a morphism F: M → N to the morphism of complexes

Composing with Q we obtain a functor A → D(A) which we denote by J. This functor J is fully faithful, and so defines an embedding A → D(A). By definition the functor Hi(−): C(A) → A inverts quasi-isomorphisms and so descends to a functor

Hi(−): D(A) → A

establishing that composite functor H0(−) ◦ J is isomorphic to the identity functor on A.

# Intuitive Algebra (Groupoid/Categorical Structure) of Open Strings As Morphisms

A geometric Dirichlet brane is a triple (L, E, ∇E) – a submanifold L ⊂ M, carrying a vector bundle E, with connection ∇E.

The real dimension of L is also often brought into the nomenclature, so that one speaks of a Dirichlet p-brane if p = dimRL.

An open string which stretches from a Dirichlet brane (L, E, ∇E) to a Dirichlet brane (K, F, ∇F), is a map X from an interval I ≅ [0,1] to M, such that X(0) ∈ L and X(1) ∈ K. An “open string history” is a map from R into open strings, or equivalently a map from a two-dimensional surface with boundary, say Σ ≡ I × R, to M , such that the two boundaries embed into L and K.

The quantum theory of these open strings is defined by a functional integral over these histories, with a weight which depends on the connections ∇E and ∇F. It describes the time evolution of an open string state which is a wave function in a Hilbert space HB,B′ labelled by the two choices of brane B = (L, E, ∇E) and B′ = (K, F, ∇F).

Distinct Dirichlet branes can embed into the same submanifold L. One way to represent this would be to specify the configurations of Dirichlet branes as a set of submanifolds with multiplicity. However, we can also represent this choice by using the choice of bundle E. Thus, a set of N identical branes will be represented by tensoring the bundle E with CN. The connection is also obtained by tensor product. An N-fold copy of the Dirichlet brane (L, E, ∇E) is thus a triple (L, E ⊗CN, ∇E ⊗ idN).

In physics, one visualizes this choice by labelling each open string boundary with a basis vector of CN, which specifies a choice among the N identical branes. These labels are called Chan-Paton factors. One then uses them to constrain the interactions between open strings. If we picture such an interaction as the joining of two open strings to one, the end of the first to the beginning of the second, we require not only the positions of the two ends to agree, but also the Chan-Paton factors. This operation is the intuitive algebra of open strings.

Mathematically, an algebra of open strings can always be tensored with a matrix algebra, in general producing a noncommutative algebra. More generally, if there is more than one possible boundary condition, then, rather than an algebra, it is better to think of this as a groupoid or categorical structure on the boundary conditions and the corresponding open strings. In the language of groupoids, particular open strings are elements of the groupoid, and the composition law is defined only for pairs of open strings with a common boundary. In the categorical language, boundary conditions are objects, and open strings are morphisms. The simplest intuitive argument that a non-trivial choice can be made here is to call upon the general principle that any local deformation of the world-sheet action should be a physically valid choice. In particular, particles in physics can be charged under a gauge field, for example the Maxwell field for an electron, the color Yang-Mills field for a quark, and so on. The wave function for a charged particle is then not complex-valued, but takes values in a bundle E.

Now, the effect of a general connection ∇E is to modify the functional integral by modifying the weight associated to a given history of the particle. Suppose the trajectory of a particle is defined by a map φ : R → M; then a natural functional on trajectories associated with a connection ∇ on M is simply its holonomy along the trajectory, a linear map from E|φ(t1) to E|φ(t2). The functional integral is now defined physically as a sum over trajectories with this holonomy included in the weight.

The simplest way to generalize this to a string is to consider the ls → 0 limit. Now the constraint of finiteness of energy is satisfied only by a string of vanishingly small length, effectively a particle. In this limit, both ends of the string map to the same point, which must therefore lie on L ∩ K.

The upshot is that, in this limit, the wave function of an open string between Dirichlet branes (L, E, ∇) and (K, F, ∇F) transforms as a section of E ⊠ F over L ∩ K, with the natural connection on the direct product. In the special case of (L, E, ∇E) ≅ (K, F, ∇F), this reduces to the statement that an open string state is a section of EndE. Open string states are sections of a graded vector bundle End E ⊗ Λ•T∗L, the degree-1 part of which corresponds to infinitesimal deformations of ∇E. In fact, these open string states are the infinitesimal deformations of ∇E, in the standard sense of quantum field theory, i.e., a single open string is a localized excitation of the field obtained by quantizing the connection ∇E. Similarly, other open string states are sections of the normal bundle of L within X, and are related in the same way to infinitesimal deformations of the submanifold. These relations, and their generalizations to open strings stretched between Dirichlet branes, define the physical sense in which the particular set of Dirichlet branes associated to a specified background X can be deduced from string theory.

# Conjuncted: Affine Schemes: How Would Functors Carry the Same Information?

If we go to the generality of schemes, the extra structure overshadows the topological points and leaves out crucial details so that we have little information, without the full knowledge of the sheaf. For example the evaluation of odd functions on topological points is always zero. This implies that the structure sheaf of a supermanifold cannot be reconstructed from its underlying topological space.

The functor of points is a categorical device to bring back our attention to the points of a scheme; however the notion of point needs to be suitably generalized to go beyond the points of the topological space underlying the scheme.

Grothendieck’s idea behind the definition of the functor of points associated to a scheme is the following. If X is a scheme, for each commutative ring A, we can define the set of the A-points of X in analogy to the way the classical geometers used to define the rational or integral points on a variety. The crucial difference is that we do not focus on just one commutative ring A, but we consider the A-points for all commutative rings A. In fact, the scheme we start from is completely recaptured only by the collection of the A-points for every commutative ring A, together with the admissible morphisms.

Let (rings) denote the category of commutative rings and (schemes) the category of schemes.

Let (|X|, OX) be a scheme and let T ∈ (schemes). We call the T-points of X, the set of all scheme morphisms {T → X}, that we denote by Hom(T, X). We then define the functor of points hX of the scheme X as the representable functor defined on the objects as

hX: (schemes)op → (sets), haX(A) = Hom(Spec A, X) = A-points of X

Notice that when X is affine, X ≅ Spec O(X) and we have

haX(A) = Hom(Spec A, O(X)) = Hom(O(X), A)

In this case the functor haX is again representable.

Consider the affine schemes X = Spec O(X) and Y = Spec O(Y). There is a one-to-one correspondence between the scheme morphisms X → Y and the ring morphisms O(X) → O(Y). Both hX and haare defined on morphisms in the natural way. If φ: T → S is a morphism and ƒ ∈ Hom(S, X), we define hX(φ)(ƒ) = ƒ ○ φ. Similarly, if ψ: A → Bis a ring morphism and g ∈ Hom(O(X), A), we define haX(ψ)(g) = ψ ○ g. The functors hX and haare for a given scheme X not really different but carry the same information. The functor of points hof a scheme X is completely determined by its restriction to the category of affine schemes, or equivalently by the functor

haX: (rings) → (sets), haX(A) = Hom(Spec A, X)

Let M = (|M|, OM) be a locally ringed space and let (rspaces) denote the category of locally ringed spaces. We define the functor of points of locally ringed spaces M as the representable functor

hM: (rspaces)op → (sets), hM(T) = Hom(T, M)

hM is defined on the manifold as

hM(φ)(g) = g ○ φ

If the locally ringed space M is a differentiable manifold, then

Hom(M, N) ≅ Hom(C(N), C(M))

This takes us to the theory of Yoneda’s Lemma.

Let C be a category, and let X, Y be objects in C and let hX: Cop → (sets) be the representable functors defined on the objects as hX(T) = Hom(T, X), and on the arrows as hX(φ)(ƒ) = ƒ . φ, for φ: T → S, ƒ ∈ Hom(T, X)

If F: Cop → (sets), then we have a one-to-one correspondence between sets:

{hX → F} ⇔ F(X)

The functor

h: C → Fun(Cop, (sets)), X ↦ hX,

is an equivalence of C with a full subcategory of functors. In particular, hX ≅ hY iff X ≅ Y and the natural transformations hX → hY are in one-to-one correspondence with the morphisms X → Y.

Two schemes (manifolds) are isomorphic iff their functors of points are isomorphic.

The advantages of using the functorial language are many. Morphisms of schemes are just maps between the sets of their A-points, respecting functorial properties. This often simplifies matters, allowing allowing for leaving the sheaves machinery in the background. The problem with such an approach, however, is that not all the functors from (schemes) to (sets) are the functors of points of a scheme, i.e., they are representable.

A functor F: (rings) → (sets) is of the form F(A) = Hom(Spec A, X) for a scheme X iff:

F is local or is a sheaf in Zariski Topology. This means that for each ring R and for every collection αi ∈ F(Rƒi), with (ƒi, i ∈ I) = R, so that αi and αj map to the same element in F(Rƒiƒj) ∀ i and j ∃ a unique element α ∈ F(R) mapping to each αi, and

F admits a cover by open affine subfunctors, which means that ∃ a family Ui of subfunctors of F, i.e. Ui(R) ⊂ F(R) ∀ R ∈ (rings), Ui = hSpec Ui, with the property that ∀ natural transformations ƒ: hSpec A  → F, the functors ƒ-1(Ui), defined as ƒ-1(Ui)(R) = ƒ-1(Ui(R)), are all representable, i.e. ƒ-1(Ui) = hVi, and the Vi form an open covering for Spec A.

This states the conditions we expect for F to be the functor of points of a scheme. Namely, locally, F must look like the functor of points of a scheme, moreover F must be a sheaf, i.e. F must have a gluing property that allows us to patch together the open affine cover.

# Functoriality in Low Dimensions. Note Quote.

Let CW be the category of CW-complexes and cellular maps, let CW0 be the full subcategory of path connected CW-complexes and let CW1 be the full subcategory of simply connected CW-complexes. Let HoCW denote the category of CW-complexes and homotopy classes of cellular maps. Let HoCWn denote the category of CW-complexes and rel n-skeleton homotopy classes of cellular maps. Dimension n = 1: It is straightforward to define a covariant truncation functor

t<n = t<1 : CW0 → HoCW together with a natural transformation

emb1 : t<1 → t<∞,

where t<∞ : CW0 → HoCW is the natural “inclusion-followed-by-quotient” functor given by t<∞(K) = K for objects K and t<∞(f) = [f] for morphisms f, such that for all objects K, emb1∗ : H0(t<1K) → H0(t<∞K) is an isomorphism and Hr(t<1K) = 0 for r ≥ 1. The details are as follows: For a path connected CW-complex K, set t<1(K) = k0, where k0 is a 0-cell of K. Let emb1(K) : t<1(K) = k0 → t<∞(K) = K be the inclusion of k0 in K. Then emb1∗ is an isomorphism on H0 as K is path connected. Clearly Hr(t<1K) = 0 for r ≥ 1. Let f : K → L be a cellular map between objects of CW0. The morphism t<1(f) : t<1(K) = k0 → l0 = t<1(L) is the homotopy class of the unique map from a point to a point. In particular, t<1(idK) = [idk0] and for a cellular map g : L → P we have t<1(gf) = t<1(g) ◦ t<1(f), so that t<1 is indeed a functor. To show that emb1 is a natural transformation, we need to see that

that is

commutes in HoCW. This is where we need the functor t<1 to have values only in HoCW, not in CW, because the square need certainly not commute in CW. (The points k0 and l0 do not know anything about f, so l0 need not be the image of k0 under f.) Since L is path connected, there is a path ω : I → L from l0 = ω(0) to f (k0) = ω(1). Then H : {k0} × I → L, H(k0, t) = ω(t), defines a homotopy from

k0 → l0 → L to k0 → K →f L.

Dimension n = 2: We will define a covariant truncation functor t<n = t<2 : CW1 → HoCW

together with a natural transformation
emb2 : t<2 → t<∞,

where t<∞ : CW1 → HoCW is as above (only restricted to simply connected spaces), such that for all objects K, emb2∗ : Hr(t<2K) → Hr(t<∞K) is an isomorphism for r = 0, 1, and Hr(t<2K) = 0 for r ≥ 2. For a simply connected CW-complex K, set t<2(K) = k0, where k0 is a 0-cell of K. Let emb2(K) : t<2(K) = k0 → t<∞(K) = K be the inclusion as in the case n = 1. It follows that emb2∗ is an isomorphism both on H0 as K is path connected and on H1 as H1(k0) = 0 = H1(K), while trivially Hr(t<2K) = 0 for r ≥ 2. On a cellular map f, t<2(f) is defined as in the case n = 1. As in the case n = 1, this yields a functor and emb2 is a natural transformation.

# Philosophical Isomorphism of Category Theory. Note Quote.

One philosophical reason for categorification is that it refines our concept of ‘sameness’ by allowing us to distinguish between isomorphism and equality. In a set, two elements are either the same or different. In a category, two objects can be ‘the same in a way’ while still being different. In other words, they can be isomorphic but not equal. Even more importantly, two objects can be the same in more than one way, since there can be different isomorphisms between them. This gives rise to the notion of the ‘symmetry group’ of an object: its group of automorphisms.

Consider, for example, the fundamental groupoid Π1(X) of a topological space X: the category with points of X as objects and homotopy classes of paths with fixed endpoints as morphisms. This category captures all the homotopy-theoretic information about X in dimensions ≤ 1. The group of automorphisms of an object x in this category is just the fundamental group π1(X,x). If we decategorify the fundamental groupoid of X, we forget how points in X are connected by paths, remembering only whether they are, and we obtain the set of components of X. This captures only the homotopy 0-type of X.

This example shows how decategorification eliminates ‘higher-dimensional information’ about a situation. Categorification is an attempt to recover this information. This example also suggests that we can keep track of the homotopy 2-type of X if we categorify further and distinguish between paths that are equal and paths that are merely isomorphic (i.e., homotopic). For this we should work with a ‘2-category’ having points of X as objects, paths as morphisms, and certain equivalence classes of homotopies between paths as 2-morphisms. In a marvelous self-referential twist, the definition of ‘2-category’ is simply the categorification of the definition of ‘category’. Like a category, a 2-category has a class of objects, but now for any pair x,y of objects there is no longer a set hom(x,y); instead, there is a category hom(x,y). Objects of hom(x,y) are called morphisms of C, and morphisms between them are called 2-morphisms of C. Composition is no longer a function, but rather a functor:

◦: hom(x, y) × hom(y, z) → hom(x, z)

For any object x there is an identity 1x ∈ hom(x,x). And now we have a choice. On the one hand, we can impose associativity and the left and right unit laws strictly, as equational laws. If we do this, we obtain the definition of ‘strict 2-category’. On the other hand, we can impose them only up to natural isomorphism, with these natural isomorphisms satisfying the coherence. This is clearly more compatible with the spirit of categorification. If we do this, we obtain the definition of ‘weak 2-category’. (Strict 2-categories are traditionally known as ‘2-categories’, while weak 2-categories are known as ‘bicategories’.)

The classic example of a 2-category is Cat, which has categories as objects, functors as morphisms, and natural transformations as 2-morphisms. The presence of 2-morphisms gives Cat much of its distinctive flavor, which we would miss if we treated it as a mere category. Indeed, Mac Lane has said that categories were originally invented, not to study functors, but to study natural transformations! A good example of two functors that are not equal, but only naturally isomorphic, are the identity functor and the ‘double dual’ functor on the category of finite-dimensional vector spaces. Given a topological space X, we can form a 2-category Π>sub>2(X) called the ‘fundamental 2-groupoid’ of X. The objects of this 2-category are the points of X. Given x, y ∈ X, the morphisms from x to y are the paths f: [0,1] → X starting at x and ending at y. Finally, given f, g ∈ hom(x, y), the 2-morphisms from f to g are the homotopy classes of paths in hom(x, y) starting at f and ending at g. Since the associative law for composition of paths holds only up to homotopy, this 2-category is a weak 2-category. If we decategorify the fundamental 2-groupoid of X, we obtain its fundamental groupoid.

From 2-categories it is a short step to dreaming of n-categories and even ω-categories — but it is not so easy to make these dreams into smoothly functioning mathematical tools. Roughly speaking, an n-category should be some sort of algebraic structure having objects, 1-morphisms between objects, 2-morphisms between 1-morphisms, and so on up to n-morphisms. There should be various ways of composing j-morphisms for 1 ≤ j ≤ n, and these should satisfy various laws. As with 2-categories, we can try to impose these laws either strictly or weakly.

Other approaches to n-categories use j-morphisms with other shapes, such as simplices, or opetopes. We believe that there is basically a single notion of weak n-category lurking behind these different approaches. If this is true, they will eventually be shown to be equivalent, and choosing among them will be merely a matter of convenience. However, the precise meaning of ‘equivalence’ here is itself rather subtle and n-categorical in flavor.

The first challenge to any theory of n-categories is to give an adequate treatment of coherence laws. Composition in an n-category should satisfy equational laws only at the top level, between n-morphisms. Any law concerning j-morphisms for j < n should hold only ‘up to equivalence’. Here a n-morphism is defined to be an ‘equivalence’ if it is invertible, while for j < n a j-morphism is recursively defined to be an equivalence if it is invertible up to equivalence. Equivalence is generally the correct substitute for the notion of equality in n-categorical mathematics. When laws are formulated as equivalences, these equivalences should in turn satisfy coherence laws of their own, but again only up to equivalence, and so on. This becomes ever more complicated and unmanageable with increasing n unless one takes a systematic approach to coherence laws.

The second challenge to any theory of n-categories is to handle certain key examples. First, for any n, there should be an (n + 1)-category nCat, whose objects are (small) n-categories, whose morphisms are suitably weakened functors between these, whose 2-morphisms are suitably weakened natural transformations, and so on. Here by ‘suitably weakened’ we refer to the fact that all laws should hold only up to equivalence. Second, for any topological space X, there should be an n-category Πn(X) whose objects are points of X, whose morphisms are paths, whose 2-morphisms are paths of paths, and so on, where we take homotopy classes only at the top level. Πn(X) should be an ‘n-groupoid’, meaning that all its j-morphisms are equivalences for 0 ≤ j ≤ n. We call Πn(X) the ‘fundamental n-groupoid of X’. Conversely, any n-groupoid should determine a topological space, its ‘geometric realization’.

In fact, these constructions should render the study of n-groupoids equivalent to that of homotopy n-types. A bit of the richness inherent in the concept of n-category becomes apparent when we make the following observation: an (n + 1)-category with only one object can be regarded as special sort of n-category. Suppose that C is an (n+1)-category with one object x. Then we can form the n-category C ̃ by re-indexing: the objects of C ̃ are the morphisms of C, the morphisms of C ̃ are the 2-morphisms of C, and so on. The n-categories we obtain this way have extra structure. In particular, since the objects of C ̃ are really morphisms in C from x to itself, we can ‘multiply’ (that is, compose) them.

The simplest example is this: if C is a category with a single object x, C ̃ is the set of endomorphisms of x. This set is actually a monoid. Conversely, any monoid can be regarded as the monoid of endomorphisms of x for some category with one object x. We summarize this situation by saying that ‘a one-object category is a monoid’. Similarly, a one-object 2-category is a monoidal category. It is natural to expect this pattern to continue in all higher dimensions; in fact, it is probably easiest to cheat and define a monoidal n-category to be an (n + 1)-category with one object.

Things get even more interesting when we iterate this process. Given an (n + k)-category C with only one object, one morphism, and so on up to one (k − 1)-morphism, we can form an n-category whose j-morphisms are the (j + k)-morphisms of C. In doing so we obtain a particular sort of n-category with extra structure and properties, which we call a ‘k-tuply monoidal’ n-category. Table below shows what we expect these to be like for low values of n and k. For example, the Eckmann-Hilton argument shows that a 2-category with one object and one morphism is a commutative monoid. Categorifying this argument, one can show that a 3-category with one object and one morphism is a braided monoidal category. Similarly, we expect that a 4-category with one object, one morphism and one 2-morphism is a symmetric monoidal category, though this has not been worked out in full detail, because of our poor understanding of 4-categories. The fact that both braided and symmetric monoidal categories appear in this table seems to explain why both are natural concepts.

In any reasonable approach to n-categories there should be an n-category nCatk whose objects are k-tuply monoidal weak n-categories. One should also be able to treat nCatk as a full sub-(n + k)-category of (n + k)Cat, though even for low n, k this is perhaps not as well known as it should be. Consider for example n = 0, k = 1. The objects of 0Cat1 are one-object categories, or monoids. The morphisms of 0Cat1 are functors between one-object categories, or monoid homomorphisms. But 0Cat1 also has 2-morphisms corresponding to natural transformations.

• Decategorification: (n, k) → (n − 1, k). Let C be a k-tuply monoidal n-category C. Then there should be a k-tuply monoidal (n − 1)-category DecatC whose j-morphisms are the same as those of C for j < n − 1, but whose (n − 1)-morphisms are isomorphism classes of (n − 1)-morphisms of C.

• Discrete categorification: (n, k) → (n + 1, k). There should be a ‘discrete’ k-tuply monoidal (n + 1)-category DiscC having the j-morphisms of C as its j-morphisms for j ≤ n, and only identity (n + 1)-morphisms. The decategorification of DiscC should be C.

• Delooping: (n, k) → (n + 1, k − 1). There should be a (k − 1)-tuply monoidal (n + 1)-category BC with one object obtained by reindexing, the j-morphisms of BC being the (j + 1)-morphisms of C. We use the notation ‘B’ and call BC the ‘delooping’ of C because of its relation to the classifying space construction in topology.

• Looping: (n, k) → (n − 1, k + 1). Given objects x, y in an n-category, there should be an (n − 1)-category hom(x, y). If x = y this should be a monoidal (n−1)-category, and we denote it as end(x). For k > 0, if 1 denotes the unit object of the k-tuply monoidal n-category C, end(1) should be a (k + 1)-tuply monoidal (n − 1)-category. We call this process ‘looping’, and denote the result as ΩC, because of its relation to loop space construction in topology. For k > 0, looping should extend to an (n + k)-functor Ω: nCatk → (n − 1)Catk+1. The case k = 0 is a bit different: we should be able to loop a ‘pointed’ n-category, one having a distinguished object x, by letting ΩC = end(x). In either case, the j-morphisms of ΩC correspond to certain (j − 1)-morphisms of C.

• Forgetting monoidal structure: (n, k) → (n, k−1). By forgetting the kth level of monoidal structure, we should be able to think of C as a (k−1)-tuply monoidal n-category FC. This should extend to an n-functor F: nCatk → nCatk−1.

• Stabilization: (n, k) → (n, k + 1). Though adjoint n-functors are still poorly understood, there should be a left adjoint to forgetting monoidal structure, which is called ‘stabilization’ and denoted by S: nCatk → nCatk+1.

• Forming the generalized center: (n,k) → (n,k+1). Thinking of C as an object of the (n+k)-category nCatk, there should be a (k+1)-tuply monoidal n-category ZC, the ‘generalized center’ of C, given by Ωk(end(C)). In other words, ZC is the largest sub-(n + k + 1)-category of (n + k)Cat having C as its only object, 1C as its only morphism, 11C as its only 2-morphism, and so on up to dimension k. This construction gets its name from the case n = 0, k = 1, where ZC is the usual center of the monoid C. Categorifying leads to the case n = 1, k = 1, which gives a very important construction of braided monoidal categories from monoidal categories. In particular, when C is the monoidal category of representations of a Hopf algebra H, ZC is the braided monoidal category of representations of the quantum double D(H).

# Catastrophe

Since, natural phenomena are continuously battered by perturbations, any thematic of the fundament that classifies critical points of smooth functions becomes the defining parameter of catastrophe. If a natural system is defined by a function of state variables, then the perturbations are represented by control parameters on which the function depends. An unfolding of a function is such a family: it is a smooth function of the state variables with the parameters satisfying a specific condition. Catastrophe’s aim is then to detect properties of a function by studying its unfoldings.

Thom studied the continuous crossing from a variety (space) to another, the connections through common boundaries and points between spaces even endowed with different dimensions (a research on the so called “cobordism” (1) which yielded him the Field Medal in 1958), until he singled out few universal forms, that is mathematical objects representing catastrophes or abrupt, although continuous, transitions of forms: specific singularities appearing when an object is submitted to bonds, such as restrictions with regard to its ordinary dimensions, that it accepts except in particular points where it offers resistance by concentrating there, so to say, its structure. The theory is used to classify how stable equilibria change when parameters are varied, with points in parameter space at which qualitative changes affects behavior termed catastrophe points. Catastrophe theory should apply to any gradient system where the force can be written as the negative gradient of a potential, and the points where the gradient vanishes are what the theory prefers degenerate points. There are seven elementary types of catastrophes or generic singularities of an application and Thom decided to study their applications in caustics, surfaces lit according to different angle shots, reflections and refractions. Initially catastrophe theory was of use just to explain caustic formation and only afterwards many other phenomena, but without yielding quantitative solutions and exact predictions, rather qualitatively framing situations that were uncontrollable by only reductionistic quantitative methods summing up elementary units. The study of forms in irregular, accidental and even chaotic situations had truly in advance led scientists like Poincaré and Hadamard, to single out structurally invariable catastrophic evolutions in the most disparate phenomena, in terms of divergences due to sensitive dependence on little variations of the initial conditions. In such cases there were not exact laws, rather evolutionary asymptotic tendencies, which did not allow exact predictions, in case only statistic ones. While when exact predictions are possible, in terms of strict laws and explicit equations, the catastrophe ceases.

For Thom, catastrophe was a methodology. He says,

Mathematicians should see catastrophe theory as just a part of the theory of local singularities of smooth morphisms, or, if they are interested in the wider ambitions of this theory, as a dubious methodology concerning the stability or instability of natural systems….the whole of qualitative dynamics, all the ‘chaos’ theories talked about so much today, depend more or less on it.

Thom gets more philosophical when it comes to the question of morphogenesis. Stability for Thom is a natural condition to place upon mathematical models for processes in nature because the conditions under which such processes take place can never be duplicated and therefore must be invariant under small perturbations and hence stable. what makes morphogenesis interesting for Thom is the fact that locally, as the transition proceeds, the parameter varies, from a stable state of a vector field to an unstable state and back to a stable state by means of a process which locally models system’s morphogenesis. Furthermore, what is observed in a process undergoing morphogenesis is precisely the shock wave and resulting configuration of chreods (2) separated by strata of the shockwave, at each interval of time and over intervals of observation time. It then follows “that to classify an observed phenomenon or to support a hypothesis about the local underlying dynamic, we need in principle only observe the process, study the observed catastrophe or discontinuity set and try to relate it to one of the finitely many universal catastrophe sets, which would become then our main object of interest. Even if a process depends on a large number of physical parameters, as long as it is described by the gradient model, its description would involve one of seven elementary catastrophes; in particular one can give a relatively simple mathematical description of such apparently complicated processes even if one does not know what the relevant physical parameters are or what the physical mechanism of the process is. According to Thom, “if we consider an unfolding, we can obtain a qualitative intelligence about the behaviors of a system in the neighborhood of an unstable equilibrium point. this idea was not accepted widely and was criticized by applied mathematicians because for them only numerical exactness allowed prediction and therefore efficient action. After the work of Grothendieck, it is known that the theory of singularity unfolding is a particular case of a general category, the theory of flat deformations of an analytic set and for flat local deformations of an analytic set only the hyper surface case has a smooth unfolding of finite dimension. For Thom, this meant the if we wanted to continue the scientific domain of calculable exact laws, we would be justified in considering the instance where an analytic process leads to a singularity of codimension one in internal variables. Might we then not expect that the process be diffused and subsequently propagated in the unfolding according to a mode that is to be defined? Such an argument allows one to think that the Wignerian domain of exact laws can be extended into a region where physical processes are no longer calculable but where analytic continuation remains qualitatively valid.

Anyway, catastrophe theory studies forms as qualitative discontinuities though on a continuous substrate. In any case forms as mental facts are immersed in a matter which is still a thought object. The more you try to analyze it the more it appears as a fog, revealing a more and more complex and inexhaustible weaving the more it refines itself through the forms it assumes. In fact complexity is more and more ascertained until a true enigma is reached when you once for all want to define reality as a universe endowed with a high number of dimensions and then object of mental experiences to which even objective phenomena are at the end concretely reduced. Concrete reality is yet more evident than a scientific explanation and naïve ontology appears more concrete than the scientific one. It is steady and universal, while the latter is always problematic and revisable. Besides, according to Bachelard, while naïve explanation is immediately reflected into the ordinary language which is accessible to everybody, the claimed scientific explanation goes with its jargon beyond immediate experience, away from the life world which only we can know immediately.

As for example, the continuous character of reality, which Thom entrusts to a world intuition as a frame of the phenomenological discontinuities themselves, is instead contradicted by the present tendency to reduce all to discrete units of information (bits) of modern computing. Of course it has a practical value: an animal individuating a prey perceives it as an entity which is absolutely distinct from its environment, just as we discretize linguistic phonemata to learn speaking without confounding them. Yet a continuous background remains, notwithstanding the tendency of our brains to discretize. Such background is for example constituted by space and time. Continuum is said an illusion as exemplified by a film which appears continuous to us, while it is made of discrete frames. Really it is an illusion but with a true mental base, otherwise it would not arise at all, and such base is just the existence of continuum. Really we perceive continuum but need discreteness, finiteness in order to keep things under control. Anyway quantum mechanics seems to introduce discreteness in absolute terms, something we do not understand but which is operatively valid, as is shown by the possibility to localize or delocalize a wave packet by simply varying the value distributions of complementary variables as position and momentum or time and energy, according to Heisenberg’s principle of indetermination. Anyway, also the apparent quantum discontinuity hides a continuity which, always according to Heisenberg’s principle, may be only obscured and not cancelled in several phenomena. It is difficult to conceive but not monstrous. The hypothesis according to which we are finite and discrete in our internal structure is afterwards false for we are more than that. We have hundreds billions of neurons, which are in continuous movement, as they are constituted by molecules continuously vibrating in the space, so giving place to infinite possible variations in a considerable dimensions number, even though we are reduced to the smallest possible number of states and dimensions to deal with the system under study, according to a technical and algorithmic thought which is operatively effective, certainly practically motivated but unidentifiable with reality.

(1) Two manifolds M and N are said to be cobordant if their disjoint union is the boundary of some other manifold. Given the extreme difficulty of the classification of manifolds it would seem very unlikely that much progress could be made in classifying manifolds up to cobordism. However, René Thom, in his remarkable, if unreadable, 1954 paper (French), gave the full solution to this problem for unoriented manifolds, as well as many powerful insights into the methods for solving it in the cases of manifolds with additional structure. The key step was the reduction of the cobordism problem to a homotopy problem, although the homotopy problem is still far from trivial. This was later generalized by Lev Pontrjagin, and this result is now known as the Thom-Pontrjagin theorem.

(2) Every natural process decomposes into structurally stable islands, the chreods. The set of chreods and the multidimensional syntax controlling their positions constitute the semantic model. When the chreod is considered as a word of this multidimensional language, the meaning (signification) of this word is precisely that of the global topology of the associated attractor (or attractors) and of the catastrophes that it (or they) undergo. In particular, the signification of a given attractor is defined by the geometry of its domain of existence on the space of external variables and the topology of the regulation catastrophes bounding that domain. One result of this is that the signification of a form (chreod) manifests itself only by the catastrophes that create or destroy it. This gives the axiom dear to the formal linguists: that the meaning of a word is nothing more than the use of the word; this is also the axiom of the “bootstrap” physicists, according to whom a particle is completely defined by the set of interactions in which it participates.