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

 

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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.

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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

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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)

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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

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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.

Metric. Part 1.

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A (semi-Riemannian) metric on a manifold M is a smooth field gab on M that is symmetric and invertible; i.e., there exists an (inverse) field gbc on M such that gabgbc = δac.

The inverse field gbc of a metric gab is symmetric and unique. It is symmetric since

gcb = gnb δnc = gnb(gnm gmc) = (gmn gnb)gmc = δmb gmc = gbc

(Here we use the symmetry of gnm for the third equality.) It is unique because if g′bc is also an inverse field, then

g′bc = g′nc δnb = g′nc(gnm gmb) = (gmn g′nc) gmb = δmc gmb = gcb = gbc

(Here again we use the symmetry of gnm for the third equality; and we use the symmetry of gcb for the final equality.) The inverse field gbc of a metric gab is smooth. This follows, essentially, because given any invertible square matrix A (over R), the components of the inverse matrix A−1 depend smoothly on the components of A.

The requirement that a metric be invertible can be given a second formulation. Indeed, given any field gab on the manifold M (not necessarily symmetric and not necessarily smooth), the following conditions are equivalent.

(1) There is a tensor field gbc on M such that gabgbc = δac.

(2) ∀ p in M, and all vectors ξa at p, if gabξa = 0, then ξa =0.

(When the conditions obtain, we say that gab is non-degenerate.) To see this, assume first that (1) holds. Then given any vector ξa at any point p, if gab ξa = 0, it follows that ξc = δac ξa = gbc gab ξa = 0. Conversely, suppose that (2) holds. Then at any point p, the map from (Mp)a to (Mp)b defined by ξa → gab ξa is an injective linear map. Since (Mp)a and (Mp)b have the same dimension, it must be surjective as well. So the map must have an inverse gbc defined by gbc(gab ξa) = ξc or gbc gab = δac.

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In the presence of a metric gab, it is customary to adopt a notation convention for “lowering and raising indices.” Consider first the case of vectors. Given a contravariant vector ξa at some point, we write gab ξa as ξb; and given a covariant vector ηb, we write gbc ηb as ηc. The notation is evidently consistent in the sense that first lowering and then raising the index of a vector (or vice versa) leaves the vector intact.

One would like to extend this notational convention to tensors with more complex index structure. But now one confronts a problem. Given a tensor αcab at a point, for example, how should we write gmc αcab? As αmab? Or as αamb? Or as αabm? In general, these three tensors will not be equal. To get around the problem, we introduce a new convention. In any context where we may want to lower or raise indices, we shall write indices, whether contravariant or covariant, in a particular sequence. So, for example, we shall write αabc or αacb or αcab. (These tensors may be equal – they belong to the same vector space – but they need not be.) Clearly this convention solves our problem. We write gmc αabc as αabm; gmc αacb as αamb; and so forth. No ambiguity arises. (And it is still the case that if we first lower an index on a tensor and then raise it (or vice versa), the result is to leave the tensor intact.)

We claimed in the preceding paragraph that the tensors αabc and αacb (at some point) need not be equal. Here is an example. Suppose ξ1a, ξ2a, … , ξna is a basis for the tangent space at a point p. Further suppose αabc = ξia ξjb ξkc at the point. Then αacb = ξia ξjc ξkb. Hence, lowering indices, we have αabc =ξia ξjb ξkc but αacb =ξia ξjc ξib at p. These two will not be equal unless j = k.

We have reserved special notation for two tensor fields: the index substiution field δba and the Riemann curvature field Rabcd (associated with some derivative operator). Our convention will be to write these as δab and Rabcd – i.e., with contravariant indices before covariant ones. As it turns out, the order does not matter in the case of the first since δab = δba. (It does matter with the second.) To verify the equality, it suffices to observe that the two fields have the same action on an arbitrary field αb:

δbaαb = (gbngamδnmb = gbnganαb = gbngnaαb = δabαb

Now suppose gab is a metric on the n-dimensional manifold M and p is a point in M. Then there exists an m, with 0 ≤ m ≤ n, and a basis ξ1a, ξ2a,…, ξna for the tangent space at p such that

gabξia ξib = +1 if 1≤i≤m

gabξiaξib = −1 if m<i≤n

gabξiaξjb = 0 if i ≠ j

Such a basis is called orthonormal. Orthonormal bases at p are not unique, but all have the same associated number m. We call the pair (m, n − m) the signature of gab at p. (The existence of orthonormal bases and the invariance of the associated number m are basic facts of linear algebraic life.) A simple continuity argument shows that any connected manifold must have the same signature at each point. We shall henceforth restrict attention to connected manifolds and refer simply to the “signature of gab

A metric with signature (n, 0) is said to be positive definite. With signature (0, n), it is said to be negative definite. With any other signature it is said to be indefinite. A Lorentzian metric is a metric with signature (1, n − 1). The mathematics of relativity theory is, to some degree, just a chapter in the theory of four-dimensional manifolds with Lorentzian metrics.

Suppose gab has signature (m, n − m), and ξ1a, ξ2a, . . . , ξna is an orthonormal basis at a point. Further, suppose μa and νa are vectors there. If

μa = ∑ni=1 μi ξia and νa = ∑ni=1 νi ξia, then it follows from the linearity of gab that

gabμa νb = μ1ν1 +…+ μmνm − μ(m+1)ν(m+1) −…−μnνn.

In the special case where the metric is positive definite, this comes to

gabμaνb = μ1ν1 +…+ μnνn

And where it is Lorentzian,

gab μaνb = μ1ν1 − μ2ν2 −…− μnνn

Metrics and derivative operators are not just independent objects, but, in a quite natural sense, a metric determines a unique derivative operator.

Suppose gab and ∇ are both defined on the manifold M. Further suppose

γ : I → M is a smooth curve on M with tangent field ξa and λa is a smooth field on γ. Both ∇ and gab determine a criterion of “constancy” for λa. λa is constant with respect to ∇ if ξnnλa = 0 and is constant with respect to gab if gab λa λb is constant along γ – i.e., if ξnn (gab λa λb = 0. It seems natural to consider pairs gab and ∇ for which the first condition of constancy implies the second. Let us say that ∇ is compatible with gab if, for all γ and λa as above, λa is constant w.r.t. gab whenever it is constant with respect to ∇.

Truncation Functors

Let A be an abelian category, and let D = D(A) be the derived category. For any complex A• in A, and n ∈ Z, we let τ≤nA• be the truncated complex

··· → An−2 → An−1 → ker(An → An+1)→ 0 → 0 → ··· , and dually we let τ≥nA be the complex

··· → 0 → 0 → coker(An−1 → An) → An+1 → An+2 → ···

Note that

Hm≤nA•) = Hm(A•) if m ≤ n

= 0 if m > n

and that

Hm≥nA•) = Hm(A•)  if m ≥ n

= 0 if m < n

One checks that τ≥n (respectively τ≤n) extends naturally to an additive functor of complexes which preserves homotopy and takes quasi-isomorphisms to quasi-isomorphisms, and hence induces an additive functor D → D. In fact if D≤n (respectively D≥n) is the full subcategory of D whose objects are the complexes A• such that Hm(A•) = 0 for m > n (respectively m < n) then we have additive functors

τ≤n : D → D≤n ⊂ D

τ≥n : D → D≥n ⊂ D

together with obvious functorial maps

inA : τ≤n A• → A•

jnA : A• → τ≥n A•

The preceding inA , jnA induce functorial isomorphisms

HomD≤n (B•,τ≤nA•) →~ HomD(B•, A•) (B• ∈ D≤n) —– (1)

HomD≥n≥nA•,C•) →~ HomD(A•,C• ) (C• ∈ D≥n) —– (2)

Bijectivity of (1) means that any map φ : B• → A• (in D) with B• ∈ D≤n factors uniquely via iA := inA

Given φ, we have a commutative diagram

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and since B• ∈ D≤n, therefore iB is an isomorphism in D, so we can write

φ = i ◦ (τ≤nφ ◦ i−1B),

and thus (1) is surjective.

To prove that (1) is also injective, we assume that iA ◦ τ≤n φ = 0 and deduce that τ≤n φ = 0. The assumption means that there is a commutative diagram in K(A)

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where s and s′′ are quasi-isomorphisms, and f/s = τ≤nφ

Applying the (idempotent) functor τ≥n, we get a commutative diagram

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Since τ≤ns and τ≤ns′′ are quasi-isomorphisms, we have

τ≤nφ = τ≤n f/τ≤ns = 0/τ≤ns′′ = 0

as desired.

Category-Theoretic Sinks

The concept dual to that of source is called sink. Whereas the concepts of sources and sinks are dual to each other, frequently sources occur more naturally than sinks.

A sink is a pair ((fi)i∈I, A), sometimes denoted by (fi,A)I or (Aifi A)I consisting of an object A (the codomain of the sink) and a family of morphisms fi : Ai → A indexed by some class I. The family (Ai)i∈I is called the domain of the sink. Composition of sinks is defined in the (obvious) way dual to that of composition of sources.

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In Set, a sink (Aifi A)I is an epi-sink if and only if it is jointly surjective, i.e., iff A = ∪i∈I fi[Ai]. In every construct, all jointly surjective sinks are epi-sinks. The converse implication holds, e.g., in Vec, Pos, Top, and Σ-Seq. A category A is thin if and only if every sink in A is an epi-sink.

Classical Theory of Fields

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Galilean spacetime consists in a quadruple (M, ta, hab, ∇), where M is the manifold R4; ta is a one form on M; hab is a smooth, symmetric tensor field of signature (0, 1, 1, 1), and ∇ is a flat covariant derivative operator. We require that ta and hab be compatible in the sense that tahab = 0 at every point, and that ∇ be compatible with both tensor fields, in the sense that ∇atb = 0 and ∇ahbc = 0.

The points of M represent events in space and time. The field ta is a “temporal metric”, assigning a “temporal length” |taξa| to vectors ξa at a point p ∈ M. Since R4 is simply connected, ∇atb = 0 implies that there exists a smooth function t : M → R such that ta = ∇at. We may thus define a foliation of M into constant – t hypersurfaces representing collections of simultaneous events – i.e., space at a time. We assume that each of these surfaces is diffeomorphic to R3 and that hab restricted these surfaces is (the inverse of) a flat, Euclidean, and complete metric. In this sense, hab may be thought of as a spatial metric, assigning lengths to spacelike vectors, all of which are tangent to some spatial hypersurface. We represent particles propagating through space over time by smooth curves whose tangent vector ξa, called the 4-velocity of the particle, satisfies ξata = 1 along the curve. The derivative operator ∇ then provides a standard of acceleration for particles, which is given by ξnnξa. Thus, in Galilean spacetime we have notions of objective duration between events; objective spatial distance between simultaneous events; and objective acceleration of particles moving through space over time.

However, Galilean spacetime does not support an objective notion of the (spatial) velocity of a particle. To get this, we move to Newtonian spacetime, which is a quintuple (M, ta, hab, ∇, ηa). The first four elements are precisely as in Galilean spacetime, with the same assumptions. The final element, ηa, is a smooth vector field satisfying ηata = 1 and ∇aηb = 0. This field represents a state of absolute rest at every point—i.e., it represents “absolute space”. This field allows one to define absolute velocity: given a particle passing through a point p with 4-velocity ξa, the (absolute, spatial) velocity of the particle at p is ξa − ηa.

There is a natural sense in which Newtonian spacetime has strictly more structure than Galilean spacetime: after all, it consists of Galilean spacetime plus an additional element. This judgment may be made precise by observing that the automorphisms of Newtonian spacetime – that is, its spacetime symmetries – form a proper subgroup of the automorphisms of Galilean spacetime. The intuition here is that if a structure has more symmetries, then there must be less structure that is preserved by the maps. In the case of Newtonian spacetime, these automorphisms are diffeomorphisms θ : M → M that preserve ta, hab, ∇, and ηa. These will consist in rigid spatial rotations, spatial translations, and temporal translations (and combinations of these). Automorphisms of Galilean spacetime, meanwhile, will be diffeomorphisms that preserve only the metrics and derivative operator. These include all of the automorphisms of Newtonian spacetime, plus Galilean boosts.

It is this notion of “more structure” that is captured by the forgetful functor approach. We define two categories, Gal and New, which have Galilean and Newtonian spacetime as their (essentially unique) objects, respectively, and have automorphisms of these spacetimes as their arrows. Then there is a functor F : New → Gal that takes arrows of New to arrows of Gal generated by the same automorphism of M. This functor is clearly essentially surjective and faithful, but it is not full, and so it forgets only structure. Thus the criterion of structural comparison may be seen as a generalization of the latter to cases where one is comparing collections of models of a theory, rather than individual spacetimes.

To see this last point more clearly, let us move to another well-trodden example. There are two approaches to classical gravitational theory: (ordinary) Newtonian gravitation (NG) and geometrized Newtonian gravitation (GNG), sometimes known as Newton-Cartan theory. Models of NG consist of Galilean spacetime as described above, plus a scalar field φ, representing a gravitational potential. This field is required to satisfy Poisson’s equation, ∇aaφ = 4πρ, where ρ is a smooth scalar field representing the mass density on spacetime. In the presence of a gravitational potential, massive test point particles will accelerate according to ξnnξa = −∇aφ, where ξa is the 4-velocity of the particle. We write models as (M, ta, hab, ∇, φ).

The models of GNG, meanwhile, may be written as quadruples (M,ta,hab,∇ ̃), where we assume for simplicity that M, ta, and hab are all as described above, and where ∇ ̃ is a covariant derivative operator compatible with ta and hab. Now, however, we allow ∇ ̃ to be curved, with Ricci curvature satisfying the geometrized Poisson equation, Rab = 4πρtatb, again for some smooth scalar field ρ representing the mass density. In this theory, gravitation is not conceived as a force: even in the presence of matter, massive test point particles traverse geodesics of ∇ ̃ — where now these geodesics depend on the distribution of matter, via the geometrized Poisson equation.

There is a sense in which NG and GNG are empirically equivalent: a pair of results due to Trautman guarantee that (1) given a model of NG, there always exists a model of GNG with the same mass distribution and the same allowed trajectories for massive test point particles, and (2), with some further assumptions, vice versa. But in an, Clark Glymour has argued that these are nonetheless inequivalent theories, because of an asymmetry in the relationship just described. Given a model of NG, there is a unique corresponding model of GNG. But given a model of GNG, there are typically many corresponding models of NG. Thus, it appears that NG makes distinctions that GNG does not make (despite the empirical equivalence), which in turn suggests that NG has more structure than GNG.

This intuition, too, may be captured using a forget functor. Define a category NG whose objects are models of NG (for various mass densities) and whose arrows are automorphisms of M that preserve ta, hab, ∇, and φ; and a category GNG whose objects are models of GNG and whose arrows are automorphisms of M that preserve ta, hab, and ∇ ̃. Then there is a functor F : NG → GNG that takes each model of NG to the corresponding model, and takes each arrow to an arrow generated by the same diffeomorphism. This results in implying

F : NG → GNG forgets only structure.

Philosophizing Forgetful Functors: This Functor Forgets only Properties: Namely, the Property of Being Abelian + This Functor Forgets Both Structure (the generating set) and Properties (the property of being a free group).

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forgetful functor is a functor which is defined by ‘forgetting’ something. For example, the forgetful functor from Grp to Set forgets the group structure of a group, remembering only the underlying set.

In common parlance, the term ‘forgetful functor’ has no precise definition, being simply used whenever a functor is obviously defined by forgetting something. Many forgetful functors of this sort have left or right adjoints (and many are actually monadic or comonadic), leading to the paradigmatic adjunction “free ⊣ forgetful.”

On the other hand, from the perspective of stuff, structure, propertyevery functor is regarded as a forgetful functor and classified by how much it forgets (namely, stuff, structure, or properties). From this perspective, the forgetful functor from GrpGrp to SetSet forgets the structure of a group and the property of admitting a group structure, as usual; but its left adjoint (the free group functor) is also forgetful: if you identify SetSet with the category of free groups with specified generators, then it forgets the structure of a set of free generators and the property of being free.

There are many cases in which we want to say that one kind of mathematical object has more structure than another kind of mathematical object. For instance, a topological space has more structure than a set. A Lie group has more structure than a smooth manifold. A ring has more structure than a group. And so on. In each of these cases, there is a sense in which the first sort of object – say, a topological space – results by taking an instance of the second sort – say, a set – and adding something more – in this case, a topology. In other cases, we want to say that two different kinds of mathematical objects have the same amount of structure. For instance, given a Boolean algebra, one can construct a special kind of topological space, known as a Stone space, from which one can uniquely reconstruct the original Boolean algebra; and vice-versa.

These sorts of relationships between mathematical objects are naturally captured in the language of category theory, via the notion of a forgetful functor. For instance, there is a functor F : Top → Set from the category Top, whose objects are topological spaces and whose arrows are continuous maps, to the category Set, whose objects are sets and whose arrows are functions. This functor takes every topological space to its underlying set, and it takes every continuous function to its underlying function. We say this functor is forgetful because, intuitively speaking, it forgets something: namely the choice of topology on a given set.

The idea of a forgetful functor is made precise by a classification of functors due to Baez et al. (2004). This requires some machinery. A functor F : C → D is said to be full if for every pair of objects A, B of C, the map F : hom(A, B) → hom(F (A), F (B)) induced by F is surjective, where hom(A, B) is the collection of arrows from A to B. Likewise, F is faithful if this induced map is injective for every such pair of objects. Finally, a functor is essentially surjective if for every object X of D, there exists some object A of C such that F(A) is isomorphic to X.

If a functor is full, faithful, and essentially surjective, we will say that it forgets nothing. A functor F : C → D is full, faithful, and essentially surjective if and only if it is essentially invertible, i.e., there exists a functor G : D → C such that G ◦ F : C → C is naturally isomorphic to 1C, the identity functor on C, and F ◦ G : D → D is naturally isomorphic to 1D. (Note, then, that G is also essentially invertible, and thus G also forgets nothing.) This means that for each object A of C, there is an isomorphism ηA : G ◦ F (A) → A such that for any arrow f : A → B in C, ηB ◦ G ◦ F(f) = f ◦ ηA, and similarly for every object of D. When two categories are related by a functor that forgets nothing, we say the categories are equivalent and that the pair F, G realizes an equivalence of categories.

Conversely, any functor that fails to be full, faithful, and essentially surjective forgets something. But functors can forget in different ways. A functor F : C → D forgets structure if it is not full; properties if it is not essentially surjective; and stuff if it is not faithful. Of course, “structure”, “property”, and “stuff” are technical terms in this context. But they are intended to capture our intuitive ideas about what it means for one kind of object to have more structure (resp., properties, stuff) than another. We can see this by considering some examples.

For instance, the functor F : Top → Set described above is faithful and essentially surjective, but not full, because not every function is continuous. So this functor forgets only structure – which is just the verdict we expected. Likewise, there is a functor G : AbGrp → Grp from the category AbGrp whose objects are Abelian groups and whose arrows are group homomorphisms to the category Grp whose objects are (arbitrary) groups and whose arrows are group homomorphisms. This functor acts as the identity on the objects and arrows of AbGrp. It is full and faithful, but not essentially surjective because not every group is Abelian. So this functor forgets only properties: namely, the property of being Abelian. Finally, consider the unique functor H : Set → 1, where 1 is the category with one object and one arrow. This functor is full and essentially surjective, but it is not faithful, so it forgets only stuff – namely all of the elements of the sets, since we may think of 1 as the category whose only object is the empty set, which has exactly one automorphism.

In what follows, we will say that one sort of object has more structure (resp. properties, stuff) than another if there is a functor from the first category to the second that forgets structure (resp. properties, stuff). It is important to note, however, that comparisons of this sort must be relativized to a choice of functor. In many cases, there is an obvious functor to choose – i.e., a functor that naturally captures the standard of comparison in question. But there may be other ways of comparing mathematical objects that yield different verdicts.

For instance, there is a natural sense in which groups have more structure than sets, since any group may be thought of as a set of elements with some additional structure. This relationship is captured by a forgetful functor F : Grp → Set that takes groups to their underlying sets and group homomorphisms to their underlying functions. But any set also uniquely determines a group, known as the free group generated by that set; likewise, functions generate group homomorphisms between free groups. This relationship is captured by a different functor, G : Set → Grp, that takes every set to the free group generated by it and every function to the corresponding group homomorphism. This functor forgets both structure (the generating set) and properties (the property of being a free group). So there is a sense in which sets may be construed to have more structure than groups.

Galois Theor(y)/(em)

The most significant discovery of Galois is that under some hypotheses, there is a one-to-one correspondence between

1. subgroups of the Galois group Gal(E/F)

2. subfields M of E such that F ⊆ M.

The correspondence goes as follows:

To each intermediate subfield M, associate the group Gal(E/M) of all M-automorphisms of E:

G = Gal : {intermediate fields} → {subgroups of Gal(E/F)}

M → G(M) = Gal(E/M)

To each subgroup H of Gal(E/F), associate the fixed subfield F(H):

F : {subgroups of Gal(E/F )} → {intermediate fields}

H → F(H)

We will prove that, under the right hypotheses, we actually have a bijection (namely G is the inverse of F). For example.

Consider the field extension E = Q(i, √5)/Q. It has four Q-automorphisms, given by (it is enough to describe their actions on i and √5):
σ1 : i →i, √5 →√5
σ2 : i →−i, √5 →√5
σ3 : i →i, √5 →−√5
σ4 : i →−i, √5 →−√5
thus
Gal(E/Q) = {σ1, σ2, σ3, σ4}. The proper subgroups of Gal(E/Q) are {σ1}, {σ1, σ2}, {σ1, σ3}, {σ1, σ4} and their corresponding subfields are E, Q(√5), Q(i), Q(i√5). This yields the following diagram:
img_20170305_082221
Theorem: Let E/F be a finite Galois extension with Galois group G.
  1. The map F is a bijection from subgroups to intermediate fields, with inverse G.
  2. Consider the intermediate field K = F(H) which is fixed by H, and σ ∈ G.Then the intermediate fieldσK = {σ(x), x∈K}

    is fixed by σHσ−1, namely σK = F(σHσ−1)

    Proof: 1. We first consider the composition of maps H → F(H) → GF(H).

    We need to prove that GF(H) = H. Take σ in H, then σ fixes F(H) by definition and σ ∈ Gal(E/F(H)) = G(F(H)), showing that

    H ⊆ GF(H).

    To prove equality, we need to rule out the strict inclusion. If H were a proper subgroup of G(F(H)), by the above proposition the fixed field F(H) of H should properly contain the fixed field of GF(H) which is F(H) itself, a contradiction, showing that

    H = GF(H)

    Now consider the reverse composition of maps K → G(K) → FG(K)

    This time we need to prove that K = FG(K). But FG(K) = fixed field by Gal(E/K) which is exactly K by the above proposition (its first point). It is enough to compute F(σHσ−1) and show that it is actually equal to

    σK = σF(H).

    F(σHσ−1) = {x ∈ E, στσ−1(x) = x ∀ τ ∈ H} = {x ∈ E, τσ−1(x)=σ−1(x) ∀ τ ∈ H}

    =  {x ∈ E, σ−1(x) ∈ F(H)}

    =  {x ∈ E, x ∈ σ(F(H))} = σ(F(H))

    We now look at subextensions of the finite Galois extension E/F and ask about their respective Galois group.

    Theorem: Let E/F be a finite Galois extension with Galois group G. Let K be an intermediate subfield, fixed by the subgroup H.

    1. The extension E/K is Galois.

    2. The extension K/F is normal if and only if H is a normal subgroup of G.

    3. If H is a normal subgroup of G, then

    Gal(K/F ) ≃ G/H = Gal(E/F )/Gal(E/K).

    4. Whether K/F is normal or not, we have

    [K : F] = [G : H]

    Proof:

    That E/K is Galois is immediate from the fact that a subextension E/K/F inherits normality and separability from E/F.

    First note that σ is an F-monomorphism of K into E if and only if σ is the restriction to K of an element of G: if σ is an F -monomorphism of K into E, it can be extended to an F-monomorphism of E into itself thanks to the normality of E. Conversely, if τ is an F-automorphism of E, then σ = τ|K is surely a F-monomorphism of K into E.

    Now, this time by a characterization of a normal extension, we have

    K/F normal ⇐⇒ σ(K) = K ∀ σ ∈ G

    Since K = F(H), we just rewrite

    K/F normal ⇐⇒ σ(F(H)) = F(H) ∀ σ ∈ G.

    Now by the above theorem, we know that σ(F(H)) = F(σHσ−1), and we have

    K/F normal ⇐⇒ F(σHσ−1) = F(H) for all σ ∈ G

    We now use again the above theorem that tells us that F is invertible, with inverse G, to get the conclusion:

    K/F normal ⇐⇒ σHσ−1 =H ∀ σ ∈ G

    To prove this isomorphism, we will use the 1st isomorphism Theorem for groups. Consider the group homomorphism

    Gal(E/F)→Gal(K/F), σ →σ|K.

    This map is surjective and its kernel is given by

    Ker={σ, σ|K =1}=H =Gal(E/K).

    Applying the first isomorphism Theorem for groups, we get

    Gal(K/F ) ≃ Gal(E/F )/Gal(E/K)

    Finally, by multiplicativity of the degrees:

    [E :F]=[E :K][K :F]

    Since E/F and E/K are Galois, we can rewrite |G| = |H|[K : F]. We conclude by Lagrange Theorem:

    [G:H]=|G|/|H|=[K :F]

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:

img_20170208_203625

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

img_20170208_203609

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:

img_20170208_204652_hdr

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:

img_20170208_205811

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:

img_20170208_212435_hdr

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:

img_20170209_073201

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:

img_20170209_074057

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…..