Why Can’t There Be Infinite Descending Chain Of Quotient Representations? – Part 3

For a quiver Q, the category Rep(Q) of finite-dimensional representations of Q is abelian. A morphism f : V → W in the category Rep(Q) defined by a collection of morphisms fi : Vi → Wi is injective (respectively surjective, an isomorphism) precisely if each of the linear maps fi is.

There is a collection of simple objects in Rep(Q). Indeed, each vertex i ∈ Q0 determines a simple object Si of Rep(Q), the unique representation of Q up to isomorphism for which dim(Vj) = δij. If Q has no directed cycles, then these so-called vertex simples are the only simple objects of Rep(Q), but this is not the case in general.

If Q is a quiver, then the category Rep(Q) has finite length.

Given a representation E of a quiver Q, then either E is simple, or there is a nontrivial short exact sequence

0 → A → E → B → 0

Now if B is not simple, then we can break it up into pieces. This process must halt, as every representation of Q consists of finite-dimensional vector spaces. In the end, we will have found a simple object S and a surjection f : E → S. Take E1 ⊂ E to be the kernel of f and repeat the argument with E1. In this way we get a filtration

… ⊂ E3 ⊂ E2 ⊂ E1 ⊂ E

with each quotient object Ei−1/Ei simple. Once again, this filtration cannot continue indefinitely, so after a finite number of steps we get En = 0. Renumbering by setting Ei := En−i for 1 ≤ i ≤ n gives a Jordan-Hölder filtration for E. The basic reason for finiteness is the assumption that all representations of Q are finite-dimensional. This means that there can be no infinite descending chains of subrepresentations or quotient representations, since a proper subrepresentation or quotient representation has strictly smaller dimension.

In many geometric and algebraic contexts, what is of interest in representations of a quiver Q are morphisms associated to the arrows that satisfy certain relations. Formally, a quiver with relations (Q, R) is a quiver Q together with a set R = {ri} of elements of its path algebra, where each ri is contained in the subspace A(Q)aibi of A(Q) spanned by all paths p starting at vertex aiand finishing at vertex bi. Elements of R are called relations. A representation of (Q, R) is a representation of Q, where additionally each relation ri is satisfied in the sense that the corresponding linear combination of homomorphisms from Vai to Vbi is zero. Representations of (Q, R) form an abelian category Rep(Q, R).

A special class of relations on quivers comes from the following construction, inspired by the physics of supersymmetric gauge theories. Given a quiver Q, the path algebra A(Q) is non-commutative in all but the simplest examples, and hence the sub-vector space [A(Q), A(Q)] generated by all commutators is non-trivial. The vector space quotientA(Q)/[A(Q), A(Q)] is seen to have a basis consisting of the cyclic paths anan−1 · · · a1 of Q, formed by composable arrows ai of Q with h(an) = t(a1), up to cyclic permutation of such paths. By definition, a superpotential for the quiver Q is an element W ∈ A(Q)/[A(Q), A(Q)] of this vector space, a linear combination of cyclic paths up to cyclic permutation.

Indecomposable Objects – Part 1

An object X in a category C with an initial object is called indecomposable if X is not the initial object and X is not isomorphic to a coproduct of two noninitial objects. A group G is called indecomposable if it cannot be expressed as the internal direct product of two proper normal subgroups of G. This is equivalent to saying that G is not isomorphic to the direct product of two nontrivial groups.

A quiver Q is a directed graph, specified by a set of vertices Q0, a set of arrows Q1, and head and tail maps

h, t : Q1 → Q0

We always assume that Q is finite, i.e., the sets Q0 and Q1 are finite.

A (complex) representation of a quiver Q consists of complex vector spaces Vi for i ∈ Qand linear maps

φa : Vt(a) → Vh(a)

for a ∈ Q1. A morphism between such representations (V, φ) and (W, ψ) is a collection of linear maps fi : Vi → Wi for i ∈ Q0 such that the diagram

commutes ∀ a ∈ Q1. A representation of Q is finite-dimensional if each vector space Vi is. The dimension vector of such a representation is just the tuple of non-negative integers (dim Vi)i∈Q0.

Rep(Q) is the category of finite-dimensional representations of Q. This category is additive; we can add morphisms by adding the corresponding linear maps fi, the trivial representation in which each Vi = 0 is a zero object, and the direct sum of two representations is obtained by taking the direct sums of the vector spaces associated to each vertex. If Q is the one-arrow quiver, • → •, then the classification of indecomposable objects of Rep(Q), yields the objects E ∈ Rep(Q) which do not have a non-trivial direct sum decomposition E = A ⊕ B. An object of Rep(Q) is just a linear map of finite-dimensional vector spaces f: V1 → V2. If W = im(f) is a nonzero proper subspace of V2, then the splitting V2 = U ⊕ W, and the corresponding object of Rep(Q) splits as a direct sum of the two representations

V1 →ƒ W and 0 → W

Thus if an object f: V1 → V2 of Rep(Q) is indecomposable, the map f must be surjective. Similarly, if ƒ is nonzero, then it must also be injective. Continuing in this way, one sees that Rep(Q) has exactly three indecomposable objects up to isomorphism:

C → 0, 0 → C, C →id C

Every other object of Rep(Q) is a direct sum of copies of these basic representations.

Ringed Spaces (2)

Let |M| be a topological space. A presheaf of commutative algebras F on X is an assignment

U ↦ F(U), U open in |M|, F(U) is a commutative algebra, such that the following holds,

(1) If U ⊂ V are two open sets in |M|, ∃ a morphism rV, U: F(V) → F(U), called the restriction morphism and often denoted by rV, U(ƒ) = ƒ|U, such that

(i) rU, U = id,

(ii) rW, U = rV, U ○ rW, V

A presheaf ƒ is called a sheaf if the following holds:

(2) Given an open covering {Ui}i∈I of U and a family {ƒi}i∈I, ƒi ∈ F(Ui) such that ƒi|Ui ∩ Uj = ƒj|Ui ∩ Uj ∀ i, j ∈ I, ∃ a unique ƒ ∈ F(U) with ƒ|Ui = ƒi

The elements in F(U) are called sections over U, and with U = |M|, these are termed global sections.

The assignments U ↦ C(U), U open in the differentiable manifold M and U ↦ OX(U), U open in algebraic variety X are examples of sheaves of functions on the topological spaces |M| and |X| underlying the differentiable manifold M and the algebraic variety X respectively.

In the language of categories, the above definition says that we have defined a functor, F, from top(M) to (alg), where top(M) is the category of the open sets in the topological space |M|, the arrows given by the inclusions of open sets while (alg) is the category of commutative algebras. In fact, the assignment U ↦ F(U) defines F on the objects while the assignment

U ⊂ V ↦ rV, U: F(V) → F(U)

defines F on the arrows.

Let |M| be a topological space. We define a presheaf of algebras on |M| to be a functor

F: top(M)op → (alg)

The suffix “op” denotes as usual the opposite category; in other words, F is a contravariant functor from top(M) to (alg). A presheaf is a sheaf if it satisfies the property (2) of the above definition.

If F is a (pre)sheaf on |M| and U is open in |M|, we define F|U, the (pre)sheaf F restricted to U, as the functor F restricted to the category of open sets in U (viewed as a topological space itself).

Let F be a presheaf on the topological space |M| and let x be a point in |M|. We define the stalk Fx of F, at the point x, as the direct limit

lim F(U)

where the direct limit is taken ∀ the U open neighbourhoods of x in |M|. Fx consists of the disjoint union of all pairs (U, s) with U open in |M|, x ∈ U, and s ∈ F(U), modulo the equivalence relation: (U, s) ≅ (V, t) iff ∃ a neighbourhood W of x, W ⊂ U ∩ V, such that s|W = t|W.

The elements in Fx are called germs of sections.

Let F and G be presheaves on |M|. A morphism of presheaves φ: F → G, for each open set U in |M|, such that ∀ V ⊂ U, the following diagram commutes

Equivalently and more elegantly, one can also say that a morphism of presheaves is a natural transformation between the two presheaves F and G viewed as functors.

A morphism of sheaves is just a morphism of the underlying presheaves.

Clearly any morphism of presheaves induces a morphism on the stalks: φx: Fx → Gx. The sheaf property, i.e., property (2) in the above definition, ensures that if we have two morphisms of sheaves φ and ψ, such that φx = ψx ∀ x, then φ = ψ.

We say that the morphism of sheaves is injective (resp. surjective) if x is injective (resp. surjective).

On the notion of surjectivity, however, one should exert some care, since we can have a surjective sheaf morphism φ: F → G such that φU: F(U) → G(U) is not surjective for some open sets U. This strange phenomenon is a consequence of the following fact. While the assignment U ↦ ker(φ(U)) always defines a sheaf, the assignment

U ↦ im( φ(U)) = F(U)/G(U)

defines in general only a presheaf and not all the presheaves are sheaves. A simple example is given by the assignment associating to an open set U in R, the algebra of constant real functions on U. Clearly this is a presheaf, but not a sheaf.

We can always associate, in a natural way, to any presheaf a sheaf called its sheafification. Intuitively, one may think of the sheafification as the sheaf that best “approximates” the given presheaf. For example, the sheafification of the presheaf of constant functions on open sets in R is the sheaf of locally constant functions on open sets in R. We construct the sheafification of a presheaf using the étalé space, which we also need in the sequel, since it gives an equivalent approach to sheaf theory.

Let F be a presheaf on |M|. We define the étalé space of F to be the disjoint union ⊔x∈|M| Fx. Let each open U ∈ |M| and each s ∈ F(U) define the map šU: U ⊔x∈|U| Fx, šU(x) = sx. We give to the étalé space the finest topology that makes the maps š continuous, ∀ open U ⊂ |M| and all sections s ∈ F(U). We define Fet to be the presheaf on |M|:

U ↦ Fet(U) = {šU: U → ⊔x∈|U| Fx, šU(x) = sx ∈ Fx}

Let F be a presheaf on |M|. A sheafification of F is a sheaf F~, together with a presheaf morphism α: F → Fsuch that

(1) any presheaf morphism ψ: F → G, G a sheaf factors via α, i.e. ψ: F →α F~ → G,

(2) F and Fare locally isomorphic, i.e., ∃ an open cover {Ui}i∈I of |M| such that F(Ui) ≅ F~(Ui) via α.

Let F and G be sheaves of rings on some topological space |M|. Assume that we have an injective morphism of sheaves G → F such that G(U) ⊂ F(U) ∀ U open in |M|. We define the quotient F/G to be the sheafification of the image presheaf: U ↦ F(U)/G(U). In general F/G (U) ≠ F(U)/G(U), however they are locally isomorphic.

Ringed space is a pair M = (|M|, F) consisting of a topological space |M| and a sheaf of commutative rings F on |M|. This is a locally ringed space, if the stalk Fx is a local ring ∀ x ∈ |M|. A morphism of ringed spaces φ: M = (|M|, F) → N = (|N|, G) consists of a morphism |φ|: |M| → |N| of the topological spaces and a sheaf morphism φ*: ON → φ*OM, where φ*OM is a sheaf on |N| and defined as follows:

*OM)(U) = OM-1(U)) ∀ U open in |N|

Morphism of ringed spaces induces a morphism on the stalks for each

x ∈ |M|: φx: ON,|φ|(x) → OM,x

If M and N are locally ringed spaces, we say that the morphism of ringed spaces φ is a morphism of locally ringed spaces if φx is local, i.e. φ-1x(mM,x) = mN,|φ|(x), where mN,|φ|(x) and mM,x are the maximal ideals in the local rings ON,|φ|(x) and OM,x respectively.

Homotopically Truncated Spaces.

The Eckmann–Hilton dual of the Postnikov decomposition of a space is the homology decomposition (or Moore space decomposition) of a space.

A Postnikov decomposition for a simply connected CW-complex X is a commutative diagram

such that pn∗ : πr(X) → πr(Pn(X)) is an isomorphism for r ≤ n and πr(Pn(X)) = 0 for r > n. Let Fn be the homotopy fiber of qn. Then the exact sequence

πr+1(PnX) →qn∗ πr+1(Pn−1X) → πr(Fn) → πr(PnX) →qn∗ πr(Pn−1X)

shows that Fn is an Eilenberg–MacLane space K(πnX, n). Constructing Pn+1(X) inductively from Pn(X) requires knowing the nth k-invariant, which is a map of the form kn : Pn(X) → Yn. The space Pn+1(X) is then the homotopy fiber of kn. Thus there is a homotopy fibration sequence

K(πn+1X, n+1) → Pn+1(X) → Pn(X) → Yn

This means that K(πn+1X, n+1) is homotopy equivalent to the loop space ΩYn. Consequently,

πr(Yn) ≅ πr−1(ΩYn) ≅ πr−1(K(πn+1X, n+1) = πn+1X, r = n+2,

= 0, otherwise.

and we see that Yn is a K(πn+1X, n+2). Thus the nth k-invariant is a map kn : Pn(X) → K(πn+1X, n+2)

Note that it induces the zero map on all homotopy groups, but is not necessarily homotopic to the constant map. The original space X is weakly homotopy equivalent to the inverse limit of the Pn(X).

Applying the paradigm of Eckmann–Hilton duality, we arrive at the homology decomposition principle from the Postnikov decomposition principle by changing:

• the direction of all arrows
• π to H
• loops Ω to suspensions S
• fibrations to cofibrations and fibers to cofibers
• Eilenberg–MacLane spaces K(G, n) to Moore spaces M(G, n)
• inverse limits to direct limits

A homology decomposition (or Moore space decomposition) for a simply connected CW-complex X is a commutative diagram

such that jn∗ : Hr(X≤n) → Hr(X) is an isomorphism for r ≤ n and Hr(X≤n) = 0 for

r > n. Let Cn be the homotopy cofiber of in. Then the exact sequence

Hr(X≤n−1) →in∗ Hr(X≤n) → Hr(Cn) →in∗ Hr−1(X≤n−1) → Hr−1(X≤n)

shows that Cn is a Moore space M(HnX, n). Constructing X≤n+1 inductively from X≤n requires knowing the nth k-invariant, which is a map of the form kn : Yn → X≤n.

The space X≤n+1 is then the homotopy cofiber of kn. Thus there is a homotopy cofibration sequence

Ynkn X≤nin+1 X≤n+1 → M(Hn+1X, n+1)

This means that M(Hn+1X, n+1) is homotopy equivalent to the suspension SYn. Consequently,

H˜r(Yn) ≅ Hr+1(SYn) ≅ Hr+1(M(Hn+1X, n+1)) = Hn+1X, r = n,

= 0, otherwise

and we see that Yn is an M(Hn+1X, n). Thus the nth k-invariant is a map kn : M(Hn+1X, n) → X≤n

It induces the zero map on all reduced homology groups, which is a nontrivial statement to make in degree n:

kn∗ : Hn(M(Hn+1X, n)) ∼= Hn+1(X) → Hn(X) ∼= Hn(X≤n)

The original space X is homotopy equivalent to the direct limit of the X≤n. The Eckmann–Hilton duality paradigm, while being a very valuable organizational principle, does have its natural limitations. Postnikov approximations possess rather good functorial properties: Let pn(X) : X → Pn(X) be a stage-n Postnikov approximation for X, that is, pn(X) : πr(X) → πr(Pn(X)) is an isomorphism for r ≤ n and πr(Pn(X)) = 0 for r > n. If Z is a space with πr(Z) = 0 for r > n, then any map g : X → Z factors up to homotopy uniquely through Pn(X). In particular, if f : X → Y is any map and pn(Y) : Y → Pn(Y) is a stage-n Postnikov approximation for Y, then, taking Z = Pn(Y) and g = pn(Y) ◦ f, there exists, uniquely up to homotopy, a map pn(f) : Pn(X) → Pn(Y) such that

homotopy commutes. Let X = S22 e3 be a Moore space M(Z/2,2) and let Y = X ∨ S3. If X≤2 and Y≤2 denote stage-2 Moore approximations for X and Y, respectively, then X≤2 = X and Y≤2 = X. We claim that whatever maps i : X≤2 → X and j : Y≤2 → Y such that i : Hr(X≤2) → Hr(X) and j : Hr(Y≤2) → Hr(Y) are isomorphisms for r ≤ 2 one takes, there is always a map f : X → Y that cannot be compressed into the stage-2 Moore approximations, i.e. there is no map f≤2 : X≤2 → Y≤2 such that

commutes up to homotopy. We shall employ the universal coefficient exact sequence for homotopy groups with coefficients. If G is an abelian group and M(G, n) a Moore space, then there is a short exact sequence

0 → Ext(G, πn+1Y) →ι [M(G, n), Y] →η Hom(G, πnY) → 0,

where Y is any space and [−,−] denotes pointed homotopy classes of maps. The map η is given by taking the induced homomorphism on πn and using the Hurewicz isomorphism. This universal coefficient sequence is natural in both variables. Hence, the following diagram commutes:

Here we will briefly write E2(−) = Ext(Z/2,−) so that E2(G) = G/2G, and EY (−) = Ext(−, π3Y). By the Hurewicz theorem, π2(X) ∼= H2(X) ∼= Z/2, π2(Y) ∼= H2(Y) ∼= Z/2, and π2(i) : π2(X≤2) → π2(X), as well as π2(j) : π2(Y≤2) → π2(Y), are isomorphisms, hence the identity. If a homomorphism φ : A → B of abelian groups is onto, then E2(φ) : E2(A) = A/2A → B/2B = E2(B) remains onto. By the Hurewicz theorem, Hur : π3(Y) → H3(Y) = Z is onto. Consequently, the induced map E2(Hur) : E23Y) → E2(H3Y) = E2(Z) = Z/2 is onto. Let ξ ∈ E2(H3Y) be the generator. Choose a preimage x ∈ E23Y), E2(Hur)(x) = ξ and set [f] = ι(x) ∈ [X,Y]. Suppose there existed a homotopy class [f≤2] ∈ [X≤2, Y≤2] such that

j[f≤2] = i[f].

Then

η≤2[f≤2] = π2(j)η≤2[f≤2] = ηj[f≤2] = ηi[f] = π2(i)η[f] = π2(i)ηι(x) = 0.

Thus there is an element ε ∈ E23Y≤2) such that ι≤2(ε) = [f≤2]. From ιE2π3(j)(ε) = jι≤2(ε) = j[f≤2] = i[f] = iι(x) = ιEY π2(i)(x)

we conclude that E2π3(j)(ε) = x since ι is injective. By naturality of the Hurewicz map, the square

commutes and induces a commutative diagram upon application of E2(−):

It follows that

ξ = E2(Hur)(x) = E2(Hur)E2π3(j)(ε) = E2H3(j)E2(Hur)(ε) = 0,

a contradiction. Therefore, no compression [f≤2] of [f] exists.

Given a cellular map, it is not always possible to adjust the extra structure on the source and on the target of the map so that the map preserves the structures. Thus the category theoretic setup automatically, and in a natural way, singles out those continuous maps that can be compressed into homologically truncated spaces.

Hypersurfaces

Let (S, CS) and (M, CM) be manifolds of dimension k and n, respectively, with 1 ≤ k ≤ n. A smooth map : S → M is said to be an imbedding if it satisfies the following three conditions.

(I1) Ψ is injective.

(I2) At all points p in S, the associated (push-forward) linear map (Ψp) : Sp → MΨ(p) is injective.

(I3) ∀ open sets O1 in S, Ψ[O1] = [S] ∩ O2 for some open set O2 in M. (Equivalently, the inverse map Ψ−1 : Ψ[S] → S is continuous with respect to the relative topology on [S].)

Several comments about the definition are in order. First, given any point p in S, (I2) implies that (Ψp)[Sp] is a k-dimensional subspace of MΨ(p). So the condition cannot be satisfied unless k ≤ n. Second, the three conditions are independent of one another. For example, the smooth map Ψ : R → R2 defined by (s) = (cos(s), sin(s)) satisfies (I2) and (I3) but is not injective. It wraps R round and round in a circle. On the other hand, the smooth map : R → R defined by (s) = s3 satisfies (I1) and (I3) but is not an imbedding because (Ψ0) : R0 → R0 is not injective. (Here R0 is the tangent space to the manifold R at the point 0). Finally, a smooth map : S → M can satisfy (I1) and (I2) but still have an image that “bunches up on itself.” It is precisely this possibility that is ruled out by condition (I3). Consider, for example, a map : R → R2 whose image consists of part of the image of the curve y = sin(1/x) smoothly joined to the segment {(0, y) : y < 1}, as in the figure below. It satisfies conditions (I1) and (I2) but is not an imbedding because we can find an open interval O1 in R such that given any open set O2 in R2, Ψ[O1] ≠ O2 ∩ Ψ[R].

Suppose(S, CS) and (M, CM) are manifolds with S ⊆ M. We say that (S, CS) is an imbedded submanifold of (M, CM) if the identity map id: S → M is an imbedding. If, in addition, k = n − 1 (where k and n are the dimensions of the two manifolds), we say that (S, CS) is a hypersurface in (M, CM). Let (S, CS) be a k-dimensional imbedded submanifold of the n-dimensional manifold (M, CM), and let p be a point in S. We need to distinguish two senses in which one can speak of “tensors at p.” There are tensors over the vector space Sp (call them S-tensors at p) and ones over the vector space Mp (call them M-tensors at p). So, for example, an S-vector ξ ̃a at p makes assignments to maps of the form f ̃: O ̃ → R where O ̃ is a subset of S that is open in the topology induced by CS, and f ̃ is smooth relative to CS. In contrast, an M-vector ξa at p makes assignments to maps of the form f : O → R where O is a subset of M that is open in the topology induced by CM, and f is smooth relative to CM. Our first task is to consider the relation between S-tensors at p and M-tensors there.

Let us say that ξa ∈ (Mp)a is tangent to S if ξa ∈ (idp)[(Sp)a]. (This makes sense. We know that (idp)[(Sp)a] is a k-dimensional subspace of (Mp)a; ξa either belongs to that subspace or it does not.) Let us further say that ηa in (Mp)a is normal to S if ηaξa =0 ∀ ξa ∈ (Mp)a that are tangent to S. Each of these classes of vectors has a natural vector space structure. The space of vectors ξa ∈ (Mp)a tangent to S has dimension k. The space of co-vectors ηa ∈ (Mp)a normal to S has dimension (n − k).

Grothendieck’s Universes and Wiles Proof (Fermat’s Last Theorem). Thought of the Day 77.0

In formulating the general theory of cohomology Grothendieck developed the concept of a universe – a collection of sets large enough to be closed under any operation that arose. Grothendieck proved that the existence of a single universe is equivalent over ZFC to the existence of a strongly inaccessible cardinal. More precisely, 𝑈 is the set 𝑉𝛼 of all sets with rank below 𝛼 for some uncountable strongly inaccessible cardinal.

Colin McLarty summarised the general situation:

Large cardinals as such were neither interesting nor problematic to Grothendieck and this paper shares his view. For him they were merely legitimate means to something else. He wanted to organize explicit calculational arithmetic into a geometric conceptual order. He found ways to do this in cohomology and used them to produce calculations which had eluded a decade of top mathematicians pursuing the Weil conjectures. He thereby produced the basis of most current algebraic geometry and not only the parts bearing on arithmetic. His cohomology rests on universes but weaker foundations also suffice at the loss of some of the desired conceptual order.

The applications of cohomology theory implicitly rely on universes. Most number theorists regard the applications as requiring much less than their ‘on their face’ strength and in particular believe the large cardinal appeals are ‘easily eliminable’. There are in fact two issues. McLarty writes:

Wiles’s proof uses hard arithmetic some of which is on its face one or two orders above PA, and it uses functorial organizing tools some of which are on their face stronger than ZFC.

There are two current programs for verifying in detail the intuition that the formal requirements for Wiles proof of Fermat’s last theorem can be substantially reduced. On the one hand, McLarty’s current work aims to reduce the ‘on their face’ strength of the results in cohomology from large cardinal hypotheses to finite order Peano. On the other hand Macintyre aims to reduce the ‘on their face’ strength of results in hard arithmetic to Peano. These programs may be complementary or a full implementation of Macintyre’s might avoid the first.

McLarty reduces

1. ‘ all of SGA (Revêtements Étales et Groupe Fondamental)’ to Bounded Zermelo plus a Universe.
2. “‘the currently existing applications” to Bounded Zermelo itself, thus the con-sistency strength of simple type theory.’ The Grothendieck duality theorem and others like it become theorem schema.

The essential insight of the McLarty’s papers on cohomology is the role of replacement in giving strength to the universe hypothesis. A 𝑍𝐶-universe is defined to be a transitive set U modeling 𝑍𝐶 such that every subset of an element of 𝑈 is itself an element of 𝑈. He remarks that any 𝑉𝛼 for 𝛼 a limit ordinal is provable in 𝑍𝐹𝐶 to be a 𝑍𝐶-universe. McLarty then asserts the essential use of replacement in the original Grothendieck formulation is to prove: For an arbitrary ring 𝑅 every module over 𝑅 embeds in an injective 𝑅-module and thus injective resolutions exist for all 𝑅-modules. But he gives a proof in a system with the proof theoretic strength of finite order arithmetic that every sheaf of modules on any small site has an infinite resolution.

Angus Macintyre dismisses with little comment the worries about the use of ‘large-structure’ tools in Wiles proof. He begins his appendix,

At present, all roads to a proof of Fermat’s Last Theorem pass through some version of a Modularity Theorem (generically MT) about elliptic curves defined over Q . . . A casual look at the literature may suggest that in the formulation of MT (or in some of the arguments proving whatever version of MT is required) there is essential appeal to higher-order quantification, over one of the following.

He then lists such objects as C, modular forms, Galois representations …and summarises that a superficial formulation of MT would be 𝛱1m for some small 𝑚. But he continues,

I hope nevertheless that the present account will convince all except professional sceptics that MT is really 𝛱01.

There then follows a 13 page highly technical sketch of an argument for the proposition that MT can be expressed by a sentence in 𝛱01 along with a less-detailed strategy for proving MT in PA.

Macintyre’s complexity analysis is in traditional proof theoretic terms. But his remark that ‘genus’ is more a useful geometric classification of curves than the syntactic notion of degree suggests that other criteria may be relevant. McLarty’s approach is not really a meta-theorem, but a statement that there was only one essential use of replacement and it can be eliminated. In contrast, Macintyre argues that ‘apparent second order quantification’ can be replaced by first order quantification. But the argument requires deep understanding of the number theory for each replacement in a large number of situations. Again, there is no general theorem that this type of result is provable in PA.

General Philosophy Of Category Theory, i.e., We Should Only Care About Objects Up To Isomorphism. Part 7.

In this section we will prove that adjoint functors determine each other up to isomorphism. The key tool is the concept of an “embedding of categories”. In particular, the hom bifunctor Cop × C → Set induces two “Yoneda embeddings”

H(−) ∶ Cop → SetC and H(−) ∶ C → SetCop

These are analogous to the two embeddings of a vector space V into its dual space that are induced by a non-degenerate bilinear function ⟨−, −⟩ ∶ V × V → K.

Embedding of Categories: Recall that a functor F ∶ C → D consists of:

• An object function F ∶ Obj(C) → Obj(D),

• For each pair of objects c1, c2 ∈ C, a hom set function:

F ∶ HomC(c1,c2) → HomD(F(c1),F(c2))

We say that F is a full functor when the hom set functions are surjective, and we say that F is a faithful functor when the hom set functions are injective. If the hom set functions are bijective then we say that F is a fully faithful functor, or an embedding of categories.

An embedding is in some sense the correct notion of an “injective functor”. If F ∶ C → D is an embedding, then the object function F ∶ Obj(C) → Obj(D) is not necessarily injective, but it is “injective up to isomorphism”. This agrees with the general philosophy of category theory, i.e., that we should only care about objects up to isomorphism.

Embedding Lemma: Let F ∶ C → D be an embedding of categories. Then F is essentially injective in the sense that for all objects c1, c2 ∈ C we have

c1 ≅ c2 in C ⇐⇒ F(c1) ≅ F(c2) in D

Furthermore, F is essentially monic6 in the sense that for all functors G1, G2 ∶ B → C we have G1 ≅ G2 in CB ⇐⇒ F ○ G1 ≅ F ○ G2 in DB

Proof: Let F ∶ C → D be full and faithful, i.e., bijective on hom sets.

To prove that F is essentially injective, suppose that α ∶ c1 ↔ c2 ∶ β is an isomorphism in C and apply F to obtain arrows F (α) ∶ F (c1) ⇄ F (c2) ∶ F (β) in D. Then by the functoriality of F we have

F (α) ○ F (β) = F (α ○ β) = F (idc2 ) = idF(c2), F (β) ○ F (α) = F (β ○ α) = F (idc1) = idF(c1)

which implies that F (α) ∶ F (c1) ↔ F (c2) ∶ F (β) is an isomorphism in D. Conversely, suppose that α′ ∶ F (c1) ↔ F (c2) ∶ β′ is an isomorphism in D. By the fullness of F there exist arrows α ∶ c1 ⇄ c2 ∶ β such that F(α)=α′ and F(β)=β′, and by the functoriality of F we have

F (α ○ β) = F (α) ○ F(β) = α′ ○ β′ =idF(c2) = F(idc2), F (β ○ α) = F (β) ○ F (α) = β′ ○ α′ = idF(c1) = F(idc1)

Then by the faithfulness of F we have α ○ β = idc2 and β ○ α = idc1, which implies that α ∶ c1 ↔ c2 ∶ β is an isomorphism in C.

To prove that F is essentially monic, let G, G ∶ B → C be any functors and suppose that

we have a natural isomorphism Φ ∶ G1~ G2. This means that for each object b ∈ B we

have an isomorphism Φb ∶ G1(b) → G2(b) in C and for each arrow β ∶ b1 → b2 in B we have a commutative square:

Recall from the previous argument that any functor sends isomorphisms to isomorphisms, thus by the functoriality of F we obtain another commutative square:

in which the horizontal arrows are isomorphisms in D. In other words, the assignment F (Φ)b ∶= F(Φb) defines a natural isomorphism F(Φ) ∶ F ○ G1 ⇒ F ○ G2

Conversely, suppose that we have a natural isomorphism Φ’ ∶ F ○ G1~ F ○ G2, meaning that for each object b ∈ B we have an isomorphism Φb ∶ F (G1(b)) → F (G2(b)) in C, and for each arrow β ∶ b1 → b2 in B we have a commutative square:

Since F is fully faithful, we know from the previous result that for each b ∈ B ∃ an isomorphism Φb ∶ G1(b) →~ G2(b) in C with the property Φb = F (Φ’b). Then by the functoriality of F and the commutativity of the above square we have,

F(Φb2 ○ G1(β)) = F(Φb2) ○ F(G1(β))

=Φ′b2 ○ F(G1(β))

= F (G2(β)) ○ Φ′b1

=F (G2(β)) ○ F′(Φb1)

= F (G2(β) ○ Φb1),

and by the faithfulness of F it follows that Φb2 ○ G1(β) = G2(β) = Φb1. We conclude that the following square commutes:

In other words, the arrows Φb assemble into a natural isomorphism Φ ∶ G1 ⇒~ G2.

Lemma (The Yoneda Embeddings): Let C be a category and recall that for each object c ∈ C we have two hom functors

Hc =HomC(c,−) ∶ C → Set and Hc ∶ HomC(−,c) ∶ Cop → Set

The mappings c ↦ Hc and c ↦ Hc define two embeddings of categories:

H(−) ∶ Cop → SetC and H(−) ∶ C → SetCop

We will prove that H(−)  is an embedding. Then the fact that H(−) is an embedding follows by substituting Cop in place of C.

Proof:

Step 1: H(−) is a Functor. For each arrow γ ∶ c1 → c2 in Cop (i.e., for each arrow γ ∶ c2 → c1 in C) we must define a natural transformation H(−)(γ) ∶ H(−)(c1) ⇒ H(−)(c2), i.e., a natural transformation Hγ ∶ Hc1 ⇒ Hc2. And this means that for each object d ∈ C we must define an arrow (Hγ)d ∶ Hc1(d) → Hc2(d), i.e., a function (Hγ)d ∶ HomC(c1,d) → HomC(c2,d). Note that the only possible choice is to send each arrow α ∶ c1 → d to the arrow α ○ γ ∶ c2 → d. In other words, ∀ d ∈ C we define,

(Hγ)d ∶= (−) ○ γ

To check that this is indeed a natural transformation Hγ ∶ Hc1 ⇒ Hc2

δ ∶ d1 → d2 in C and observe that the following diagram commutes:

Indeed, the commutativity of this square is just the associative axiom for composition. Thus we have defined the action of H(−) on arrows in Cop. To see that this defines a functor Cop → SetC, we need to show that for any composible arrows γ1, γ2 ∈ Arr(C) we have Hγ1 ○ γ2 = Hγ2 ○ Hγ1. So consider any arrows γ1 ∶ c2 → c1 and γ2 ∶ c3 → c2. Then ∀ objects d ∈ C and for all arrows δ ∶ c1 → d we have

[Hγ2 ○ Hγ1]d(δ) = [(Hγ2)d ○ (Hγ1)d] (δ)

= (Hγ2)d [(Hγ1)d(δ)]

= (Hγ2)d (δ ○ γ1)

= (δ ○ γ1) ○ γ2

= δ ○ (γ1 ○ γ2)

= (Hγ1 ○ γ2)d(δ)

Since this holds ∀ δ ∈ Hc1(d) we have [Hγ2 ○ Hγ1]d = (Hγ1 ○ γ2)d, and then since this holds ∀ d ∈ C we conclude that Hγ1 ○ γ2 = Hγ2 ○ Hγ1 as desired.

Step 2:

H(−) is Faithful. For each pair of objects c1,c2 ∈ C we want to show that the function H(−) ∶ HomCop (c1, c2) → HomSetC (Hc1 , Hc2)

defined in part (1) is injective. So consider any two arrows α, β ∶ c2 → c1 in C and suppose that we have Hα = Hβ as natural transformations. In this case we want to show that α = β.

Recall that ∀ objects d ∈ C and all arrows δ ∈ Hc1(d) we have defined (Hα)d(δ) = δ ○ α. Since Hα = Hβ, this means that

δ ○ α = (Hα)d(δ) = (Hβ)d(δ) = δ ○ β. Now we just take d = c1 and δ = idc1 to obtain

α = (idc1 ○ α) = (idc1 ○ β) = β

as desired.

Step 3:

H(−) is Full. For each pair of objects c1, c2 ∈ C we want to show that the function

H(−) ∶ HomCop (c1, c2) → HomSetC (Hc1 , Hc2 )
is surjective. So consider any natural transformation Φ ∶ Hc1 ⇒ Hc2. In this case we want to find an arrow φ ∶ c2 → c1 with the property Hφ = Φ. Where can we find such an arrow? By definition of “natural transformation” we have a function Φd ∶ Hc1(d) → Hc2(d) for each object d ∈ C, and for each arrow δ ∶ d1 → d2 we know that the following square commutes:

Note that the category C might have very few arrows. (Indeed, C might be a discrete category, i.e., with only the identity arrows.) This suggests that our only possible choice is to evaluate the function Φc1 ∶ Hc1 (c1) → Hc2 (c1) at the identity arrow to obtain an arrow φ ∶= Φc1 (idc1) ∈ Hc2 (c1). Now hopefully we have Hφ = Φ (otherwise the theorem is not true). To check this, consider any element d ∈ C and any arrow δ ∶ c1 → d. Substituting this δ into the above diagram gives a commutative square:

Then by following the arrow idc1 ∈ H c1 (c1) around the square in two different ways, and by using the definition (Hφ)d(δ) ∶= δ ○ φ from part (1), we obtain

Φd(δ ○ idc1) = δ ○ Φc1 (idc1) Φd(δ) = δ ○ φ

Φd(δ) = (Hφ)d(δ)

Since this holds for all arrows δ ∈ Hc1(d) we have Φd = (Hφ)d, and then since this holds for

all objects d ∈ C we conclude that Φ = Hφ as desired.

Let’s pause to apply the Embedding Lemma to the Yoneda embedding H(−) ∶ Cop → SetC. The fact that H(−) is “essentially injective” means that for all objects c1, c2 ∈ C we have c1 ≅ cin C ⇐⇒ Hc1 ≅ Hc2 in SetC.

[Note that c1 ≅ c2 in C if and only if c1 ≅ c2 in Cop.] This useful fact is the starting point for many areas of modern mathematics. It tells us that if we know all the information about arrows pointing to (or from) an object c ∈ C, then we know the object up to isomorphism. In some sense this is a justification for the philosophy of category theory. The Embedding Lemma also implies that the Yoneda embedding is “essentially monic,” i.e., “left-cancellable up to natural isomorphism”. We will use this fact to prove the uniqueness of adjoints.

Uniqueness of Adjoints: Let L ∶ C ⇄ D ∶ R be an adjunction of categories. Then each of L and R determines the other up to natural isomorphism.

Proof: We will prove that R determines L. The other direction is similar. So suppose that L′ ∶ C ⇄ D ∶ R is another adjunction. Then we have two bijections

HomD(L(c),d) ≅ HomC(c,R(d)) ≅ HomD(L′(c),d)

that are natural in (c, d) ∈ Cop × D, and by composing them we obtain a bijection

HomD(L(c),d) ≅ HomD(L′(c),d)

that is natural in (c,d) ∈ Cop × D

Naturality in d ∈ D means that for each c ∈ Cop we have a natural isomorphism of functors HomD(L(c),−) ≅ HomD(L′(c),−) in the category SetD.

Now let us compose the functor L ∶ Cop → Dop  with the Yoneda embedding H(−) ∶ Dop → SetD to obtain a functor (H(−) ○ L) ∶ Cop → SetD. Observe that if we apply the functor H(−) ○ L to an object c ∈ Cop then we obtain the functor

(H(−) ○ L)(c) = HomD(L(c),−) ∈ SetD

Thus, naturality in c ∈ Cop means exactly that we have a natural isomorphism of functors (H(−) ○ L) ≅ (H(−) ○ L′) in the category (SetD)Cop. Finally, since the “Yoneda embedding” H(−) is an embedding of categories, the Embedding Lemma tells us that we can cancel H(−) on the left to obtain a natural isomorphism:

(H(−) ○ L) ≅ (H(−) ○ L′) in (SetD)Cop ⇒ L ≅ L′ in (Dop)Cop

In other words, we have L ≅ L′ in DC…..