# Interleaves Many important spaces in topology and algebraic geometry have no odd-dimensional homology. For such spaces, functorial spatial homology truncation simplifies considerably. On the theory side, the simplification arises as follows: To define general spatial homology truncation, we used intermediate auxiliary structures, the n-truncation structures. For spaces that lack odd-dimensional homology, these structures can be replaced by a much simpler structure. Again every such space can be embedded in such a structure, which is the analogon of the general theory. On the application side, the crucial simplification is that the truncation functor t<n will not require that in truncating a given continuous map, the map preserve additional structure on the domain and codomain of the map. In general, t<n is defined on the category CWn⊃∂, meaning that a map must preserve chosen subgroups “Y ”. Such a condition is generally necessary on maps, for otherwise no truncation exists. So arbitrary continuous maps between spaces with trivial odd-dimensional homology can be functorially truncated. In particular the compression rigidity obstructions arising in the general theory will not arise for maps between such spaces.

Let ICW be the full subcategory of CW whose objects are simply connected CW-complexes K with finitely generated even-dimensional homology and vanishing odd-dimensional homology for any coefficient group. We call ICW the interleaf category.

For example, the space K = S22 e3 is simply connected and has vanishing integral homology in odd dimensions. However, H3(K;Z/2) = Z/2 ≠ 0.

Let X be a space whose odd-dimensional homology vanishes for any coefficient group. Then the even-dimensional integral homology of X is torsion-free.

Taking the coefficient group Q/Z, we have

Tor(H2k(X),Q/Z) = H2k+1(X) ⊗ Q/Z ⊕ Tor(H2k(X),Q/Z) = H2k+1(X;Q/Z) = 0.

Thus H2k(X) is torsion-free, since the group Tor(H2k(X),Q/Z) is isomorphic to the torsion subgroup of H2k(X).

Any simply connected closed 4-manifold is in ICW. Indeed, such a manifold is homotopy equivalent to a CW-complex of the form

Vi=1kSi2ƒe4

where the homotopy class of the attaching map ƒ : S3 → Vi=1k Si2 may be viewed as a symmetric k × k matrix with integer entries, as π3(Vi=1kSi2) ≅ M(k), with M(k) the additive group of such matrices.

Any simply connected closed 6-manifold with vanishing integral middle homology group is in ICW. If G is any coefficient group, then H1(M;G) ≅ H1(M) ⊗ G ⊕ Tor(H0M,G) = 0, since H0(M) = Z. By Poincaré duality,

0 = H3(M) ≅ H3(M) ≅ Hom(H3M,Z) ⊕ Ext(H2M,Z),

so that H2(M) is free. This implies that Tor(H2M,G) = 0 and hence H3(M;G) ≅ H3(M) ⊗ G ⊕ Tor(H2M,G) = 0. Finally, by G-coefficient Poincaré duality,

H5(M;G) ≅ H1(M;G) ≅ Hom(H1M,G) ⊕ Ext(H0M,G) = Ext(Z,G) = 0

Any smooth, compact toric variety X is in ICW: Danilov’s Theorem implies that H(X;Z) is torsion-free and the map A(X) → H(X;Z) given by composing the canonical map from Chow groups to homology, Ak(X) = An−k(X) → H2n−2k(X;Z), where n is the complex dimension of X, with Poincaré duality H2n−2k(X;Z) ≅ H2k(X;Z), is an isomorphism. Since the odd-dimensional cohomology of X is not in the image of this map, this asserts in particular that Hodd(X;Z) = 0. By Poincaré duality, Heven(X;Z) is free and Hodd(X;Z) = 0. These two statements allow us to deduce from the universal coefficient theorem that Hodd(X;G) = 0 for any coefficient group G. If we only wanted to establish Hodd(X;Z) = 0, then it would of course have been enough to know that the canonical, degree-doubling map A(X) → H(X;Z) is onto. One may then immediately reduce to the case of projective toric varieties because every complete fan Δ has a projective subdivision Δ, the corresponding proper birational morphism X(Δ) → X(Δ) induces a surjection H(X(Δ);Z) → H(X(Δ);Z) and the diagram commutes.

Let G be a complex, simply connected, semisimple Lie group and P ⊂ G a connected parabolic subgroup. Then the homogeneous space G/P is in ICW. It is simply connected, since the fibration P → G → G/P induces an exact sequence

1 = π1(G) → π1(G/P) → π0(P) → π0(G) = 0,

which shows that π1(G/P) → π0(P) is a bijection. Accordingly, ∃ elements sw(P) ∈ H2l(w)(G/P;Z) (“Schubert classes,” given geometrically by Schubert cells), indexed by w ranging over a certain subset of the Weyl group of G, that form a basis for H(G/P;Z). (For w in the Weyl group, l(w) denotes the length of w when written as a reduced word in certain specified generators of the Weyl group.) In particular Heven(G/P;Z) is free and Hodd(G/P;Z) = 0. Thus Hodd(G/P;G) = 0 for any coefficient group G.

The linear groups SL(n, C), n ≥ 2, and the subgroups S p(2n, C) ⊂ SL(2n, C) of transformations preserving the alternating bilinear form

x1yn+1 +···+ xny2n −xn+1y1 −···−x2nyn

on C2n × C2n are examples of complex, simply connected, semisimple Lie groups. A parabolic subgroup is a closed subgroup that contains a Borel group B. For G = SL(n,C), B is the group of all upper-triangular matrices in SL(n,C). In this case, G/B is the complete flag manifold

G/B = {0 ⊂ V1 ⊂···⊂ Vn−1 ⊂ Cn}

of flags of subspaces Vi with dimVi = i. For G = Sp(2n,C), the Borel subgroups B are the subgroups preserving a half-flag of isotropic subspaces and the quotient G/B is the variety of all such flags. Any parabolic subgroup P may be described as the subgroup that preserves some partial flag. Thus (partial) flag manifolds are in ICW. A special case is that of a maximal parabolic subgroup, preserving a single subspace V. The corresponding quotient SL(n, C)/P is a Grassmannian G(k, n) of k-dimensional subspaces of Cn. For G = Sp(2n,C), one obtains Lagrangian Grassmannians of isotropic k-dimensional subspaces, 1 ≤ k ≤ n. So Grassmannians are objects in ICW. The interleaf category is closed under forming fibrations.

# Causal Isomorphism as Homeomorphism, or Diffeomorphism or a Conformal Isometry? Drunken Risibility.

Let (M, gab) and (M′, g′ab) be (temporally oriented) relativistic spacetimes.

We say that a bijection φ : M → M′ between their underlying point sets is a ≪-causal isomorphism if, ∀ p and q in M,

p ≪ q ⇐⇒ φ(p) ≪ φ(q).

Then we can ask the following: Does a ≪-causal isomorphism have to be a homeomorphism? A diffeomorphism? A conformal isometry? (We know in advance that a causal isomorphism need not be a (full) isometry because conformally equivalent metrics gab and Ω2gab on a manifold M determine the same relation ≪. The best one can ask for is that it be a conformal isometry – i.e. that it be a diffeomorphism that preserves the metric up to a conformal factor.) Without further restrictions on (M, gab) and (M′, g′ab), the answer is certainly “no” to all three questions. Unless the “causal structure” of a spacetime (i.e., the structure determined by ≪) is reasonably well behaved, it provides no useful information at all. For example, let us say that a spacetime is causally degenerate if p ≪ q for all points p and q. Any bijection between two causally degenerate spacetimes qualifies, trivially, as a ≪-causal isomorphism. But we can certainly find causally degenerate spacetimes whose underlying manifolds have different topologies. But a suitably “rolled-up” version of Minkowski spacetime is also causally degenerate, and the latter has the manifold structure S1 × R3.

There is a hierarchy of “causality conditions” that is relevant here. Hawking and Ellis impose, with varying degrees of stringency, the requirement that there exist no closed, or “almost closed,” timelike curves. Here are three.

Chronology: There do not exist smooth closed timelike curves. (Equivalently, for all p, it is not the case that p ≪ p.)

Future (respectively, past) distinguishablity: ∀ points p, and all sufficiently small open sets O containing p, no smooth future-directed (respectively, past-directed) timelike curve that starts at p, and leaves O, ever returns to O. Strong causality: For all points p, and all sufficiently small open sets O containing p, no smooth future-directed timelike curve that starts in O, and leaves O, ever returns to O.

It is clear that strong causality implies both future distinguishability and past distinguishability, and that each of the distinguishability conditions (alone) implies chronology.

The names “future distinguishability” and “past distinguishability” are easily explained. For any p, let I+(p) be the set {q: p ≪ q} and let I(p) be the set {q : q ≪ p}. It turns out that future distinguishability is equivalent to the requirement that, ∀ p and q,

I+(p) = I+(q) =⇒ p = q.

And the counterpart requirement with I+ replaced by I is equivalent to past distinguishability.

# Anti-Haecceitism. Thought of the Day 45.0 ; Conc is the property of being concurrent, Red is the property of definiteness, and Heavy is the property of vividness.

In the language of modern metaphysics, w and w′ above are qualitatively indiscernible. And anti-haecceitism is the doctrine which says that qualitatively indiscernible worlds are identical. So, we immediately see a problem looming.

But why accept anti-haecceitism? The best reasons focus on physics. Just as the debate between Leibniz and Newton’s followers focused on physics, the strongest arguments still against haecceitism come from physics. Anti-haecceitism as understood here concerns the identity of indiscernible (“isomorphic”) worlds or “situations” or “states”. In many areas of physics, including statistical physics, spacetime physics and quantum theory, the physics tells us that certain “indiscernible situations” are in fact literally identical.

A simple example comes from the statistical physics of “indiscernibile particles”. Consider a box, partitioned into Left-side and Right-side (L and R), and containing two indiscernible particles. One naively thinks this permits four distinct states or situations: i.e., both in L; both in R, and one in L and one in R. However, physics tells us that there are only three states, not four, and we might denote these: S2,0, S1,1, S0,2. The state S1,1, i.e., where “one is L and one is R”, is a single state; there are not two distinct possibilities. The correct description of S1,1 uses existential quantifiers:

∃x ∃y (x ≠ y ∧ Lx ∧ Ry)

One can (syntactically) introduce labels for the particles, say a, b. One can do this in two ways, to obtain:

a ≠ b ∧ La ∧ Rb

b ≠ a ∧ Lb ∧ Ra

But this labelling is purely representational, and not in any way fixed by the physical state S1,1 itself. So, there are distinct indiscernible objects in “situations” or states.

From spacetime physics, consider the principle sometimes called “Leibniz equivalence” (Norton). A formulation (but under a different name) is given in Wald’s monograph General Relativity. Wald’s formulation of Leibniz equivalence is, essentially, this:

isomorphic spacetime models represent the same physical world.

For example, let

S = (M, g, . . . )

be a spacetime model with carrier set |M| of points. (i.e., M is the underlying manifold.) Then Leibniz Equivalence implies:

If π : |M| → |M| is any bijection, then πS and S represent

the same world. There are many other examples, including examples from quantum theory. Consequently, independently of our pre-theoretic considerations concerning modality, it seems to me that our best physics – statistical physics, relativity and quantum theory – is telling us that anti-haecceitism is true: given a structure A which represents a world w, any permuted copy πA should somehow represent the same world, w.

# Philosophy of Local Time Let us hypothesize on the notion of local time.

Existence of temporal order: For each concrete basic thing x ∈ Θ, there exist a single ordering relation between their states ≤.

We now give a name to this ordering relation:

Denotation of temporal order: The set of lawful states of x is temporally ordered by the ≤ relation.

The above is a partial order relation: there are pairs of states that are not ordered by ≤; e.g. given an initial condition (x0,v0) for a moving particle, there are states (x1,v1) that are not visited by the particle.

Proper history: A totally order set of states of x is called a proper history of x.

The axiomatics do not guarantee the existence of a single proper history: they allow many of them, as in “The garden of forking paths”. The following axiom forbids such possibility.

Unicity of proper history: Each thing has one and only one proper history.

Arrow of time: The axiomatics describe a kind of “arrow of time”, although it is not related to irreversibility.

A proper history is also an ontological history. The parameter t ∈ M has not to be continuous. The following axiom, a very strong version of Heraclitus’ hypothesis Panta rhei, states that every thing is changing continuously:

Continuity: If the entire set of states of an ontological history is divided in two subsets hp and hf such that every state in hp temporally precedes any state in hf, then there exists one and only one state s0 such that s1 ≤ s0 ≤ s2, where s1 ∈ hp and s2 ∈ hf.

The axiom of continuity is stated in the Dedekind form.

Continuity in quantum mechanics: Although quantum mechanical “changes of state” are usually considered “instantaneous”, theory shows that probabilities change in a continuous way. The finite width of spectral lines also shows a continuous change in time.

Real representation: Given a unit change (s0, s1) there exists a bijection T : h ↔ R such that

h1 = {s(τ)|τ ∈ R} —– (1)
s0 = s(0) —– (2)
s1 = s(1) —– (3)

Local time: The function T is called local time. The unit change (s0, s1) is arbitary. It defines an arbitrary “unit of local time”.

The above theory of local time has an important philosophical consequence: becoming, which is usually conceived as evolution in time, is here more fundamental than time. The latter is constructed as an emergent property of a changing (i.e. a becoming) thing.

# Of Topos and Torsors

Let X be a topological space. One goal of algebraic topology is to study the topology of X by means of algebraic invariants, such as the singular cohomology groups Hn(X;G) of X with coefficients in an abelian group G. These cohomology groups have proven to be an extremely useful tool, due largely to the fact that they enjoy excellent formal properties, and the fact that they tend to be very computable. However, the usual definition of Hn(X;G) in terms of singular G-valued cochains on X is perhaps somewhat unenlightening. This raises the following question: can we understand the cohomology group Hn(X;G) in more conceptual terms?

As a first step toward answering this question, we observe that Hn(X;G) is a representable functor of X. That is, there exists an Eilenberg-MacLane space K(G,n) and a universal cohomology class η ∈ Hn(K(G,n);G) such that, for any topological space X, pullback of η determines a bijection

[X, K(G, n)] → Hn(X; G)

Here [X,K(G,n)] denotes the set of homotopy classes of maps from X to K(G,n). The space K(G,n) can be characterized up to homotopy equivalence by the above property, or by the the formula πkK(G,n)≃ ∗ if k̸ ≠ n

or

G if k = n.

In the case n = 1, we can be more concrete. An Eilenberg MacLane space K(G,1) is called a classifying space for G, and is typically denoted by BG. The universal cover of BG is a contractible space EG, which carries a free action of the group G by covering transformations. We have a quotient map π : EG → BG. Each fiber of π is a discrete topological space, on which the group G acts simply transitively. We can summarize the situation by saying that EG is a G-torsor over the classifying space BG. For every continuous map X → BG, the fiber product X~ : EG × BG X has the structure of a G-torsor on X: that is, it is a space endowed with a free action of G and a homeomorphism X~/G ≃ X. This construction determines a map from [X,BG] to the set of isomorphism classes of G-torsors on X. If X is a well-behaved space (such as a CW complex), then this map is a bijection. We therefore have (at least) three different ways of thinking about a cohomology class η ∈ H1(X; G):

(1) As a G-valued singular cocycle on X, which is well-defined up to coboundaries.

(2) As a continuous map X → BG, which is well-defined up to homotopy.

(3) As a G-torsor on X, which is well-defined up to isomorphism.

The singular cohomology of a space X is constructed using continuous maps from simplices ∆k into X. If there are not many maps into X (for example if every path in X is constant), then we cannot expect singular cohomology to tell us very much about X. The second definition uses maps from X into the classifying space BG, which (ultimately) relies on the existence of continuous real-valued functions on X. If X does not admit many real-valued functions, then the set of homotopy classes [X,BG] is also not a very useful invariant. For such spaces, the third approach is the most powerful: there is a good theory of G-torsors on an arbitrary topological space X.

There is another reason for thinking about H1(X;G) in the language of G-torsors: it continues to make sense in situations where the traditional ideas of topology break down. If X is a G-torsor on a topological space X, then the projection map X → X is a local homeomorphism; we may therefore identify X with a sheaf of sets F on X. The action of G on X determines an action of G on F. The sheaf F (with its G-action) and the space X (with its G-action) determine each other, up to canonical isomorphism. Consequently, we can formulate the definition of a G-torsor in terms of the category ShvSet(X) of sheaves of sets on X without ever mentioning the topological space X itself. The same definition makes sense in any category which bears a sufficiently strong resemblance to the category of sheaves on a topological space: for example, in any Grothendieck topos. This observation allows us to construct a theory of torsors in a variety of nonstandard contexts, such as the étale topology of algebraic varieties.

Describing the cohomology of X in terms of the sheaf theory of X has still another advantage, which comes into play even when the space X is assumed to be a CW complex. For a general space X, isomorphism classes of G-torsors on X are classified not by the singular cohomology H1sing(X;G), but by the sheaf cohomology H1sheaf(X; G) of X with coefficients in the constant sheaf G associated to G. This sheaf cohomology is defined more generally for any sheaf of groups G on X. Moreover, we have a conceptual interpretation of H1sheaf(X; G) in general: it classifies G-torsors on X (that is, sheaves F on X which carry an action of G and locally admit a G-equivariant isomorphism F ≃ G) up to isomorphism. The general formalism of sheaf cohomology is extremely useful, even if we are interested only in the case where X is a nice topological space: it includes, for example, the theory of cohomology with coefficients in a local system on X.

Let us now attempt to obtain a similar interpretation for cohomology classes η ∈ H2 (X ; G). What should play the role of a G-torsor in this case? To answer this question, we return to the situation where X is a CW complex, so that η can be identified with a continuous map X → K(G,2). We can think of K(G,2) as the classifying space of a group: not the discrete group G, but instead the classifying space BG (which, if built in a sufficiently careful way, comes equipped with the structure of a topological abelian group). Namely, we can identify K(G, 2) with the quotient E/BG, where E is a contractible space with a free action of BG. Any cohomology class η ∈ H2(X;G) determines a map X → K(G,2), and we can form the pullback X~ = E × BG X. We now think of X as a torsor over X: not for the discrete group G, but instead for its classifying space BG.

To complete the analogy with our analysis in the case n = 1, we would like to interpret the fibration X → X as defining some kind of sheaf F on the space X. This sheaf F should have the property that for each x ∈ X, the stalk Fx can be identified with the fiber X~x ≃ BG. Since the space BG is not discrete (or homotopy equivalent to a discrete space), the situation cannot be adequately described in the usual language of set-valued sheaves. However, the classifying space BG is almost discrete: since the homotopy groups πiBG vanish for i > 1, we can recover BG (up to homotopy equivalence) from its fundamental groupoid. This suggests that we might try to think about F as a “groupoid-valued sheaf” on X, or a stack (in groupoids) on X.

# Irrationality. Note Quote. To mathematics it is unique, that two absolutely contrary opinions do not logically exclude each other but exist simultaneously while there seems to be no chance to pick out a false one and to establish a remaining truth. This case is realised by the philosophy and mathematics of the infinite. While transfinite set theory is impossible without different degrees of infinity, constructivists and intuitionists deny this notion without running into inconsistencies as is admitted by some of the foremost set theorists:

It would not be astonishing if in different axiomatic systems different results were obtained with respect to peculiarities of those systems. But set theorists on one side and constructivists and intuitionists on the other are certainly believing to address the same entities when speaking of “rational numbers” or of “irrational numbers”. In spite of that, the former are convinced that there are infinitely many more irrational numbers than rational numbers while the latter deny that:

This situation yields bewildering results:

Nevertheless, the great majority of mathematicians refuse to accept the thesis that Cantor’s ideas were but a pathological fancy. Though the foundations of set theory are still somewhat shaky. Most surprising and by no means to be expected of a pupil of Fraenkel’s is that Robinson states:

Infinite totalities do not exist in any sense of the word (i.e. either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless. Nevertheless, we should act as if infinite totalities really existed. 

Does there exist a correct and an incorrect position? And, if so, who is right, who is wrong?

Following the advice of Fraenkel, namely to judge about the value and necessity of the basic axioms, in particular of the axiom of choice, by considering its consequences, in order to settle this question. These consequences will turn out to entail what, in an euphemistic way, by set theorists usually is called a “paradoxical result”, in order to avoid the term self-contradiction.

Apart from the well-ordering theorem some statements of quite different character – in particular geometrical statements – have been proved by means of the axiom of choice, which because of their paradoxical character induced some mathematicians to reject the axiom. Presumably the earliest statement of this kind is Hausdorff’s discovery that half of the sphere’s surface is congruent to a third of it. … It may surprise scholars working in the field … that even after more than half a century of utilising the axiom of choice and well-ordering theorem, a number of first-rate mathematicians (especially French) have not essentially changed their distrustful attitude.

Transfinite set theory arises from Cantor’s observation that the set of all irrational numbers has infinitely many more members than the set of all rational numbers. While the latter has the same cardinality χ0 as the set N of all natural numbers n, the cardinality χ of the set of all irrational numbers is larger, χ = 2χ0. It is proven to be uncountable, i.e., any bijection with N can be excluded.

# 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: 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]

# RAPL (Right Adjoint Preserve Limits) Theorem. Part 8b/End Part.

Fix a small category I and a diagram D ∶ I → C of shape I. Then the limit of D, if it exists, consists of an object l ∈ C and a natural isomorphism

Cone ∶ HomC(−,l) ≅ HomCI((−)I,D) ∶ Uni

in the category SetCop.

This is intuitively plausible if we recall the definition of limits. Recall that a cone under D consists of an object l ∈ C and a natural transformation Λ ∶ lI ⇒ D. We say that the cone (l,Λ) is the the limit of D if, for any other cone Φ ∶ cI ⇒ D, there exists a unique arrow υ ∶ c → l making the following diagram in CI commute: The map sending the cone Φ ∶ cI ⇒ D to the unique arrow υ ∶ c → l is the desired function HomCI (cI,D) → HomC(c,l). Furthermore, it’s clear that this function is a bijection since we can pull back any arrow α ∶ c → l to the cone Λ ○ αI ∶ cI ⇒ D. The main difficulty is to show that the data of naturality for these bijections is equivalent to the data of the canonical cone Λ ∶ lI ⇒ D.

Proof: First assume that the limit of D exists and is given by the cone (limID,Λ). In this case we want to define a family of bijections

Unic ∶HomcI(cI,D) →~ Hom (c,limID)

that is natural in c ∈ Cop. (Then the inverse Cone ∶= Uni−1 is automatically natural. So consider any element Φ ∈ HomCI(cI,D), i.e., any cone Φ ∶ cI ⇒ D. By the definition of limits we know that there exists a unique arrow υ ∶ c → limID making the following diagram commute: Therefore the assignment Unic(Φ) ∶= υ defines an injective function (recall that the functor (−)I is faithful, so that υ1I = υ2I implies υ1 = υ2). To see that Unic is surjective, consider any arrow α ∶ c → limID in C. We want to define a cone Φα ∶ cI ⇒ D with the property that Unicα) = α. By definition of Unic this means that we must have Φα ∶= Λ ○ αI — in other words, we must have α)i ∶= Λi ○ α indices i ∈ I. And note that this does define a natural transformation Φα ∶ cI ⇒ D since for all arrows δ ∶ i ∈ j in I we have

D(δ) ○ (Φα)i =D(δ) ○ (Λi ○ α)

= (D(δ) ○ Λi) ○ α

= Λj ○ α (Naturality of Λ)

= (Φα)j

We conclude that Unic is a bijection. To see that Unic is natural in c ∈ Cop, consider any arrow γ ∶ c1 → c2 in C (i.e., any arrow γ ∶ c2 → c1 in C). We want to show that the following diagram commutes: And to see this, consider any cone Φ ∶ cI1 ⇒ D. By composing with the natural transformation γI ∶ cI2 ⇒ cI1 we obtain the following commutative diagram in CI: Since the diagonal embedding (−)I ∶ C → CI is a functor, the bottom arrow is given by

(Unic1 (Φ))I ○ γI = (Unic1 (Φ) ○ γ)I

But by the definition of the function Unic2 this arrow also equals (Unic2(Φ ○ γI))I

Then since (−)I is a faithful functor we conclude that

Unic2 (Φ ○ γI) = Unic1 (Φ) ○ γ

and hence the desired square commutes. Conversely, consider an object l ∈ C and suppose that we have a bijection

Conec ∶HomC(c,l) ←→ HomCI(cI,D) ∶ Unic

that is natural in c ∈ Cop. In other words, suppose that for each arrow γ ∶ c1 → c2 in Cop (i.e., for each arrow γ ∶ c2 → c1 in C) we have a commutative square: We want to show that this determines a unique cone Λ ∶ lI ⇒ D such that (l, Λ) is the limit of D. The only possible choice is to define Λ ∶= Conel(idl). Now given any cone Φ ∶ cI ⇒ D we want to show that there exists a unique arrow υ ∶ c → l with the property Λ ○ υI = Φ.

So suppose that there exists some arrow υ ∶ c → l with the property Λ ○ υI = Φ. By substituting γ ∶= υ into the above diagram we obtain a commutative square: Then following the arrow idl ∈ HomC(l, l) around the square in two different ways gives

idl ○ υ = Unic(Conel(idl) ○ υI)

υ = Unic(Λ ○ υI)

υ = Unic(Φ)

Thus there exists at most one such arrow υ. To show that there exists at least one such arrow, we must check that the arrow Unic(Φ) actually does satisfy Λ ○ (Unic(Φ))I = Φ. Indeed, by substituting υ ∶= Unic(Φ) into the above diagram we obtain a commutative square: Then following the arrow idl ∈ HomC(l,l) around the

Conel(idl) ○ (Unic(Φ))I) = Conec(idl ○ Unic(Φ)) Λ ○ (Unic(Φ))I

= Conec(Unic(Φ))

Λ ○ (Unic(Φ))I = Φ

square in two ways gives as desired.

[Remark: We have proved that the limit of a diagram D ∶ I → C, if it exists, consists of an object limID ∈ C and a natural isomorphism

HomC(−, limID) ≅ HomCI((−)I,D) of functors Cop → Set. It turns out that if all limits of shape I exist in C then there is a unique way to extend this to a natural isomorphism

HomC(−,limI−) ≅ HomCI((−)I,−)

of functors Cop × CI → Set, and hence that we have an adjunction (−)I ∶ C ⇄ CI ∶ limI. However, we don’t need this result right now so we won’t prove it. Dually, the colimit of D, if it exists, consists of an object colimID ∈ C and a natural isomorphism HomC(colimID, −) ≅ HomCI(D, (−)I) of functors C → Set. If all colimits of shape I exist in C then this extends uniquely to an adjunction colimI ∶ CI ⇄ C ∶ (−)I. This explains the title of the previous lemma.]

Theorem (RAPL):

Let L ∶ C ⇄ D ∶ R be an adjunction and consider a diagram D ∶ I → D of shape I in D. If the diagram D ∶ I → D has a limit cone Λ ∶ lI ⇒ D then the composite diagram RI(D) ∶ I → C also has a limit cone, which is given by RI(Λ) ∶ R(l)I ⇒ RI(D).

Proof:

In this proof we will write limID ∶= l ∈ D, and we will just assume that the limit object limI RI(D) ∈ C exists. Now we want to show that the following objects are isomorphic in C : R(limID) ≅ limI RI (D). (We will ignore the data of the limit cone.)

So assume that L ∶ C ⇄ D ∶ R is an adjunction. Then we have the following sequence of bijections, each of which is natural in c ∈ Cop:

Homc(c, R(limID)) →~ Homc(L(c), limID) (L ⊣ R)

~ HomDI(L(c),D) (Diagonal ⊣ Limit)

~ HomDI (LI (cI),D)

~ HomCI(c,RI(D))

~ Homc(c,limIRI(D)) (Diagonal ⊣ Limit)

By composing these we obtain a family of bijections

Homc(c,R(limID)) →~ Homc(c,limI RI(D))

that is natural in c ∈ Cop. In other words, we obtain an isomorphism of hom functors HR(limI(D)) ≅ HlimI RI(D) in the category SetCop. Then since the Yoneda embedding H(−) : C → SetCop is essentially injective (from the Embedding Lemma), we obtain an isomorphism of objects R(limID) ≅ limI RI(D) in the category C.

# RAPL (Right Adjoint Preserve Limits) Theorem. Part 8a.

To prove the RAPL theorem we must first translate the definition of limit/colimit into a language that is compatible with the definition of adjoint functors.

Recall that a diagram is a functor D ∶ I → C from a small category I. If C is locally small then we have a locally small category CI consisting of diagrams and natural transformations between them. For each object c ∈ C we also have the constant diagram cI ∶ I → C that sends each object i ∈ Obj(I) to cI(i) ∶= c and each arrow δ ∈ Arr(I) to cI(δ) ∶= idc.

It is a general phenomenon that many categorical properties of CI are inherited from C. The next lemma collects a few of these properties that we will need later.

Diagram Lemma. Fix a small category I and locally small categories C, D. Then:

(i) For any category C, the mapping c ↦ cI defines a fully faithful functor (−)I ∶ C → CI which we call the diagonal embedding.

(ii)  For any functor F ∶ C → D the mapping FI(D)∶= F ○ D defines a functor FI ∶ CI → DI with the property that F (−)I = FI((−)I).

(iii)  Any adjunction L ∶ C ⇄ D ∶ R induces an adjunction LI ∶ CI ⇄ DI ∶ RI That is, we have a natural isomorphism of bifunctors

HomCI (−, RI(−)) ≅ HomDI (LI(−), −)

from (CI)op × DI to Set.

(iv) In particular, naturality in DI tells us that for all objects l ∈ C and all natural transformations Λ ∶ lI ⇒ D we have a commutative square: Proof:

(i): For any arrow α ∶ c1 → c2 in C we want to define a natural transformation of diagrams αI ∶ cI1 ⇒ cI2, and there is only one way to do this. Since (cI1)i = c1 and (cI2)i = c2 ∀ i ∈ I, the arrow I)i ∶= (cI1)i → (cI2)i must be defined by I)i ∶= α. Then for any arrow δ ∶ i → j in I we have cI1(δ) = idc1 and cI2(δ) = idc2, so that

I)i ○ (cI1) (δ) = (α ○ idc1) = (idc2 ○ α) = (cI2)i (δ) ○ (αI)i

and hence we obtain a natural transformation αI ∶ cI1 ⇒ cI2. The assignment α ↦ αI is functorial since for all arrows α, β such that α ○ β exists and ∀ i ∈ I we have (α ○ β)Ii = α ○ β = (αI)i ○ (βI)i = (αI ○ βI)i,

and hence (α ○ β)I = αI ○ βI. Finally, note that we have a bijection of hom sets HomC (c1, c2) ↔ HomCI (cI1, cI2given by α ↔ αI, and hence the functor (−)I ∶ C → CI is fully faithful.

(ii): Let F ∶ C → D be any functor. Then for any diagram D ∶ I → C we obtain a diagram FI(D) ∶ I → D by composition: FI(D) ∶= F ○ D. This assignment is functorial in D ∈ CI. To see this, consider any natural transformation Φ ∶ D1 ⇒ D2 in the category CI. Then for any arrow δ ∶ i → j in I we can apply F to the naturality square for Φ to obtain another commutative square: If we define FI(Φ)i ∶= F(Φi) ∀ i ∈ I then this second commutative square says that FI(Φ) ∶ FI(D1) ⇒ FI(D2) is a natural transformation in DI. If Φ and Ψ are two arrows (natural transformations) in CI such that Φ ○ Ψ is defined, then ∀ i ∈ I we have FI(Φ ○ Ψ)i = F((Φ ○ Ψ)i) = F(Φi ○ Ψi) = F(Φi) ○ F(Ψi) = FI(Φ)i ○ FI(Ψ)i = (FI(Φ) ○ FI(Ψ))i and hence FI(Φ ○ Ψ) = FI(Φ) ○ FI(Ψ). Thus we have defined a functor FI ∶ CI → DI. Finally, note that ∀ i ∈ I, c ∈ C, and α ∈ Arr(C) we have

FI(cI)i = F((cI)i) = F(c) = ((F(c))I)i FII)i = F((αI)i) = F(α) = ((F(α))I)i

and hence we have an equality of functors FI((−)I) = F (−)I from C to DI

(iii): Let L ∶ C ⇄ D ∶ R be any adjunction. We will denote each bijection HomC(−,R(−)) ↔ HomC(L(−), −) by φ ↦ φ, so that φ= = φ. Now we want to define a natural family of bijections HomCI (−, RI (−)) ≅ HomDI (LI(−), −)

To do this, consider diagrams C ∈ CI, D ∈ DI, and a natural transformation Φ ∶ C ⇒ RI(D). Then for each index i ∈ I we have an arrow Φi ∶ C(i) → R(D(i)), which determines an arrow Φi ∶ L(C(i)) → D(i) by adjunction. The arrows Φi assemble into a natural transformation Φ ∶ LI(C) ⇒ D. To see this, consider any arrow δ ∶ i ∈ j in I. Then from the naturality of Φ and the adjunction L ⊣ R we have

D(δ) ○ Φi = (R(D(δ)) ○ Φi)                             naturality of L ⊣ R

= (Φj ○ C(δ))                                                        naturality of Φ

= Φj ○ L(C(δ))                                                     naturality of L ⊣ R

as desired. In a similar way one can check that for each natural transformation Ψ ∶ LI(C) ⇒ D, the arrows Ψi ∶ C(i) → R(D(i)) assemble into a natural transformation Ψ ∶ C ⇒ RI(D). Thus we have established the desired bijection of hom sets HomCI (C, RI (D)) ↔ HomDI (LI (C), D).

To prove that this bijection is natural in (C, D) ∈ (CI)op × DI, consider any pair of natural transformations Γ∶ C2 ⇒ C1 in CI and ∆ ∶ D1 ⇒ D2 in DI. We need to show that a certain cube of functions commutes. For a fixed diagram C ∈ CI the following square commutes: First, recall that the natural transformation RI(∆) ∶ RI(D1) ⇒ RI(D2) is defined pointwise by RI(∆)i ∶= R(∆i) ∶ R(D1(i)) → R(D2(i)). Now consider any Φ ∶ C ⇒ RI(D1). The naturality of the original adjunction tells us that (R(∆i) ○ Φi) = ∆i ○ Φi, and hence we have

((RI(∆) ○ Φ)i) = (RI(∆)i ○ Φi)

= (R(∆i) ○ Φi)

= ∆i ○ Φi

= (∆ ○ Φ)i

∀ i ∈ I. By definition this means that (RI(∆) ○ Φ) = ∆ ○ Φ, and hence the desired square commutes. It remains only to check that the cube is natural in (CI)op. This follows from a similar pointwise computation.

(iv): Now fix an element l ∈ C, a diagram D ∈ DI, and a natural transformation Λ ∶ lI ⇒ D. By substituting C = R(l)I, D1 = lI, D2 = D, and ∆ = Λ into the above commutative square and using part (ii), we obtain the commutative square from the statement of the lemma. In particular, following the identity arrow idIR(l) around the square in two ways gives

(RI(Λ) ○ idIR(l)) = Λ ○ (idIR(l))

Finally, one can check pointwise that (idIR(l)) = ((idR(l))I) and hence we obtain the identity

(RI(Λ) ○ idIR(l)) = ((idR(l))I)

Now we will reformulate the definition of limit/colimit in terms of adjoint functors. If all limits/colimits of shape I exist in some category C then it turns out (surprisingly) that we can think of limits/colimits as right/left adjoints to the diagonal embedding (−)I ∶ C → CI : colimI ⊣(−)I ⊣ limI

In the next section/part’s lemma we will prove something slightly more general. We will characterize a specific limit/colimit of shape I, without assuming that all limits/colimits of shape I exist.

# Functors. Part 5.

We have called L ∶ P ⇄ Q ∶ R an adjoint pair of functions, but of course they more than just functions. If L ⊣ R is an adjunction, then property (2) of Galois connections says that ∀ p1, p2 ∈ P and q1, q2 ∈ Q we have

p1 ≤ P p2 ⇒ L(p1) ≤ Q L(p2) and q1 ≤ Q q2 ⇒ R(q1) ≤ P R(q2)

That is, the functions L ∶ P → Q and R ∶ Q → P are actually homomorphisms of posets.

Definition of Functor: Let C and D be categories. A functor F ∶ C → D consists a family of functions:

• A function on objects F ∶ Obj(C) → Obj(D),

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

F ∶ HomC(c1,c2) → HomD(F(c1),F(c2)). These functions must preserve the category structure:

(i) Identity: For all objects c ∈ C we have F (idc) = idF(c).

(ii) Composition: For all arrows α, β ∈ C such that β ○ α is defined, we have

F (β ○ α) = F (β) ○ F (α)

Functors compose in an associative way, and for each category C there is a distinguished identity functor idC ∶ C → C. In other words, the collection of all categories with functors between them forms a (very big) category, which we denote by Cat.

This definition is not surprising. It basically says that a functor F ∶ C → D sends commutative diagrams in C to commutative diagrams in D. That is, for each diagram D ∶ I → C in C we have a diagram FI(D) ∶ I → D in D (defined by FI(D) ∶= F ○ D), which is commutative if and only if D is.

Now let’s try to guess the definition of an “adjunction of categories”. Let C and D be categories and let L ∶ C ⇄ D ∶ R be any two functors. When C is a poset, recall that ∀ x, y ∈ C we have ∣HomC (x, y)∣ ∈ {0, 1} and we use the notations

x ≤ y ⇐⇒ ∣HomC(x,y)∣ = 1

x ≤/ y ⇐⇒ ∣HomC(x,y)∣ = 0

Thus if C and D are posets, we can rephrase the definition of a poset adjunction L ∶ C ⇄ D ∶ R by stating that ∀ objects c ∈ C and d ∈ D there exists a bijection of hom sets:

HomC (c, R(d)) ←→ HomD (L(d), c)

In this form the definition now applies to any pair of functors between categories.

However, if we want to preserve the important theorems (uniqueness of adjoints and RAPL) then we need to impose some “naturality” condition on the family of bijections between hom sets. This condition is automatic for posets, so we will have to look elsewhere for motivation.

Let C and D be categories. We say that a pair of functors L ∶ C ⇄ D ∶ R is an adjunction if for all objects c ∈ C and d ∈ D there exists a bijection of hom sets

HomC (c, R(d)) ←→ HomD (L(d), c)

Furthermore, we require that these bijections fit together in the following “natural” way. For each arrow γ ∶ c2 → c1 in C and each arrow δ ∶ d1 → d2 in D we require that the following cube of functions commutes: Natural Transformation:

Let C and D be categories and consider two parallel functors F1,F2 ∶ C → D. A natural transformation Φ ∶ F ⇒ G consists of a family of arrows Φc ∶F(c) → R(c), one for each object c ∈ C, such that for each arrow γ ∶ c1 → c2 in C the following square commutes: The figure below the square is called a “2-cell diagram”. It hints at the close relationship between category theory and topology.

Let DC denote the collection of all functors from C to D and natural transformations between them. One can check that this forms a category (called a functor category). Given F1, F2 ∈ DC we say that F1 and F2 are naturally isomorphic if they are isomorphic in DC, i.e., if there exists a pair of natural transformations Φ ∶ F1 ⇒ F2 and Ψ ∶ F2 ⇒ F1 such that Ψ ○ Φ = idF1 and Ψ ○ Φ = idF2 are the identity natural transformations. In this case we will write F1 ≅ F2 and we will say that Φ and Φ−1 ∶= Ψ are natural isomorphisms.

To develop some intuition for this definition, let I be a small category and let C be any category. We have previously referred to functors D ∶ I → C as “diagrams of shape I in C“. Now we can think of CI as a category of diagrams. Given two such diagrams D1, D2 ∈ CI , we visualize a natural transformation Φ ∶ D1 ⇒ D2 as a “cylinder”: The diagrams D1 and D2 need not be commutative, but if they are then the whole cylinder is commutative.

Limit/Colimit:

Consider a diagram D ∶ I → C. The limit of D, if it exists, consists of an object limID ∈ C and a canonical natural transformation Λ ∶ (limID)I ⇒ D such that for each object c ∈ C and natural transformation Φ ∶ cI ⇒ D there exists a unique natural transformation υI ∶ cI ⇒ (limID)I making the following diagram in CI commute: Hom Functors:

Let C be a category. For each object c ∈ C the mapping d ↦ HomC(c, d) defines a functor from C to the category of sets Set. We denote it by

Hc ∶= HomC(c,−) ∶ C → Set

To define the action of Hc on arrows, consider any δ ∶ d1 → d2 in C. Then we must have a function Hc(δ) ∶ Hc(d1) → Hc(d2), i.e., a function Hc(δ) ∶ HomC(c,d1) → HomC(c,d2). There is only one way to define this:

H(δ) (φ) ∶= δ ○ φ

Similarly, for each arrow δ ∶ c1 → c2 we can define a function Hc(δ) ∶ HomC(d2,c) → HomC(d1,c) by H(δ) (φ) ∶= φ ○ δ. This again defines a functor into sets, but this time it is from the opposite category Cop (which is defined by reversing all arrows in C):

Hc ∶= HomC(−,c) ∶ Cop → Set

Finally, we can put these two functors together to obtain the hom bifunctor

HomC(−,−)∶ Cop ×C →Set

which sends each pair of arrows (γ ∶ c2 → c1, δ ∶ d1 → d2) to the function

HomC(γ,δ) ∶ HomC(c1,d1) → HomC(c2,d2)

defined by φ ↦ δ ○ φ ○ γ. The product category Cop × C is defined in the most obvious way.