Grothendieckian Construction of K-Theory with a Bundle that is Topologically Trivial and Class that is Torsion.


All relativistic quantum theories contain “antiparticles,” and allow the process of particle-antiparticle annihilation. This inspires a physical version of the Grothendieck construction of K-theory. Physics uses topological K-theory of manifolds, whose motivation is to organize vector bundles over a space into an algebraic invariant, that turns out to be useful. Algebraic K-theory started from Ki defined for i, with relations to classical constructions in algebra and number theory, followed by Quillen’s homotopy-theoretic definition ∀ i. The connections to algebra and number theory often persist for larger values of i, but in ways that are subtle and conjectural, such as special values of zeta- and L-functions.

One could also use the conserved charges of a configuration which can be measured at asymptotic infinity. By definition, these are left invariant by any physical process. Furthermore, they satisfy quantization conditions, of which the prototype is the Dirac condition on allowed electric and magnetic charges in Maxwell theory.

There is an elementary construction which, given a physical theory T, produces an abelian group of conserved charges K(T). Rather than considering the microscopic dynamics of the theory, all that is needed to be known is a set S of “particles” described by T, and a set of “bound state formation/decay processes” by which the particles combine or split to form other particles. These are called “binding processes.” Two sets of particles are “physically equivalent” if some sequence of binding processes convert the one to the other. We then define the group K(T) as the abelian group ZS of formal linear combinations of particles, quotiented by this equivalence relation.

Suppose T contains the particles S = {A,B,C}.

If these are completely stable, we could clearly define three integral conserved charges, their individual numbers, so K(T) ≅ Z3.

Introducing a binding process

A + B ↔ C —– (1)

with the bidirectional arrow to remind us that the process can go in either direction. Clearly K(T) ≅ Z2 in this case.

One might criticize this proposal on the grounds that we have assumed that configurations with a negative number of particles can exist. However, in all physical theories which satisfy the constraints of special relativity, charged particles in physical theories come with “antiparticles,” with the same mass but opposite charge. A particle and antiparticle can annihilate (combine) into a set of zero charge particles. While first discovered as a prediction of the Dirac equation, this follows from general axioms of quantum field theory, which also hold in string theory.

Thus, there are binding processes

B + B̄ ↔ Z1 + Z2 + · · · .

where B̄ is the antiparticle to a particle B, and Zi are zero charge particles, which must appear by energy conservation. To define the K-theory, we identify any such set of zero charge particles with the identity, so that

B + B̄ ↔ 0

Thus the antiparticles provide the negative elements of K(T).

Granting the existence of antiparticles, this construction of K-theory can be more simply rephrased as the Grothendieck construction. We can define K(T) as the group of pairs (E, F) ∈ (ZS, ZS), subject to the relations (E, F) ≅ (E+B, F +B) ≅ (E+L, F +R) ≅ (E+R, F +L), where (L, R) are the left and right hand side of a binding process (1).

Thinking of these as particles, each brane B must have an antibrane, which we denote by B̄. If B wraps a submanifold L, one expects that B̄ is a brane which wraps a submanifold L of opposite orientation. A potential problem is that it is not a priori obvious that the orientation of L actually matters physically, especially in degenerate cases such as L a point.

Now, let us take X as a Calabi-Yau threefold for definiteness. A physical A-brane, which are branes of the A-model topological string and thereby a TQFT shadow of the D-branes of the superstring, is specified by a pair (L, E) of a special Lagrangian submanifold L with a flat bundle E. The obvious question could be: When are (L1, E1) and (L2, E2) related by a binding process? A simple heuristic answer to this question is given by the Feynman path integral. Two configurations are connected, if they are connected by a continuous path through the configuration space; any such path (or a small deformation of it) will appear in the functional integral with some non-zero weight. Thus, the question is essentially topological. Ignoring the flat bundles for a moment, this tells us that the K-theory group for A-branes is H3(Y, Z), and the class of a brane is simply (rank E)·[L] ∈ H3(Y, Z). This is also clear if the moduli space of flat connections on L is connected.

But suppose it is not, say π1(L) is torsion. In this case, we need deeper physical arguments to decide whether the K-theory of these D-branes is H3(Y, Z), or some larger group. But a natural conjecture is that it will be K1(Y), which classifies bundles on odd-dimensional submanifolds. Two branes which differ only in the choice of flat connection are in fact connected in string theory, consistent with the K-group being H3(Y, Z). For Y a simply connected Calabi-Yau threefold, K1(Y) ≅ H3(Y, Z), so the general conjecture is borne out in this case

There is a natural bilinear form on H3(Y, Z) given by the oriented intersection number

I(L1, L2) = #([L1] ∩ [L2]) —– (2)

It has symmetry (−1)n. In particular, it is symplectic for n = 3. Furthermore, by Poincaré duality, it is unimodular, at least in our topological definition of K-theory.

D-branes, which are extended objects defined by mixed Dirichlet-Neumann boundary conditions in string theory, break half of the supersymmetries of the type II superstring and carry a complete set of electric and magnetic Ramond-Ramond charges. The product of the electric and magnetic charges is a single Dirac unit, and that the quantum of charge takes the value required by string duality. Saying that a D-brane has RR-charge means that it is a source for an “RR potential,” a generalized (p + 1)-form gauge potential in ten-dimensional space-time, which can be verified from its world-volume action that contains a minimal coupling term,

∫C(p + 1) —–(3)

where C(p + 1) denotes the gauge potential, and the integral is taken over the (p+1)-dimensional world-volume of the brane. For p = 0, C(1) is a one-form or “vector” potential (as in Maxwell theory), and thus the D0-brane is an electrically charged particle with respect to this 10d Maxwell theory. Upon further compactification, by which, the ten dimensions are R4 × X, and a Dp-brane which wraps a p-dimensional cycle L; in other words its world-volume is R × L where R is a time-like world-line in R4. Using the Poincaré dual class ωL ∈ H2n−p(X, R) to L in X, to rewrite (3) as an integral

R × X C(p + 1) ∧ ωL —– (4)

We can then do the integral over X to turn this into the integral of a one-form over a world-line in R4, which is the right form for the minimal electric coupling of a particle in four dimensions. Thus, such a wrapped brane carries a particular electric charge which can be detected at asymptotic infinity. Summarizing the RR-charge more formally,

LC = ∫XC ∧ ωL —– (5)

where C ∈ H∗(X, R). In other words, it is a class in Hp(X, R).

In particular, an A-brane (for n = 3) carries a conserved charge in H3(X, R). Of course, this is weaker than [L] ∈ H3(X, Z). To see this physically, we would need to see that some of these “electric” charges are actually “magnetic” charges, and study the Dirac-Schwinger-Zwanziger quantization condition between these charges. This amounts to showing that the angular momentum J of the electromagnetic field satisfies the quantization condition J = ħn/2 for n ∈ Z. Using an expression from electromagnetism, J⃗ = E⃗ × B⃗ , this is precisely the condition that (2) must take an integer value. Thus the physical and mathematical consistency conditions agree. Similar considerations apply for coisotropic A-branes. If X is a genuine Calabi-Yau 3-fold (i.e., with strict SU(3) holonomy), then a coisotropic A-brane which is not a special Lagrangian must be five-dimensional, and the corresponding submanifold L is rationally homologically trivial, since H5(X, Q) = 0. Thus, if the bundle E is topologically trivial, the homology class of L and thus its K-theory class is torsion.

If X is a torus, or a K3 surface, the situation is more complicated. In that case, even rationally the charge of a coisotropic A-brane need not lie in the middle-dimensional cohomology of X. Instead, it takes its value in a certain subspace of ⊕p Hp(X, Q), where the summation is over even or odd p depending on whether the complex dimension of X is even or odd. At the semiclassical level, the subspace is determined by the condition

(L − Λ)α = 0, α ∈ ⊕p Hp(X, Q)

where L and Λ are generators of the Lefschetz SL(2, C) action, i.e., L is the cup product with the cohomology class of the Kähler form, and Λ is its dual.

The Closed String Cochain Complex C is the String Theory Substitute for the de Rham Complex of Space-Time. Note Quote.


In closed string theory the central object is the vector space C = CS1 of states of a single parameterized string. This has an integer grading by the “ghost number”, and an operator Q : C → C called the “BRST operator” which raises the ghost number by 1 and satisfies Q2 = 0. In other words, C is a cochain complex. If we think of the string as moving in a space-time M then C is roughly the space of differential forms defined along the orbits of the action of the reparametrization group Diff+(S1) on the free loop space LM (more precisely, square-integrable forms of semi-infinite degree). Similarly, the space C of a topologically-twisted N = 2 supersymmetric theory, is a cochain complex which models the space of semi-infinite differential forms on the loop space of a Kähler manifold – in this case, all square-integrable differential forms, not just those along the orbits of Diff+(S1). In both kinds of example, a cobordism Σ from p circles to q circles gives an operator UΣ,μ : C⊗p → C⊗q which depends on a conformal structure μ on Σ. This operator is a cochain map, but its crucial feature is that changing the conformal structure μ on Σ changes the operator UΣ,μ only by a cochain homotopy. The cohomology H(C) = ker(Q)/im(Q) – the “space of physical states” in conventional string theory – is therefore the state space of a topological field theory.

A good way to describe how the operator UΣ,μ varies with μ is as follows:

If MΣ is the moduli space of conformal structures on the cobordism Σ, modulo diffeomorphisms of Σ which are the identity on the boundary circles, then we have a cochain map

UΣ : C⊗p → Ω(MΣ, C⊗q)

where the right-hand side is the de Rham complex of forms on MΣ with values in C⊗q. The operator UΣ,μ is obtained from UΣ by restricting from MΣ to {μ}. The composition property when two cobordisms Σ1 and Σ2 are concatenated is that the diagram


commutes, where the lower horizontal arrow is induced by the map MΣ1 × MΣ2 → MΣ2 ◦ Σ1 which expresses concatenation of the conformal structures.

For each pair a, b of boundary conditions we shall still have a vector space – indeed a cochain complex – Oab, but it is no longer the space of morphisms from b to a in a category. Rather, what we have is an A-category. Briefly, this means that instead of a composition law Oab × Obc → Oac we have a family of ways of composing, parametrized by the contractible space of conformal structures on the surface of the figure:


In particular, any two choices of a composition law from the family are cochain homotopic. Composition is associative in the sense that we have a contractible family of triple compositions Oab × Obc × Ocd → Oad, which contains all the maps obtained by choosing a binary composition law from the given family and bracketing the triple in either of the two possible ways.

This is not the usual way of defining an A-structure. According to Stasheff’s original definition, an A-structure on a space X consists of a sequence of choices: first, a composition law m2 : X × X → X; then, a choice of a map

m3 : [0, 1] × X × X × X → X which is a homotopy between

(x, y, z) ↦ m2(m2(x, y), z) and (x, y, z) ↦ m2(x, m2(y, z)); then, a choice of a map

m4 : S4 × X4 → X,

where S4 is a convex plane polygon whose vertices are indexed by the five ways of bracketing a 4-fold product, and m4|((∂S4) × X4) is determined by m3; and so on. There is an analogous definition – applying to cochain complexes rather than spaces.

Apart from the composition law, the essential algebraic properties are the non-degenerate inner product, and the commutativity of the closed algebra C. Concerning the latter, when we pass to cochain theories the multiplication in C will of course be commutative up to cochain homotopy, but, the moduli space MΣ of closed string multiplications i.e., the moduli space of conformal structures on a pair of pants Σ, modulo diffeomorphisms of Σ which are the identity on the boundary circles, is not contractible: it has the homotopy type of the space of ways of embedding two copies of the standard disc D2 disjointly in the interior of D2 – this space of embeddings is of course a subspace of MΣ. In particular, it contains a natural circle of multiplications in which one of the embedded discs moves like a planet around the other, and there are two different natural homotopies between the multiplication and the reversed multiplication. This might be a clue to an important difference between stringy and classical space-times. The closed string cochain complex C is the string theory substitute for the de Rham complex of space-time, an algebra whose multiplication is associative and (graded)commutative on the nose. Over the rationals or the real or complex numbers, such cochain algebras model the category of topological spaces up to homotopy, in the sense that to each such algebra C, we can associate a space XC and a homomorphism of cochain algebras from C to the de Rham complex of XC which is a cochain homotopy equivalence. If we do not want to ignore torsion in the homology of spaces we can no longer encode the homotopy type in a strictly commutative cochain algebra. Instead, we must replace commutative algebras with so-called E-algebras, i.e., roughly, cochain complexes C over the integers equipped with a multiplication which is associative and commutative up to given arbitrarily high-order homotopies. An arbitrary space X has an E-algebra CX of cochains, and conversely one can associate a space XC to each E-algebra C. Thus we have a pair of adjoint functors, just as in rational homotopy theory. The cochain algebras of closed string theory have less higher commutativity than do E-algebras, and this may be an indication that we are dealing with non-commutative spaces that fits in well with the interpretation of the B-field of a string background as corresponding to a bundle of matrix algebras on space-time. At the same time, the non-degenerate inner product on C – corresponding to Poincaré duality – seems to show we are concerned with manifolds, rather than more singular spaces.

Let us consider the category K of cochain complexes of finitely generated free abelian groups and cochain homotopy classes of cochain maps. This is called the derived category of the category of finitely generated abelian groups. Passing to cohomology gives us a functor from K to the category of Z-graded finitely generated abelian groups. In fact the subcategory K0 of K consisting of complexes whose cohomology vanishes except in degree 0 is actually equivalent to the category of finitely generated abelian groups. But the category K inherits from the category of finitely generated free abelian groups a duality functor with properties as ideal as one could wish: each object is isomorphic to its double dual, and dualizing preserves exact sequences. (The dual C of a complex C is defined by (C)i = Hom(C−i, Z).) There is no such nice duality in the category of finitely generated abelian groups. Indeed, the subcategory K0 is not closed under duality, for the dual of the complex CA corresponding to a group A has in general two non-vanishing cohomology groups: Hom(A,Z) in degree 0, and in degree +1 the finite group Ext1(A,Z) Pontryagin-dual to the torsion subgroup of A. This follows from the exact sequence:

0 → Hom(A, Z) → Hom(FA, Z) → Hom(RA, Z) → Ext1(A, Z) → 0

derived from an exact sequence

0 → RA → FA → A → 0

The category K also has a tensor product with better properties than the tensor product of abelian groups, and, better still, there is a canonical cochain functor from (locally well-behaved) compact spaces to K which takes Cartesian products to tensor products.

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


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.




If e0 ∈ R1+1 is a future-directed timelike unit vector, and if e1 is the unique spacelike unit vector with e0e1 = 0 that “points to the right,” then coordinates x0 and x1 on R1+1 are defined by x0(q) := qe0 and x1(q) := qe1. The partial differential operator

x : = ∂2x0 − ∂2x1

does not depend on the choice of e0.

The Fourier transform of the Klein-Gordon equation

(□ + m2)u = 0 —– (1)

where m > 0 is a given mass, is

(−p2 + m2)û(p) = 0 —– (2)

As a consequence, the support of û has to be a subset of the hyperbola Hm ⊂ R1+1 specified by the condition p2 = m2. One connected component of Hm consists of positive-energy vectors only; it is called the upper mass shell Hm+. The elements of Hm+ are the 4-momenta of classical relativistic point particles.

Denote by L1 the restricted Lorentz group, i.e., the connected component of the Lorentz group containing its unit element. In 1 + 1 dimensions, L1 coincides with the one-parameter Abelian group B(χ), χ ∈ R, of boosts. Hm+ is an orbit of L1 without fixed points. So if one chooses any point p′ ∈ Hm+, then there is, for each p ∈ Hm+, a unique χ(p) ∈ R with p = B(χ(p))p′. By construction, χ(B(ξ)p) = χ(p) + ξ, so the measure dχ on Hm+ is invariant under boosts and does note depend on the choice of p′.

For each p ∈ Hm+, the plane wave q ↦ e±ipq on R1+1 is a classical solution of the Klein-Gordon equation. The Klein-Gordon equation is linear, so if a+ and a are, say, integrable functions on Hm+, then

F(q) := ∫Hm+ (a+(p)e-ipq + a(p)eipq dχ(p) —– (3)

is a solution of the Klein-Gordon equation as well. If the functions a± are not integrable, the field F may still be well defined as a distribution. As an example, put a± ≡ (2π)−1, then

F(q) = (2π)−1 Hm+ (e-ipq + eipq) dχ(p) = π−1Hm+ cos(pq) dχ(p) =: Φ(q) —– (4)

and for a± ≡ ±(2πi)−1, F equals

F(q) = (2πi)−1Hm+ (e-ipq – eipq) dχ(p) = π−1Hm+ sin(pq) dχ(p) =: ∆(q) —– (5)

Quantum fields are obtained by “plugging” classical field equations and their solutions into the well-known second quantization procedure. This procedure replaces the complex (or, more generally speaking, finite-dimensional vector) field values by linear operators in an infinite-dimensional Hilbert space, namely, a Fock space. The Hilbert space of the hermitian scalar field is constructed from wave functions that are considered as the wave functions of one or several particles of mass m. The single-particle wave functions are the elements of the Hilbert space H1 := L2(Hm+, dχ). Put the vacuum (zero-particle) space H0 equal to C, define the vacuum vector Ω := 1 ∈ H0, and define the N-particle space HN as the Hilbert space of symmetric wave functions in L2((Hm+)N, dNχ), i.e., all wave functions ψ with

ψ(pπ(1) ···pπ(N)) = ψ(p1 ···pN)

∀ permutations π ∈ SN. The bosonic Fock space H is defined by

H := ⊕N∈N HN.

The subspace

D := ∪M∈N ⊕0≤M≤N HN is called a finite particle space.

The definition of the N-particle wave functions as symmetric functions endows the field with a Bose–Einstein statistics. To each wave function φ ∈ H1, assign a creation operator a+(φ) by

a+(φ)ψ := CNφ ⊗s ψ, ψ ∈ D,

where ⊗s denotes the symmetrized tensor product and where CN is a constant.

(a+(φ)ψ)(p1 ···pN) = CN/N ∑v φ(pν)ψ(pπ(1) ···p̂ν ···pπ(N)) —– (6)

where the hat symbol indicates omission of the argument. This defines a+(φ) as a linear operator on the finite-particle space D.

The adjoint operator a(φ) := a+(φ) is called an annihilation operator; it assigns to each ψ ∈ HN, N ≥ 1, the wave function a(φ)ψ ∈ HN−1 defined by

(a(φ)ψ)(p1 ···pN) := CN ∫Hm+ φ(p)ψ(p1 ···pN−1, p) dχ(p)

together with a(φ)Ω := 0, this suffices to specify a(φ) on D. Annihilation operators can also be defined for sharp momenta. Namely, one can define to each p ∈ Hm+ the annihilation operator a(p) assigning to

each ψ ∈ HN, N ≥ 1, the wave function a(p)ψ ∈ HN−1 given by

(a(p)ψ)(p1 ···pN−1) := Cψ(p, p1 ···pN−1), ψ ∈ HN,

and assigning 0 ∈ H to Ω. a(p) is, like a(φ), well defined on the finite-particle space D as an operator, but its hermitian adjoint is ill-defined as an operator, since the symmetric tensor product of a wave function by a delta function is no wave function.

Given any single-particle wave functions ψ, φ ∈ H1, the commutators [a(ψ), a(φ)] and [a+(ψ), a+(φ)] vanish by construction. It is customary to choose the constants CN in such a fashion that creation and annihilation operators exhibit the commutation relation

[a(φ), a+(ψ)] = ⟨φ, ψ⟩ —– (7)

which requires CN = N. With this choice, all creation and annihilation operators are unbounded, i.e., they are not continuous.

When defining the hermitian scalar field as an operator valued distribution, it must be taken into account that an annihilation operator a(φ) depends on its argument φ in an antilinear fashion. The dependence is, however, R-linear, and one can define the scalar field as a C-linear distribution in two steps.

For each real-valued test function φ on R1+1, define

Φ(φ) := a(φˆ|Hm+) + a+(φˆ|Hm+)

then one can define for an arbitrary complex-valued φ

Φ(φ) := Φ(Re(φ)) + iΦ(Im(φ))

Referring to (4), Φ is called the hermitian scalar field of mass m.

Thereafter, one could see

[Φ(q), Φ(q′)] = i∆(q − q′) —– (8)

Referring to (5), which is to be read as an equation of distributions. The distribution ∆ vanishes outside the light cone, i.e., ∆(q) = 0 if q2 < 0. Namely, the integrand in (5) is odd with respect to some p′ ∈ Hm+ if q is spacelike. Note that pq > 0 for all p ∈ Hm+ if q ∈ V+. The consequence of this is called microcausality: field operators located in spacelike separated regions commute (for the hermitian scalar field).

Homotopically Truncated Spaces.

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

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


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

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

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

K(πn+1X, n+1) → Pn+1(X) → Pn(X) → Yn

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

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

= 0, otherwise.

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

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

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

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

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


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

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

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

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

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

Ynkn X≤nin+1 X≤n+1 → M(Hn+1X, n+1)

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

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

= 0, otherwise

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

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

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

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


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


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

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

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


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

j[f≤2] = i[f].


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

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

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


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


It follows that

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

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

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

Abelian Categories, or Injective Resolutions are Diagrammatic. Note Quote.


Jean-Pierre Serre gave a more thoroughly cohomological turn to the conjectures than Weil had. Grothendieck says

Anyway Serre explained the Weil conjectures to me in cohomological terms around 1955 – and it was only in these terms that they could possibly ‘hook’ me …I am not sure anyone but Serre and I, not even Weil if that is possible, was deeply convinced such [a cohomology] must exist.

Specifically Serre approached the problem through sheaves, a new method in topology that he and others were exploring. Grothendieck would later describe each sheaf on a space T as a “meter stick” measuring T. The cohomology of a given sheaf gives a very coarse summary of the information in it – and in the best case it highlights just the information you want. Certain sheaves on T produced the Betti numbers. If you could put such “meter sticks” on Weil’s arithmetic spaces, and prove standard topological theorems in this form, the conjectures would follow.

By the nuts and bolts definition, a sheaf F on a topological space T is an assignment of Abelian groups to open subsets of T, plus group homomorphisms among them, all meeting a certain covering condition. Precisely these nuts and bolts were unavailable for the Weil conjectures because the arithmetic spaces had no useful topology in the then-existing sense.

At the École Normale Supérieure, Henri Cartan’s seminar spent 1948-49 and 1950-51 focussing on sheaf cohomology. As one motive, there was already de Rham cohomology on differentiable manifolds, which not only described their topology but also described differential analysis on manifolds. And during the time of the seminar Cartan saw how to modify sheaf cohomology as a tool in complex analysis. Given a complex analytic variety V Cartan could define sheaves that reflected not only the topology of V but also complex analysis on V.

These were promising for the Weil conjectures since Weil cohomology would need sheaves reflecting algebra on those spaces. But understand, this differential analysis and complex analysis used sheaves and cohomology in the usual topological sense. Their innovation was to find particular new sheaves which capture analytic or algebraic information that a pure topologist might not focus on.

The greater challenge to the Séminaire Cartan was, that along with the cohomology of topological spaces, the seminar looked at the cohomology of groups. Here sheaves are replaced by G-modules. This was formally quite different from topology yet it had grown from topology and was tightly tied to it. Indeed Eilenberg and Mac Lane created category theory in large part to explain both kinds of cohomology by clarifying the links between them. The seminar aimed to find what was common to the two kinds of cohomology and they found it in a pattern of functors.

The cohomology of a topological space X assigns to each sheaf F on X a series of Abelian groups HnF and to each sheaf map f : F → F′ a series of group homomorphisms Hnf : HnF → HnF′. The definition requires that each Hn is a functor, from sheaves on X to Abelian groups. A crucial property of these functors is:

HnF = 0 for n > 0

for any fine sheaf F where a sheaf is fine if it meets a certain condition borrowed from differential geometry by way of Cartan’s complex analytic geometry.

The cohomology of a group G assigns to each G-module M a series of Abelian groups HnM and to each homomorphism f : M →M′ a series of homomorphisms HnF : HnM → HnM′. Each Hn is a functor, from G-modules to Abelian groups. These functors have the same properties as topological cohomology except that:

HnM = 0 for n > 0

for any injective module M. A G-module I is injective if: For every G-module inclusion N M and homomorphism f : N → I there is at least one g : M → I making this commute


Cartan could treat the cohomology of several different algebraic structures: groups, Lie groups, associative algebras. These all rest on injective resolutions. But, he could not include topological spaces, the source of the whole, and still one of the main motives for pursuing the other cohomologies. Topological cohomology rested on the completely different apparatus of fine resolutions. As to the search for a Weil cohomology, this left two questions: What would Weil cohomology use in place of topological sheaves or G-modules? And what resolutions would give their cohomology? Specifically, Cartan & Eilenberg defines group cohomology (like several other constructions) as a derived functor, which in turn is defined using injective resolutions. So the cohomology of a topological space was not a derived functor in their technical sense. But a looser sense was apparently current.

Grothendieck wrote to Serre:

I have realized that by formulating the theory of derived functors for categories more general than modules, one gets the cohomology of spaces at the same time at small cost. The existence follows from a general criterion, and fine sheaves will play the role of injective modules. One gets the fundamental spectral sequences as special cases of delectable and useful general spectral sequences. But I am not yet sure if it all works as well for non-separated spaces and I recall your doubts on the existence of an exact sequence in cohomology for dimensions ≥ 2. Besides this is probably all more or less explicit in Cartan-Eilenberg’s book which I have not yet had the pleasure to see.

Here he lays out the whole paper, commonly cited as Tôhoku for the journal that published it. There are several issues. For one thing, fine resolutions do not work for all topological spaces but only for the paracompact – that is, Hausdorff spaces where every open cover has a locally finite refinement. The Séminaire Cartan called these separated spaces. The limitation was no problem for differential geometry. All differential manifolds are paracompact. Nor was it a problem for most of analysis. But it was discouraging from the viewpoint of the Weil conjectures since non-trivial algebraic varieties are never Hausdorff.

Serre replied using the same loose sense of derived functor:

The fact that sheaf cohomology is a special case of derived func- tors (at least for the paracompact case) is not in Cartan-Sammy. Cartan was aware of it and told [David] Buchsbaum to work on it, but he seems not to have done it. The interest of it would be to show just which properties of fine sheaves we need to use; and so one might be able to figure out whether or not there are enough fine sheaves in the non-separated case (I think the answer is no but I am not at all sure!).

So Grothendieck began rewriting Cartan-Eilenberg before he had seen it. Among other things he preempted the question of resolutions for Weil cohomology. Before anyone knew what “sheaves” it would use, Grothendieck knew it would use injective resolutions. He did this by asking not what sheaves “are” but how they relate to one another. As he later put it, he set out to:

consider the set13 of all sheaves on a given topological space or, if you like, the prodigious arsenal of all the “meter sticks” that measure it. We consider this “set” or “arsenal” as equipped with its most evident structure, the way it appears so to speak “right in front of your nose”; that is what we call the structure of a “category”…From here on, this kind of “measuring superstructure” called the “category of sheaves” will be taken as “incarnating” what is most essential to that space.

The Séminaire Cartan had shown this structure in front of your nose suffices for much of cohomology. Definitions and proofs can be given in terms of commutative diagrams and exact sequences without asking, most of the time, what these are diagrams of.  Grothendieck went farther than any other, insisting that the “formal analogy” between sheaf cohomology and group cohomology should become “a common framework including these theories and others”. To start with, injectives have a nice categorical sense: An object I in any category is injective if, for every monic N → M and arrow f : N → I there is at least one g : M → I such that


Fine sheaves are not so diagrammatic.

Grothendieck saw that Reinhold Baer’s original proof that modules have injective resolutions was largely diagrammatic itself. So Grothendieck gave diagrammatic axioms for the basic properties used in cohomology, and called any category that satisfies them an Abelian category. He gave further diagrammatic axioms tailored to Baer’s proof: Every category satisfying these axioms has injective resolutions. Such a category is called an AB5 category, and sometimes around the 1960s a Grothendieck category though that term has been used in several senses.

So sheaves on any topological space have injective resolutions and thus have derived functor cohomology in the strict sense. For paracompact spaces this agrees with cohomology from fine, flabby, or soft resolutions. So you can still use those, if you want them, and you will. But Grothendieck treats paracompactness as a “restrictive condition”, well removed from the basic theory, and he specifically mentions the Weil conjectures.

Beyond that, Grothendieck’s approach works for topology the same way it does for all cohomology. And, much further, the axioms apply to many categories other than categories of sheaves on topological spaces or categories of modules. They go far beyond topological and group cohomology, in principle, though in fact there were few if any known examples outside that framework when they were given.

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…..