# Indecomposable Objects – Part 1

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

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

h, t : Q1 → Q0

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

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

φa : Vt(a) → Vh(a)

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

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

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

V1 →ƒ W and 0 → W

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

C → 0, 0 → C, C →id C

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

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

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

… → Mi-1 →di-1 Mi →di → Mi+1 → …

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

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

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

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

Hi(M) = ker(di)/im(di−1)

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

Hi(−) : C(A) → A,

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

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

M = ⊕i Mi

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

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

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

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

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

M ← P → N

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

0 → S →f M0 →d0 M1 →d1 M2 → …

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

is a quasi-isomorphism.

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

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

with the following two properties:

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

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

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

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

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

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

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

J: A → C(A)

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

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

Hi(−): D(A) → A

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

# Categories of Pointwise Convergence Topology: Theory(ies) of Bundles.

Let H be a fixed, separable Hilbert space of dimension ≥ 1. Lets denote the associated projective space of H by P = P(H). It is compact iff H is finite-dimensional. Let PU = PU(H) = U(H)/U(1) be the projective unitary group of H equipped with the compact-open topology. A projective bundle over X is a locally trivial bundle of projective spaces, i.e., a fibre bundle P → X with fibre P(H) and structure group PU(H). An application of the Banach-Steinhaus theorem shows that we may identify projective bundles with principal PU(H)-bundles and the pointwise convergence topology on PU(H).

If G is a topological group, let GX denote the sheaf of germs of continuous functions G → X, i.e., the sheaf associated to the constant presheaf given by U → F(U) = G. Given a projective bundle P → X and a sufficiently fine good open cover {Ui}i∈I of X, the transition functions between trivializations P|Ui can be lifted to bundle isomorphisms gij on double intersections Uij = Ui ∩ Uj which are projectively coherent, i.e., over each of the triple intersections Uijk = Ui ∩ Uj ∩ Uk the composition gki gjk gij is given as multiplication by a U(1)-valued function fijk : Uijk → U(1). The collection {(Uij, fijk)} defines a U(1)-valued two-cocycle called a B-field on X,which represents a class BP in the sheaf cohomology group H2(X, U(1)X). On the other hand, the sheaf cohomology H1(X, PU(H)X) consists of isomorphism classes of principal PU(H)-bundles, and we can consider the isomorphism class [P] ∈ H1(X,PU(H)X).

There is an isomorphism

H1(X, PU(H)X) → H2(X, U(1)X) provided by the

boundary map [P] ↦ BP. There is also an isomorphism

H2(X, U(1)X) → H3(X, ZX) ≅ H3(X, Z)

The image δ(P) ∈ H3(X, Z) of BP is called the Dixmier-Douady invariant of P. When δ(P) = [H] is represented in H3(X, R) by a closed three-form H on X, called the H-flux of the given B-field BP, we will write P = PH. One has δ(P) = 0 iff the projective bundle P comes from a vector bundle E → X, i.e., P = P(E). By Serre’s theorem every torsion element of H3(X,Z) arises from a finite-dimensional bundle P. Explicitly, consider the commutative diagram of exact sequences of groups given by

where we identify the cyclic group Zn with the group of n-th roots of unity. Let P be a projective bundle with structure group PU(n), i.e., with fibres P(Cn). Then the commutative diagram of long exact sequences of sheaf cohomology groups associated to the above commutative diagram of groups implies that the element BP ∈ H2(X, U(1)X) comes from H2(X, (Zn)X), and therefore its order divides n.

One also has δ(P1 ⊗ P2) = δ(P1) + δ(P2) and δ(P) = −δ(P). This follows from the commutative diagram

and the fact that P ⊗ P = P(E) where E is the vector bundle of Hilbert-Schmidt endomorphisms of P . Putting everything together, it follows that the cohomology group H3(X, Z) is isomorphic to the group of stable equivalence classes of principal PU(H)-bundles P → X with the operation of tensor product.

We are now ready to define the twisted K-theory of the manifold X equipped with a projective bundle P → X, such that Px = P(H) ∀ x ∈ X. We will first give a definition in terms of Fredholm operators, and then provide some equivalent, but more geometric definitions. Let H be a Z2-graded Hilbert space. We define Fred0(H) to be the space of self-adjoint degree 1 Fredholm operators T on H such that T2 − 1 ∈ K(H), together with the subspace topology induced by the embedding Fred0(H) ֒→ B(H) × K(H) given by T → (T, T2 − 1) where the algebra of bounded linear operators B(H) is given the compact-open topology and the Banach algebra of compact operators K = K(H) is given the norm topology.

Let P = PH → X be a projective Hilbert bundle. Then we can construct an associated bundle Fred0(P) whose fibres are Fred0(H). We define the twisted K-theory group of the pair (X, P) to be the group of homotopy classes of maps

K0(X, H) = [X, Fred0(PH)]

The group K0(X, H) depends functorially on the pair (X, PH), and an isomorphism of projective bundles ρ : P → P′ induces a group isomorphism ρ∗ : K0(X, H) → K0(X, H′). Addition in K0(X, H) is defined by fibre-wise direct sum, so that the sum of two elements lies in K0(X, H2) with [H2] = δ(P ⊗ P(C2)) = δ(P) = [H]. Under the isomorphism H ⊗ C2 ≅ H, there is a projective bundle isomorphism P → P ⊗ P(C2) for any projective bundle P and so K0(X, H2) is canonically isomorphic to K0(X, H). When [H] is a non-torsion element of H3(X, Z), so that P = PH is an infinite-dimensional bundle of projective spaces, then the index map K0(X, H) → Z is zero, i.e., any section of Fred0(P) takes values in the index zero component of Fred0(H).

Let us now describe some other models for twisted K-theory which will be useful in our physical applications later on. A definition in algebraic K-theory may given as follows. A bundle of projective spaces P yields a bundle End(P) of algebras. However, if H is an infinite-dimensional Hilbert space, then one has natural isomorphisms H ≅ H ⊕ H and

End(H) ≅ Hom(H ⊕ H, H) ≅ End(H) ⊕ End(H)

as left End(H)-modules, and so the algebraic K-theory of the algebra End(H) is trivial. Instead, we will work with the Banach algebra K(H) of compact operators on H with the norm topology. Given that the unitary group U(H) with the compact-open topology acts continuously on K(H) by conjugation, to a given projective bundle PH we can associate a bundle of compact operators EH → X given by

EH = PH ×PU K

with δ(EH) = [H]. The Banach algebra AH := C0(X, EH) of continuous sections of EH vanishing at infinity is the continuous trace C∗-algebra CT(X, H). Then the twisted K-theory group K(X, H) of X is canonically isomorphic to the algebraic K-theory group K(AH).

We will also need a smooth version of this definition. Let AH be the smooth subalgebra of AH given by the algebra CT(X, H) = C(X, L1PH),

where L1PH = PH ×PUL1. Then the inclusion CT(X, H) → CT(X, H) induces an isomorphism KCT(X, H) → KCT(X, H) of algebraic K-theory groups. Upon choosing a bundle gerbe connection, one has an isomorphism KCT(X, H) ≅ K(X, H) with the twisted K-theory defined in terms of projective Hilbert bundles P = PH over X.

Finally, we propose a general definition based on K-theory with coefficients in a sheaf of rings. It parallels the bundle gerbe approach to twisted K-theory. Let B be a Banach algebra over C. Let E(B, X) be the category of continuous B-bundles over X, and let C(X, B) be the sheaf of continuous maps X → B. The ring structure in B equips C(X, B) with the structure of a sheaf of rings over X. We can therefore consider left (or right) C(X, B)-modules, and in particular the category LF C(X, B) of locally free C(X, B)-modules. Using the functor in the usual way, for X an equivalence of additive categories

E(B, X) ≅ LF (C(X, B))

Since these are both additive categories, we can apply the Grothendieck functor to each of them and obtain the abelian groups K(LF(C(X, B))) and K(E(B, X)). The equivalence of categories ensures that there is a natural isomorphism of groups

K(LF (C(X, B))) ≅ K(E(B, X))

This motivates the following general definition. If A is a sheaf of rings over X, then we define the K-theory of X with coefficients in A to be the abelian group

K(X, A) := K LF(A)

For example, consider the case B = C. Then C(X, C) is just the sheaf of continuous functions X → C, while E(C, X) is the category of complex vector bundles over X. Using the isomorphism of K-theory groups we then have

K(X, C(X,C)) := K(LF (C(X, C))) ≅ K (E(C, X)) = K0(X)

The definition of twisted K-theory uses another special instance of this general construction. For this, we define an Azumaya algebra over X of rank m to be a locally trivial algebra bundle over X with fibre isomorphic to the algebra of m × m complex matrices over C, Mm(C). An example is the algebra End(E) of endomorphisms of a complex vector bundle E → X. We can define an equivalence relation on the set A(X) of Azumaya algebras over X in the following way. Two Azumaya algebras A, A′ are called equivalent if there are vector bundles E, E′ over X such that the algebras A ⊗ End(E), A′ ⊗ End(E′) are isomorphic. Then every Azumaya algebra of the form End(E) is equivalent to the algebra of functions C(X) on X. The set of all equivalence classes is a group under the tensor product of algebras, called the Brauer group of X and denoted Br(X). By Serre’s theorem there is an isomorphism

δ : Br(X) → tor(H3(X, Z))

where tor(H3(X, Z)) is the torsion subgroup of H3(X, Z).

If A is an Azumaya algebra bundle, then the space of continuous sections C(X, A) of X is a ring and we can consider the algebraic K-theory group K(A) := K0(C(X,A)) of equivalence classes of projective C(X, A)-modules, which depends only on the equivalence class of A in the Brauer group. Under the equivalence, we can represent the Brauer group Br(X) as the set of isomorphism classes of sheaves of Azumaya algebras. Let A be a sheaf of Azumaya algebras, and LF(A) the category of locally free A-modules. Then as above there is an isomorphism

K(X, C(X, A)) ≅ K Proj (C(X, A))

where Proj (C(X, A)) is the category of finitely-generated projective C(X, A)-modules. The group on the right-hand side is the group K(A). For given [H] ∈ tor(H3(X, Z)) and A ∈ Br(X) such that δ(A) = [H], this group can be identified as the twisted K-theory group K0(X, H) of X with twisting A. This definition is equivalent to the description in terms of bundle gerbe modules, and from this construction it follows that K0(X, H) is a subgroup of the ordinary K-theory of X. If δ(A) = 0, then A is equivalent to C(X) and we have K(A) := K0(C(X)) = K0(X). The projective C(X, A)-modules over a rank m Azumaya algebra A are vector bundles E → X with fibre Cnm ≅ (Cm)⊕n, which is naturally an Mm(C)-module.

# Of Magnitudes, Metrization and Materiality of Abstracto-Concrete Objects.

The possibility of introducing magnitudes in a certain domain of concrete material objects is by no means immediate, granted or elementary. First of all, it is necessary to find a property of such objects that permits to compare them, so that a quasi-serial ordering be introduced in their set, that is a total linear ordering not excluding that more than one object may occupy the same position in the series. Such an ordering must then undergo a metrization, which depends on finding a fundamental measuring procedure permitting the determination of a standard sample to which the unit of measure can be bound. This also depends on the existence of an operation of physical composition, which behaves additively with respect to the quantity which we intend to measure. Only if all these conditions are satisfied will it be possible to introduce a magnitude in a proper sense, that is a function which assigns to each object of the material domain a real number. This real number represents the measure of the object with respect to the intended magnitude. This condition, by introducing an homomorphism between the domain of the material objects and that of the positive real numbers, transforms the language of analysis (that is of the concrete theory of real numbers) into a language capable of speaking faithfully and truly about those physical objects to which it is said that such a magnitude belongs.

Does the success of applying mathematics in the study of the physical world mean that this world has a mathematical structure in an ontological sense, or does it simply mean that we find in mathematics nothing but a convenient practical tool for putting order in our representations of the world? Neither of the answers to this question is right, and this is because the question itself is not correctly raised. Indeed it tacitly presupposes that the endeavour of our scientific investigations consists in facing the reality of “things” as it is, so to speak, in itself. But we know that any science is uniquely concerned with a limited “cut” operated in reality by adopting a particular point of view, that is concretely manifested by adopting a restricted number of predicates in the discourse on reality. Several skilful operational manipulations are needed in order to bring about a homomorphism with the structure of the positive real numbers. It is therefore clear that the objects that are studied by an empirical theory are by no means the rough things of everyday experience, but bundles of “attributes” (that is of properties, relations and functions), introduced through suitable operational procedures having often the explicit and declared goal of determining a concrete structure as isomorphic, or at least homomorphic, to the structure of real numbers or to some other mathematical structure. But now, if the objects of an empirical theory are entities of this kind, we are fully entitled to maintain that they are actually endowed with a mathematical structure: this is simply that structure which we have introduced through our operational procedures. However, this structure is objective and real and, with respect to it, the mathematized discourse is far from having a purely conventional and pragmatic function, with the goal of keeping our ideas in order: it is a faithful description of this structure. Of course, we could never pretend that such a discourse determines the structure of reality in a full and exhaustive way, and this for two distinct reasons: In the first place, reality (both in the sense of the totality of existing things, and of the ”whole” of any single thing), is much richer than the particular “slide” that it is possible to cut out by means of our operational manipulations. In the second place, we must be aware that a scientific object, defined as a structured set of attributes, is an abstract object, is a conceptual construction that is perfectly defined just because it is totally determined by a finite list of predicates. But concrete objects are by no means so: they are endowed with a great deal of attributes of an indefinite variety, so that they can at best exemplify with an acceptable approximation certain abstract objects that are totally encoding a given set of attributes through their corresponding predicates. The reason why such an exemplification can only be partial is that the different attributes that are simultaneously present in a concrete object are, in a way, mutually limiting themselves, so that this object does never fully exemplify anyone of them. This explains the correct sense of such common and obvious remarks as: “a rigid body, a perfect gas, an adiabatic transformation, a perfect elastic recoil, etc, do not exist in reality (or in Nature)”. Sometimes this remark is intended to vehiculate the thesis that these are nothing but intellectual fictions devoid of any correspondence with reality, but instrumentally used by scientists in order to organize their ideas. This interpretation is totally wrong, and is simply due to a confusion between encoding and exemplifying: no concrete thing encodes any finite and explicit number of characteristics that, on the contrary, can be appropriately encoded in a concept. Things can exemplify several concepts, while concepts (or abstract objects) do not exemplify the attributes they encode. Going back to the distinction between sense on the one hand, and reference or denotation on the other hand, we could also say that abstract objects belong to the level of sense, while their exemplifications belong to the level of reference, and constitute what is denoted by them. It is obvious that in the case of empirical sciences we try to construct conceptual structures (abstract objects) having empirical denotations (exemplified by concrete objects). If one has well understood this elementary but important distinction, one is in the position of correctly seeing how mathematics can concern physical objects. These objects are abstract objects, are structured sets of predicates, and there is absolutely nothing surprising in the fact that they could receive a mathematical structure (for example, a structure isomorphic to that of the positive real numbers, or to that of a given group, or of an abstract mathematical space, etc.). If it happens that these abstract objects are exemplified by concrete objects within a certain degree of approximation, we are entitled to say that the corresponding mathematical structure also holds true (with the same degree of approximation) for this domain of concrete objects. Now, in the case of physics, the abstract objects are constructed by isolating certain ontological attributes of things by means of concrete operations, so that they actually refer to things, and are exemplified by the concrete objects singled out by means of such operations up to a given degree of approximation or accuracy. In conclusion, one can maintain that mathematics constitutes at the same time the most exact language for speaking of the objects of the domain under consideration, and faithfully mirrors the concrete structure (in an ontological sense) of this domain of objects. Of course, it is very reasonable to recognize that other aspects of these things (or other attributes of them) might not be treatable by means of the particular mathematical language adopted, and this may imply either that these attributes could perhaps be handled through a different available mathematical language, or even that no mathematical language found as yet could be used for handling them.

# Category Theory of a Sketch. Thought of the Day 50.0

If a sketch can be thought of as an abstract concept, a model of a sketch is not so much an interpretation of a sketch, but a concrete or particular instantiation or realization of it. It is tempting to adopt a Kantian terminology here and say that a sketch is an abstract concept, a functor between a sketch and a category C a schema and the models of a sketch the constructions in the “intuition” of the concept.

The schema is not unique since a sketch can be realized in many different categories by many different functors. What varies from one category to the other is not the basic structure of the realizations, but the types of morphisms of the underlying category, e.g., arbitrary functions, continuous maps, etc. Thus, even though a sketch captures essential structural ingredients, others are given by the “environment” in which this structure will be realized, which can be thought of as being itself another structure. Hence, the “meaning” of some concepts cannot be uniquely given by a sketch, which is not to say that it cannot be given in a structuralist fashion.

We now distinguish the group as a structure, given by the sketch for the theory of groups, from the structure of groups, given by a category of groups, that is the category of models of the sketch for groups in a given category, be it Set or another category, e.g., the category of topological spaces with continuous maps. In the latter case, the structure is given by the exactness properties of the category, e.g., Cartesian closed, etc. This is an important improvement over the traditional framework in which one was unable to say whether we should talk about the structure common to all groups, usually taken to be given by the group axioms, or the structure generated by “all” groups. Indeed, one can now ask in a precise manner whether a category C of structures, e.g., the category of (small) groups, is sketchable, that is, whether there exists a sketch S such that Mod(S, Set) is equivalent as a category to C.

There is another category associated to a sketch, namely the theory of that sketch. The theory of a sketch S, denoted by Th(S), is in a sense “freely” constructed from S : the arrows of the underlying graph are freely composed and the diagrams are imposed as equations, and so are the cones and the cocones. Th(S) is in fact a model of S in the previous sense with the following universal property: for any other model M of S in a category C there is a unique functor F: Th(S) → C such that FU = M, where U: S → Th(S). Thus, for instance, the theory of groups is a category with a group object, the generic group, “freely” constructed from the sketch for groups. It is in a way the “universal” group in the sense that any other group in any category can be constructed from it. This is possible since it contains all possible arrows, i.e., all definable operations, obtained in a purely internal or abstract manner. It is debatable whether this category should be called the theory of the sketch. But that may be more a matter of terminology than anything else, since it is clear that the “free” category called the theory is there to stay in one way or another.

# Obstruction Theory

Obstruction is a concept in homotopy theory where an invariant equals zero if a corresponding problem is solvable and is non-zero otherwise. Let Y be a space, and assume for convenience that Y is n-simple for every n, that is, the action of π1(y) on πn(y) is trivial for every n. Under this hypothesis we can forget about base points for homotopy groups, and any map ƒ: S → Y determines an element of πn(Y).

Let B be a complex and A a subcomplex. Write Xn for A U Bn, where Bn denotes the n-skeleton of B. Let σ be an (n + I)-cell of B which is not in A. Let g = gσ be the attaching map σ. = Sn → Xn ⊂ B.

Given a map ƒ: Xn → Y, denote by c(ƒ) the cochain in Cn+1 (B, A; πn(Y)) given by c(ƒ): σ → [f º gσ]. Then it is clear that ƒ may be extended over Xngσ σ iff f º gσ is null-homotopic, that is, iff c(ƒ)(σ) = 0, and therefore that ƒ can be extended over Xn+1 = A ∪ Bn+1 if the cochain c(ƒ) is the zero cochain. It is a theorem of obstruction theory that c(ƒ) is a cocycle. It is called the obstruction cocycle or “the obstruction to extending ƒ over Bn+1

There are two immediate applications. First, any map of an n-dimensional complex K into an n-connected space X is null-homotopic.

Take (B, A) = (K x I, K x i) and define ƒ:A → X by the given map K → X on one piece and a constant map on the other piece; then ƒ can be extended over B because the obstructions lie in the trivial groups πi(X).

Second, as a particular case, a finite-dimensional complex K is contractible iff πi(K) is trivial for all i < dim K.

Suppose ƒ, g are two maps Xn → Y which agree on Xn-1. Then for each n-cell of B which is not in A, we get a map Sn → Y by taking ƒ and g on the two hemispheres. The resulting cochain of Cn(B, A; πn(Y)) is called the difference cochain of ƒ and g, denoted d(ƒ, g).

# Quantum Numbers as Representations of Gauge Groups

As emphasized by Klein and Weyl, a group is a collection of operations leaving a certain “object” unchanged. This amounts to classifying the symmetries of the object. When the “object” in question is the laws of Physics in a space-time with negligible gravitation-induced curvature, the symmetries can be classified as follows: (i) No point in four- dimensional space-time is privileged, hence one can shift or translate the origin of space-time arbitrarily in four directions. Noether’s theorem then implies there are four associated conserved quantities, namely the three components of space momentum and the energy. These four quantities naturally constitute the components of a 4-vector Pμ, μ = 0, 1, 2, 3. (ii) No direction is special in space; leading to three conserved quantities Ji, i = 1, 2, 3. (iii) There is no special inertial frame; the same laws of Physics hold in inertial frames moving with constant speed in any one of the three independent directions. What is generally known as Noether’s Theorem states that if the Lagrangian function for a physical system is not affected by a continuous change (transformation) in the coordinate system used to describe it, then there will be a corresponding conservation law; i.e. there is a quantity that is constant. For example, if the Lagrangian is independent of the location of the origin then the system will preserve (or conserve) linear momentum. If it is independent of the base time then energy is conserved. If it is independent of the angle of measurement then angular momentum is conserved.

As we suggested above, it is possible to get a non-mathematical insight into Noether’s theorem relating symmetries to conserved quantities. Consider a single particle moving in a completely homogeneous space. It cannot come to a stop or change its velocity because this would have to happen at some particular point, but all points being equal, it is impossible to choose one. Hence the particle has no choice but to move at constant velocity or, in other words, to conserve its linear momentum, which was anciently called “impetus”. It is easy to extend the argument to a rotating object in an isotropic space and conclude that it cannot come to a stop at any particular angle since there is no special angle; hence its angular momentum is conserved.

(ii) and (iii) amount to covariance of the laws of physics under rotations in a four-dimensional space with a metric that is not positive-definite. The squared length of a 4-vector defined via this metric must then be an important invariant independent of the orientation or the velocity of the frame. Indeed, for the 4-vector Pμ this is the squared mass m2 of the particle, and it is one of the two invariant labels used in specifying the representation. The other label is the squared length of another 4-vector called the Pauli-Lubanski vector. It then follows from the algebra of the group that this squared length takes on values s(s + 1) and that in contrast to m2, which assumes continuous values, s can only be zero, or a positive integer, or half a positive odd integer. The unitary representation of the Poincaré group for a particle of mass m and spin s provides its relativistic quantum mechanical wave function. The equation of motion the wave function must obey also comes with the representation; it is the Bargmann-Wigner equation for that spin and mass. In relativistic quantum mechanics and quantum field theory, the Bargmann–Wigner equations describe free particles of arbitrary spin j, an integer for bosons (j = 1, 2, 3 …) or half-integer for fermions (j = 123252 …). The solutions to the equations are wavefunctions, mathematically in the form of multi-component spinor fields. The procedure of second quantization then naturally promotes the wave functions to quantized field operators, and in a sense demotes the particles to quanta created or destroyed by these operators. Pauli’s spin-statistics theorem, based on a set of very general requirements such as the existence of a lowest energy vacuum state, the positivity of energy and probability, microcausality, and the invariance of the laws of Physics under the Poincaré group, leads to the result that the only acceptable quantum conditions for field operators of integer-spin particles are commutation relations, while those corresponding to half-integer spin must obey anticommutation relations. The standard terms for the two families of particles are bosons and fermions, respectively. The Pauli’s Exclusion Principle, or the impossibility of putting two electrons into the same state, is now seen to be the result of the anticommutation relation between electron creation operators: to place two fermions in the same state, the same creation operator has to be applied twice. The result must vanish, since the operator anticommutes with itself.

The symmetries of space-time are reflected in the fields which are representations of the symmetry groups; a quantum mechanical recipe called quantization then turns these fields into operators capable of creating and destroying quanta (or particles, in more common parlance) at all space-time points. Actually, the framework we have described only suffices to describe “Free fields” which do not interact with each other. In order to incorporate interactions, one has to resort to another kind of symmetry called gauge symmetry, which operates in an “internal” space attached to each point of space-time. While the identity of masses and spins of, say, electrons can be attributed to space-time symmetries, the identities of additional quantum numbers such as charge, isospin and “color” can only be explained in terms of the representations of these gauge groups.

# Marching From Galois Connections to Adjunctions. Part 4.

To make the transition from Galois connections to adjoint functors we make a slight change of notation. The change is only cosmetic but it is very important for our intuition.

Definition of Poset Adjunction. Let (P, ≤P) and (Q, ≤Q) be posets. A pair of functions L ∶ P ⇄ Q ∶ R is called an adjunction if ∀ p ∈ P and q ∈ Q we have

p ≤P R(q) ⇐⇒ L(p) ≤Q q

In this case we write L ⊣ R and call this an adjoint pair of functions. The function L is the left adjoint and R is the right adjoint.

The only difference between Galois connections and poset adjunctions is that we have reversed the partial order on Q. To be precise, we define the opposite poset Qop with the same underlying set Q, such that for all q1 , q2 ∈ Q we have

q1Qop q2 ⇐⇒ q2Q q1

Then an adjunction P ⇄ Q is just the same thing as a Galois connection P ⇄ Qop.

However, this difference is important because it breaks the symmetry. It also prepares us for the notation of an adjunction between categories, where it is more common to use an “asymmetric pair of covariant functors” as opposed to a “symmetric pair of contravariant functors”.

Uniqueness of Adjoints for Posets: Let P and Q be posets and let L ∶ P ⇄ Q ∶ R be an adjunction. Then each of the adjoint functions L ⊣ R uniquely determines the other.

Proof: To prove that R determines L, suppose that L′ ∶ P ⇄ Q ∶ R is another adjunction. Then by definition of adjunction we have for all q ∈ Q that

L(p) ≤Q q ⇐⇒ p ≤P R(q) ⇐⇒ L′(p) ≤Q q

In particular, setting q = L(p) gives

L(p) ≤Q L(p) ⇒ L′(p) ≤Q L′(p)

and setting q = L′(p) gives

L′(p) ≤Q L(p) ⇒ L(p) ≤Q L′(p)

Then by the antisymmetry of Q we have L(p) = L′(p). Since this holds for all p ∈ P we conclude that L = L′, as desired.

RAPL Theorem for Posets. Let L ∶ P ⇄ Q ∶ R be an adjunction of posets. Then for all subsets S ⊆ P and T ⊆ Q we have

L (∨P S) = ∨Q L(S) and R (∧Q T) = ∧P R(T).

In words, this could be said as “left adjoints preserve join” and “right adjoints preserve meet”.

Proof: We just have to observe that sending Q to its opposite Qop switches the definitions of join and meet: Qop = ∧Q and Qop = ∨Q.

It seems worthwhile to emphasize the new terminology with a picture. Suppose that the posets P and Q have top and bottom elements: 1P , 0P ∈ P and 1Q, 0Q ∈ Q. Then a poset adjunction L ∶ P ⇄ Q ∶ R looks like this:

In this case RL ∶ P → P is a closure operator as before, but now LR ∶ Q → Q is called an interior operator. From the case of Galois connections we also know that LRL = L and RLR = R. Since bottom elements are colmits and top elements are limits, the identities L(0P ) = 0Q and R(1Q) = 1P are special cases of the RAPL Theorem.

Just as with Galois connections, adjunctions between the Boolean lattices 2U and 2V are in bijection with relations ∼ ⊆ U × V, but this time we will view the relation as a function f ∼ ∶ U → 2V that sends each to the set f ∼ (u)∶= {v∈V ∶ u∼v}. We can also think off as a “multi-valued function” from U to V.

Adjunctions of Boolean Lattices: Let U,V be sets and consider an arbitrary function f ∶ U → 2V. Then subsets S ∈ 2U and T ∈ 2V we define

L(S) ∶= ∪s∈S f(s) ∈ 2V,

R(T) ∶= {u∈U ∶ f(u) ⊆ T} ∈ 2U

The pair of functions Lf ∶ 2U ⇄ 2V ∶ Rf is an adjunction of Boolean lattices. To see this, note  S ∈ 2U and T ∈ 2V

S ⊆ Rf (T) ⇐⇒ ∀ s∈S, s ∈ R(T)

⇐⇒ ∀ s∈S, f(s) ⊆ T

⇐⇒ ∪s∈S f(s) ⊆ T

⇐⇒ L(S) ⊆ T

Functions : Let f ∶ U → V be any function. We can extend this to a function f ∶ U → 2V by defining f(u) ∶= {f(u)} ∀ u ∈ U. In this case we denote the corresponding left and right adjoint functions by f ∶= Lf ∶ 2U → 2V and f−1 ∶= Rf ∶ 2V → 2U, so that ∀ S ∈ 2U and T ∈ 2V we have

f(S) = {f(s) ∶ s ∈ S}, f−1(T)={u∈U ∶ f(s) ∈ T}

The resulting adjunction f ∶ 2U ⇄ 2V ∶ f−1 is called the image and preimage of the function. It follows from RAPL that image preserves unions and preimage preserves intersections.

But now something surprising happens. We can restrict the preimage f−1 ∶ 2V → 2U to a function f−1 ∶ V → 2U by defining f−1(v) ∶= f−1({v}) for each v ∈ V. Then since f−1 = Lf−1 we obtain another adjunction

f−1 ∶ 2V ⇄ 2U ∶ Rf−1,
where this time f−1 is the left adjoint. The new right adjoint is defined for each S ∈ 2U by

R f−1(S) = {v∈V ∶ f−1(v) ⊆ S}

There seems to be no standard notation for this function, but people call it f! ∶= Rf−1 (the “!” is pronounced “shriek”). In summary, each function f ∶ U → V determines a triple of

adjoints f ⊣ f−1 ⊣ f! where f preserves unions, f! preserves intersections, and f−1 preserves both unions and intersections. Logicians will tell you that the functions f and f! are closely related to the existential (∃) and universal (∀) quantifiers, in the sense that for all S ∈ 2U we have

f∗ (S) = {v∈V ∶ ∃ u ∈ f−1 (v), u ∈ S}, f(S)={v ∈ V ∶ ∀ u ∈ f−1(v), u ∈ S}

Group Homomorphisms: Given a group G we let (L (G), ⊆) denote its poset of subgroups. Since the intersection of subgroups is again a subgroup, we have ∧ = ∩. Then since L (G) has arbitrary meets it also has arbitrary joins. In particular, the join of two subgroups A, B ∈ L (G) is given by

A ∨ B = ⋂ {C ∈ L(G) ∶ A ⊆ C and B ⊆ C},

which is the smallest subgroup containing the union A ∪ B. Thus L (G) is a lattice, but since A ∨ B ≠ A ∪ B (in general) it is not a sublattice of 2G.

Now let φ ∶ G → H be an arbitrary group homomorphism. One can check that the image and preimage φ ∶ 2G ⇄ 2H ∶ φ−1 send subgroups to subgroups, hence they restrict to an adjunction between subgroup lattices:

φ ∶L(G) ⇄ L(H)∶ φ−1.

The function φ! ∶ 2G → 2H does not send subgroups to subgroups, and in general the function φ−1 ∶ L(H) → L(G) does not have a right adjoint. For all subgroups A ∈ L (G) and B ∈ L (H) one can check that

φ−1φ(A)=A ∨ ker φ and φφ−1(B) = B ∧ im φ

Thus the φ−1φ-fixed subgroups of G are precisely those that contain the kernel and the φφ−1-fixed subgroups of H are precisely those contained in the image. Finally, the Fundamental Theorem gives us an order-preserving bijection as in the following picture:

…..

# Marching Along Categories, Groups and Rings. Part 2

A category C consists of the following data:

A collection Obj(C) of objects. We will write “x ∈ C” to mean that “x ∈ Obj(C)

For each ordered pair x, y ∈ C there is a collection HomC (x, y) of arrows. We will write α∶x→y to mean that α ∈ HomC(x,y). Each collection HomC(x,x) has a special element called the identity arrow idx ∶ x → x. We let Arr(C) denote the collection of all arrows in C.

For each ordered triple of objects x, y, z ∈ C there is a function

○ ∶ HomC (x, y) × HomC(y, z) → HomC (x, z), which is called composition of  arrows. If  α ∶ x → y and β ∶ y → z then we denote the composite arrow by β ○ α ∶ x → z.

If each collection of arrows HomC(x,y) is a set then we say that the category C is locally small. If in addition the collection Obj(C) is a set then we say that C is small.

Identitiy: For each arrow α ∶ x → y the following diagram commutes:

Associative: For all arrows α ∶ x → y, β ∶ y → z, γ ∶ z → w, the following diagram commutes:

We say that C′ ⊆ C is a subcategory if Obj(C′) ⊆ Obj(C) and if ∀ x,y ∈ Obj(C′) we have HomC′(x,y) ⊆ HomC(x,y). We say that the subcategory is full if each inclusion of hom sets is an equality.

Let C be a category. A diagram D ⊆ C is a collection of objects in C with some arrows between them. Repetition of objects and arrows is allowed. OR. Let I be any small category, which we think of as an “index category”. Then any functor D ∶ I → C is called a diagram of shape I in C. In either case, we say that the diagram D commutes if for all pairs of objects x,y in D, any two directed paths in D from x to y yield the same arrow under composition.

Identity arrows generalize the reflexive property of posets, and composition of arrows generalizes the transitive property of posets. But whatever happened to the antisymmetric property? Well, it’s the same issue we had before: we should really define equivalence of objects in terms of antisymmetry.

Isomorphism: Let C be a category. We say that two objects x,y ∈ C are isomorphic in C if there exist arrows α ∶ x → y and β ∶ y → x such that the following diagram commutes:

In this case we write x ≅C y, or just x ≅ y if the category is understood.

If γ ∶ y → x is any other arrow satisfying the same diagram as β, then by the axioms of identity and associativity we must have

γ = γ ○ idy = γ ○ (α ○ β) = (γ ○ α) ○ β = idx ○ β = β

This allows us to refer to β as the inverse of the arrow α. We use the notations β = α−1 and

β−1 = α.

A category with one object is called a monoid. A monoid in which each arrow is invertible is called a group. A small category in which each arrow is invertible is called a groupoid.

Subcategories of Set are called concrete categories. Given a concrete category C ⊆ Set we can think of its objects as special kinds of sets and its arrows as special kinds of functions. Some famous examples of conrete categories are:

• Grp = groups & homomorphisms
• Ab = abelian groups & homomorphisms
• Rng = rings & homomorphisms
• CRng = commutative rings & homomorphisms

Note that Ab ⊆ Grp and CRng ⊆ Rng are both full subcategories. In general, the arrows of a concrete category are called morphisms or homomorphisms. This explains our notation of HomC.

Homotopy: The most famous example of a non-concrete category is the fundamental groupoid π1(X) of a topological space X. Here the objects are points and the arrows are homotopy classes of continuous directed paths. The skeleton is the set π0(X) of path components (really a discrete category, i.e., in which the only arrows are the identities). Categories like this are the reason we prefer the name “arrow” instead of “morphism”.

Limit/Colimit: Let D ∶ I → C be a diagram in a category C (thus D is a functor and I is a small “index” category). A cone under D consists of

• an object c ∈ C,

• a collection of arrows αi ∶ x → D(i), one for each index i ∈ I,

such that for each arrow δ ∶ i → j in I we have αj = D(δ) ○ α

In visualizing this:

The cone (c,(αi)i∈I) is called a limit of the diagram D if, for any cone (z,(βi)i∈I) under D, the following picture holds:

[This picture means that there exists a unique arrow υ ∶ z → c such that, for each arrow δ ∶ i → j in I (including the identity arrows), the following diagram commutes:

When δ = idi this diagram just says that βi = αi ○ υ. We do not assume that D itself is commutative. Dually, a cone over D consists of an object c ∈ C and a set of arrows αi ∶ D(i) → c satisfying αi = αj ○ D(δ) for each arrow δ ∶ i → j in I. This cone is called a colimit of the diagram D if, for any cone (z,(βi)i∈I) over D, the following picture holds:

When the (unique) limit or colimit of the diagram D ∶ I → C exists, we denote it by (limI D, (φi)i∈I) or (colimI D, (φi)i∈I), respectively. Sometimes we omit the canonical arrows φi from the notation and refer to the object limID ∈ C as “the limit of D”. However, we should not forget that the arrows are part of the structure, i.e., the limit is really a cone.

Posets: Let P be a poset. We have already seen that the product/coproduct in P (if they exist) are the meet/join, respectively, and that the final/initial objects in P (if they exist) are the top/bottom elements, respectively. The only poset with a zero object is the one element poset.

Sets: The empty set ∅ ∈ Set is an initial object and the one point set ∗ ∈ Set is a final object. Note that two sets are isomorphic in Set precisely when there is a bijection between them, i.e., when they have the same cardinality. Since initial/final objects are unique up to isomorphism, we can identify the initial object with the cardinal number 0 and the final object with the cardinal number 1. There is no zero object in Set.

Products and coproducts exist in Set. The product of S,T ∈ Set consists of the Cartesian product S × T together with the canonical projections πS ∶ S × T → S and πT ∶ S × T → T. The coproduct of S, T ∈ Set consists of the disjoint union S ∐ T together with the canonical injections ιS ∶ S → S ∐ T and ιT ∶ T → S ∐ T. After passing to the skeleton, the product and coproduct of sets become the product and sum of cardinal numbers.

[Note: The “external disjoint union” S ∐ T is a formal concept. The familiar “internal disjoint union” S ⊔ T is only defined when there exists a set U containing both S and T as subsets. Then the union S ∪ T is the join operation in the Boolean lattice 2U ; we call the union “disjoint” when S ∩ T = ∅.]

Groups: The trivial group 1 ∈ Grp is a zero object, and for any groups G, H ∈ Grp the zero homomorphism 1 ∶ G → H sends all elements of G to the identity element 1H ∈ H. The product of groups G, H ∈ Grp is their direct product G × H and the coproduct is their free product G ∗ H, along with the usual canonical morphisms.

Let Ab ⊆ Grp be the full subcategory of abelian groups. The zero object and product are inherited from Grp, but we give them new names: we denote the zero object by 0 ∈ Ab and for any A, B ∈ Ab we denote the zero arrow by 0 ∶ A → B. We denote the Cartesian product by A ⊕ B and we rename it the direct sum. The big difference between Grp and Ab appears when we consider coproducts: it turns out that the product group A ⊕ B is also the coproduct group. We emphasize this fact by calling A ⊕ B the biproduct in Ab. It comes equipped with four canonical homomorphisms πA, πB, ιA, ιB satisfying the usual properties, as well as the following commutative diagram:

This diagram is the ultimate reason for matrix notation. The universal properties of product and coproduct tell us that each endomorphism φ ∶ A ⊕ B → A ⊕ B is uniquely determined by its four components φij ∶= πi ○ φ ○ ιj for i, j ∈ {A,B},so we can represent it as a matrix:

Then the composition of endomorphisms becomes matrix multiplication.

Rings. We let Rng denote the category of rings with unity, together with their homomorphisms. The initial object is the ring of integers Z ∈ Rng and the final object is the zero ring 0 ∈ Rng, i.e., the unique ring in which 0R = 1R. There is no zero object. The product of two rings R, S ∈ Rng is the direct product R × S ∈ Rng with component wise addition and multiplication. Let CRng ⊆ Rng be the full subcategory of commutative rings. The initial/final objects and product in CRng are inherited from Rng. The difference between Rng and CRng again appears when considering coproducts. The coproduct of R,S ∈ CRng is denoted by R ⊗Z S and is called the tensor product over Z…..