# Philosophical Identity of Derived Correspondences Between Smooth Varieties. Let there be a morphism f : X → Y between varieties. Then all the information about f is encoded in the graph Γf ⊂ X × Y of f, which (as a set) is defined as

Γf = {(x, f(x)) : x ∈ X} ⊂ X × Y —– (1)

Now consider the natural projections pX, pY from X × Y to the factors X, Y. Restricted to the subvariety Γf, pX is an isomorphism (since f is a morphism). The fibres of pY restricted to Γf are just the fibres of f; so for example f is proper iff pY | Γf is.

If H(−) is any reasonable covariant homology theory (say singular homology in the complex topology for X, Y compact), then we have a natural push forward map

f : H(X) → H(Y)

This map can be expressed in terms of the graph Γf and the projection maps as

f(α) = pY∗ (pX(α) ∪ [Γf]) —– (2)

where [Γf] ∈ H (X × Y) is the fundamental class of the subvariety [Γf]. Generalizing this construction gives us the notion of a “multi-valued function” or correspondence from X to Y, simply defined to be a general subvariety Γ ⊂ X × Y, replacing the assumption that pX be an isomorphism with some weaker assumption, such as pXf, pY | Γf finite or proper. The right hand side of (2) defines a generalized pushforward map

Γ : H(X) → H(Y)

A subvariety Γ ⊂ X × Y can be represented by its structure sheaf OΓ on X × Y. Associated to the projection maps pX, pY, we also have pullback and pushforward operations on sheaves. The cup product on homology turns out to have an analogue too, namely tensor product. So, appropriately interpreted, (2) makes sense as an operation from the derived category of X to that of Y.

A derived correspondence between a pair of smooth varieties X, Y is an object F ∈ Db(X × Y) with support which is proper over both factors. A derived correspondence defines a functor ΦF by

ΦF : Db(X) → Db(Y)
(−) ↦ RpY∗(LpX(−) ⊗L F)

where (−) could refer to both objects and morphisms in Db(X). F is sometimes called the kernel of the functor ΦF.

The functor ΦF is exact, as it is defined as a composite of exact functors. Since the projection pX is flat, the derived pullback LpX is the same as ordinary pullback pX. Given derived correspondences E ∈ Db(X × Y), F ∈ Db(Y × Z), we obtain functors Φ: Db(X) → Db(Y), Φ: Db(Y) → Db(Z), which can then be composed to get a functor

ΦF ◦ Φ: Db(X) → Db(Z)

which is a two-sided identity with respect to composition of kernels.

# Closed String Algebra as a Graded-Commutative Algebra C: Cochain Complex Differentials: Part 2, Note Quote. The most general target category we can consider is a symmetric tensor category: clearly we need a tensor product, and the axiom HY1⊔Y2 ≅ HY1 ⊗ HY2 only makes sense if there is an involutory canonical isomorphism HY1 ⊗ HY2 ≅ HY2 ⊗ HY1 .

A very common choice in physics is the category of super vector spaces, i.e., vector spaces V with a mod 2 grading V = V0 ⊕ V1, where the canonical isomorphism V ⊗ W ≅ W ⊗ V is v ⊗ w ↦ (−1)deg v deg ww ⊗ v. One can also consider the category of Z-graded vector spaces, with the same sign convention for the tensor product.

In either case the closed string algebra is a graded-commutative algebra C with a trace θ : C → C. In principle the trace should have degree zero, but in fact the commonly encountered theories have a grading anomaly which makes the trace have degree −n for some integer n.

We define topological-spinc theories, which model 2d theories with N = 2 supersymmetry, by replacing “manifolds” with “manifolds with spinc structure”.

A spinc structure on a surface with a conformal structure is a pair of holomorphic line bundles L1, L2 with an isomorphism L1 ⊗ L2 ≅ TΣ of holomorphic line bundles. A spin structure is the particular case when L1 = L2. On a 1-manifold S a spinc structure means a spinc structure on a ribbon neighbourhood of S in a surface with conformal structure. An N = 2 superconformal theory assigns a vector space HS;L1,L2 to each 1-manifold S with spinc structure, and an operator

US0;L1,L2: HS0;L1,L2 → HS1;L1,L2

to each spinc-cobordism from S0 to S1. To explain the rest of the structure we need to define the N = 2 Lie superalgebra associated to a spin1-manifold (S;L1,L2). Let G = Aut(L1) denote the group of bundle isomorphisms L1 → L1 which cover diffeomorphisms of S. (We can identify this group with Aut(L2).) It has a homomorphism onto the group Diff+(S) of orientation-preserving diffeomorphisms of S, and the kernel is the group of fibrewise automorphisms of L1, which can be identified with the group of smooth maps from S to C×. The Lie algebra Lie(G) is therefore an extension of the Lie algebra Vect(S) of Diff+(S) by the commutative Lie algebra Ω0(S) of smooth real-valued functions on S. Let Λ0S;L1,L2 denote the complex Lie algebra obtained from Lie(G) by complexifying Vect(S). This is the even part of a Lie super algebra whose odd part is Λ1S;L1,L2 = Γ(L1) ⊕ Γ(L2). The bracket Λ1 ⊗ Λ1 → Λ0 is completely determined by the property that elements of Γ(L1) and of Γ(L2) anticommute among themselves, while the composite

Γ(L1) ⊗ Γ(L2) → Λ0 → VectC(S)

takes (λ12) to λ1λ2 ∈ Γ(TS).

In an N = 2 theory we require the superalgebra Λ(S;L1,L2) to act on the vector space HS;L1,L2, compatibly with the action of the group G, and with a similar intertwining property with the cobordism operators to that of the N = 1 case. For an N = 2 theory the state space always has an action of the circle group coming from its embedding in G as the group of fibrewise multiplications on L1 and L2. Equivalently, the state space is always Z-graded.

An N = 2 theory always gives rise to two ordinary conformal field theories by equipping a surface Σ with the spinc structures (C,TΣ) and (TΣ,C). These are called the “A-model” and the “B-model” associated to the N = 2 theory. In each case the state spaces are cochain complexes in which the differential is the action of the constant section of the trivial component of the spinc-structure.

# Superconformal Spin/Field Theories: When Vector Spaces have same Dimensions: Part 1, Note Quote. A spin structure on a surface means a double covering of its space of non-zero tangent vectors which is non-trivial on each individual tangent space. On an oriented 1-dimensional manifold S it means a double covering of the space of positively-oriented tangent vectors. For purposes of gluing, this is the same thing as a spin structure on a ribbon neighbourhood of S in an orientable surface. Each spin structure has an automorphism which interchanges its sheets, and this will induce an involution T on any vector space which is naturally associated to a 1-manifold with spin structure, giving the vector space a mod 2 grading by its ±1-eigenspaces. A topological-spin theory is a functor from the cobordism category of manifolds with spin structures to the category of super vector spaces with its graded tensor structure. The functor is required to take disjoint unions to super tensor products, and additionally it is required that the automorphism of the spin structure of a 1-manifold induces the grading automorphism T = (−1)degree of the super vector space. This choice of the supersymmetry of the tensor product rather than the naive symmetry which ignores the grading is forced by the geometry of spin structures if the possibility of a semisimple category of boundary conditions is to be allowed. There are two non-isomorphic circles with spin structure: S1ns, with the Möbius or “Neveu-Schwarz” structure, and S1r, with the trivial or “Ramond” structure. A topological-spin theory gives us state spaces Cns and Cr, corresponding respectively to S1ns and S1r.

There are four cobordisms with spin structures which cover the standard annulus. The double covering can be identified with its incoming end times the interval [0,1], but then one has a binary choice when one identifies the outgoing end of the double covering over the annulus with the chosen structure on the outgoing boundary circle. In other words, alongside the cylinders A+ns,r = S1ns,r × [0,1] which induce the identity maps of Cns,r there are also cylinders Ans,r which connect S1ns,r to itself while interchanging the sheets. These cylinders Ans,r induce the grading automorphism on the state spaces. But because Ans ≅ A+ns by an isomorphism which is the identity on the boundary circles – the Dehn twist which “rotates one end of the cylinder by 2π” – the grading on Cns must be purely even. The space Cr can have both even and odd components. The situation is a little more complicated for “U-shaped” cobordisms, i.e., cylinders with two incoming or two outgoing boundary circles. If the boundaries are S1ns there is only one possibility, but if the boundaries are S1r there are two, corresponding to A±r. The complication is that there seems no special reason to prefer either of the spin structures as “positive”. We shall simply choose one – let us call it P – with incoming boundary S1r ⊔ S1r, and use P to define a pairing Cr ⊗ Cr → C. We then choose a preferred cobordism Q in the other direction so that when we sew its right-hand outgoing S1r to the left-hand incoming one of P the resulting S-bend is the “trivial” cylinder A+r. We shall need to know, however, that the closed torus formed by the composition P ◦ Q has an even spin structure. The Frobenius structure θ on C restricts to 0 on Cr.

There is a unique spin structure on the pair-of-pants cobordism in the figure below, which restricts to S1ns on each boundary circle, and it makes Cns into a commutative Frobenius algebra in the usual way. If one incoming circle is S1ns and the other is S1r then the outgoing circle is S1r, and there are two possible spin structures, but the one obtained by removing a disc from the cylinder A+r is preferred: it makes Cr into a graded module over Cns. The chosen U-shaped cobordism P, with two incoming circles S1r, can be punctured to give us a pair of pants with an outgoing S1ns, and it induces a graded bilinear map Cr × Cr → Cns which, composing with the trace on Cns, gives a non-degenerate inner product on Cr. At this point the choice of symmetry of the tensor product becomes important. Let us consider the diffeomorphism of the pair of pants which shows us in the usual case that the Frobenius algebra is commutative. When we lift it to the spin structure, this diffeomorphism induces the identity on one incoming circle but reverses the sheets over the other incoming circle, and this proves that the cobordism must have the same output when we change the input from S(φ1 ⊗ φ2) to T(φ1) ⊗ φ2, where T is the grading involution and S : Cr ⊗ Cr → Cr ⊗ Cr is the symmetry of the tensor category. If we take S to be the symmetry of the tensor category of vector spaces which ignores the grading, this shows that the product on the graded vector space Cr is graded-symmetric with the usual sign; but if S is the graded symmetry then we see that the product on Cr is symmetric in the naive sense.

There is an analogue for spin theories of the theorem which tells us that a two-dimensional topological field theory “is” a commutative Frobenius algebra. It asserts that a spin-topological theory “is” a Frobenius algebra C = (Cns ⊕ CrC) with the following property. Let {φk} be a basis for Cns, with dual basis {φk} such that θCkφm) = δmk, and let βk and βk be similar dual bases for Cr. Then the Euler elements χns := ∑ φkφk and χr = ∑ βkβk are independent of the choices of bases, and the condition we need on the algebra C is that χns = χr. In particular, this condition implies that the vector spaces Cns and Cr have the same dimension. In fact, the Euler elements can be obtained from cutting a hole out of the torus. There are actually four spin structures on the torus. The output state is necessarily in Cns. The Euler elements for the three even spin structures are equal to χe = χns = χr. The Euler element χo corresponding to the odd spin structure, on the other hand, is given by χo = ∑(−1)degβkβkβk.

A spin theory is very similar to a Z/2-equivariant theory, which is the structure obtained when the surfaces are equipped with principal Z/2-bundles (i.e., double coverings) rather than spin structures.

It seems reasonable to call a spin theory semisimple if the algebra Cns is semisimple, i.e., is the algebra of functions on a finite set X. Then Cr is the space of sections of a vector bundle E on X, and it follows from the condition χns = χr that the fibre at each point must have dimension 1. Thus the whole structure is determined by the Frobenius algebra Cns together with a binary choice at each point x ∈ X of the grading of the fibre Ex of the line bundle E at x.

We can now see that if we had not used the graded symmetry in defining the tensor category we should have forced the grading of Cr to be purely even. For on the odd part the inner product would have had to be skew, and that is impossible on a 1-dimensional space. And if both Cns and Cr are purely even then the theory is in fact completely independent of the spin structures on the surfaces.

A concrete example of a two-dimensional topological-spin theory is given by C = C ⊕ Cη where η2 = 1 and η is odd. The Euler elements are χe = 1 and χo = −1. It follows that the partition function of a closed surface with spin structure is ±1 according as the spin structure is even or odd.

The most common theories defined on surfaces with spin structure are not topological: they are 2-dimensional conformal field theories with N = 1 supersymmetry. It should be noticed that if the theory is not topological then one does not expect the grading on Cns to be purely even: states can change sign on rotation by 2π. If a surface Σ has a conformal structure then a double covering of the non-zero tangent vectors is the complement of the zero-section in a two-dimensional real vector bundle L on Σ which is called the spin bundle. The covering map then extends to a symmetric pairing of vector bundles L ⊗ L → TΣ which, if we regard L and TΣ as complex line bundles in the natural way, induces an isomorphism L ⊗C L ≅ TΣ. An N = 1 superconformal field theory is a conformal-spin theory which assigns a vector space HS,L to the 1-manifold S with the spin bundle L, and is equipped with an additional map

Γ(S,L) ⊗ HS,L → HS,L

(σ,ψ) ↦ Gσψ,

where Γ(S,L) is the space of smooth sections of L, such that Gσ is real-linear in the section σ, and satisfies G2σ = Dσ2, where Dσ2 is the Virasoro action of the vector field σ2 related to σ ⊗ σ by the isomorphism L ⊗C L ≅ TΣ. Furthermore, when we have a cobordism (Σ,L) from (S0,L0) to (S1,L1) and a holomorphic section σ of L which restricts to σi on Si we have the intertwining property

Gσ1 ◦ UΣ,L = UΣ,L ◦ Gσ0

….

# 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.

# Super Lie Algebra A super Lie algebra L is an object in the category of super vector spaces together with a morphism [ , ] : L ⊗ L → L, often called the super bracket, or simply, the bracket, which satisfies the following conditions

Anti-symmetry,

[ , ] + [ , ] ○ cL,L = 0

which is the same as

[x, y] + (-1)|x||y|[y, x] = 0 for x, y ∈ L homogenous.

Jacobi identity,

[, [ , ]] + [, [ , ]] ○ σ + [, [ , ]] ○ σ2 = 0,

where σ ∈ S3 is a three-cycle, i.e. taking the first entity of [, [ , ]] to the second, and the second to the third, and then the third to the first. So, for x, y, z ∈ L homogenous, this reads

[x + [y, z]] + (-1)|x||y| + |x||z|[y, [z, x]] + (-1)|y||z| + |x||z|[z, [x, y]] = 0

It is important to note that in the super category, these conditions are modifications of the properties of the bracket in a Lie algebra, designed to accommodate the odd variables. We can immediately extend this definition to the case where L is an A-module for A a commutative superalgebra, thus defining a Lie superalgebra in the category of A-modules. In fact, we can make any associative superalgebra A into a Lie superalgebra by taking the bracket to be

[a, b] = ab – (-1)|a||b|ba,

i.e., we take the bracket to be the difference τ – τ ○ cA,A, where τ is the multiplication morphism on A.

A left A-module is a super vector space M with a morphism A ⊗ M → M, a ⊗ m ↦ am, of super vector spaces obeying the usual identities; that is, ∀ a, b ∈ A and x, y ∈ M, we have

a (x + y) = ax + ay

(a + b)x = ax + bx

(ab)x  = a(bx)

1x = x

A right A-module is defined similarly. Note that if A is commutative, a left A-module is also a right A-module if we define (the sign rule)

m . a = (-1)|m||a|a . m

for m ∈ M and a ∈ A. Morphisms of A-modules are defined in the obvious manner: super vector space morphisms φ: M → N such that φ(am) = aφ(m) ∀ a ∈ A and m ∈ M. So, we have the category of A-modules. For A commutative, the category of A-modules admits tensor products: for M1, M2 A-modules, M1 ⊗ M2 is taken as the tensor product of M1 as a right module with M2 as a left module.

Turning our attention to free A-modules, we have the notion of super vector kp|q over k, and so we define Ap|q := A ⊗ kp|q where

(Ap|q)0 = A0 ⊗ (kp|q)0 ⊕ A1 ⊗ (kp|q)1

(Ap|q)1 = A1 ⊗ (kp|q)0 ⊕ A0 ⊗ (kp|q)1

We say that an A-module M is free if it is isomorphic (in the category of A-modules) to Ap|q for some (p, q). This is equivalent to saying that M contains p even elements {e1, …, ep} and q odd elements {ε1, …, εq} such that

M0 = spanA0{e1, …, ep} ⊕ spanA11, …, εq}

M1 = spanA1{e1, …, ep} ⊕ spanA01, …, εq}

We shall also say M as the free module generated over A by the even elements e1, …, eand the odd elements ε1, …, εq.

Let T: Ap|q → Ar|s be a morphism of free A-modules and then write ep+1, …., ep+q for the odd basis elements ε1, …, εq. Then T is defined on the basis elements {e1, …, ep+q} by

T(ej) = ∑i=1p+q eitij

Hence T can be represented as a matrix of size (r + s) x (p + q)

T = (T1 T2 T3 T4)

where T1 is an r x p matrix consisting of even elements of A, T2 is an r x q matrix of odd elements, T3 is an s x p matrix of odd elements, and T4 is an s x q matrix of even elements. When we say that T is a morphism of super A-modules, it means that it must preserve parity, and therefore the parity of the blocks, T1 & T4, which are even and T2 & T3, which are odd, is determined. When we define T on the basis elements, the basis elements precedes the coordinates tij. This is important to keep the signs in order and comes naturally from composing morphisms. In other words if the module is written as a right module with T acting from the left, composition becomes matrix product in the usual manner:

(S . T)(ej) = S(∑i eitij) = ∑i,keksiktij

hence for any x ∈ Ap|q , we can express x as the column vector x = ∑eixi and so T(x) is given by the matrix product T x.

# Category of a Quantum Groupoid For a quantum groupoid H let Rep(H) be the category of representations of H, whose objects are finite-dimensional left H -modules and whose morphisms are H -linear homomorphisms. We shall show that Rep(H) has a natural structure of a monoidal category with duality.

For objects V, W of Rep(H) set

V ⊗ W = x ∈ V ⊗k W|x = ∆(1) · x ⊂ V ⊗k W —– (1)

with the obvious action of H via the comultiplication ∆ (here ⊗k denotes the usual tensor product of vector spaces). Note that ∆(1) is an idempotent and therefore V ⊗ W = ∆(1) × (V ⊗k W). The tensor product of morphisms is the restriction of usual tensor product of homomorphisms. The standard associativity isomorphisms (U ⊗ V ) ⊗ W → U ⊗ (V ⊗ W ) are functorial and satisfy the pentagon condition, since ∆ is coassociative. We will suppress these isomorphisms and write simply U ⊗ V ⊗ W.

The target counital subalgebra Ht ⊂ H has an H-module structure given by h · z = εt(hz),where h ∈ H, z ∈ Ht.

Ht is the unit object of Rep(H).

Define a k-linear homomorphism lV : Ht ⊗ V → V by lV(1(1) · z ⊗ 1(2) · v) = z · v, z ∈ Ht, v ∈ V.

This map is H-linear, since

lV h · (1(1) · z ⊗ 1(2) · v) = lV(h(1) · z ⊗ h(2) · v) = εt(h(1)z)h(2) · v = hz · v = h · lV (1(1) · z ⊗ 1(2) · v),

∀ h ∈ H. The inverse map l−1V: → Ht ⊗ V is given by V

l−1V(v) = S(1(1)) ⊗ (1(2) · v) = (1(1) · 1) ⊗ (1(2) · v)

The collection {lV}V gives a natural equivalence between the functor Ht ⊗ (·) and the identity functor. Indeed, for any H -linear homomorphism f : V → U we have:

lU ◦ (id ⊗ f)(1(1) · z ⊗ 1(2) · v) = lU 1(1) · z ⊗ 1(2) · f(v) = z · f(v) = f(z·v) = f ◦ lV(1(1) · z ⊗ 1(2) · v)

Similarly, the k-linear homomorphism rV : V ⊗ Ht → V defined by rV(1(1) · v ⊗ 1(2) · z) = S(z) · v, z ∈ Ht, v ∈ V, has the inverse r−1V(v) = 1(1) · v ⊗ 1(2) and satisfies the necessary properties.

Finally, we can check the triangle axiom idV ⊗ lW = rV ⊗ idW : V ⊗ Ht ⊗ W → V ⊗ W ∀ objects V, W of Rep(H). For v ∈ V, w ∈ W we have

(idV ⊗ lW)(1(1) · v ⊗ 1′(1)1(2) · z ⊗ 1′(2) · w)

= 1(1) · v ⊗ 1′(2)z · w) = 1(1)S(z) · v ⊗ 1(2) · w

=(rV ⊗ idW) (1′(1) · v ⊗ 1′(2) 1(1) · z ⊗ 1(2) · w),

therefore, idV ⊗ lW = rV ⊗ idW

Using the antipode S of H, we can provide Rep(H) with a duality. For any object V of Rep(H), define the action of H on V = Homk(V, k) by

(h · φ)(v) = φ S(h) · v —– (2)

where h ∈ H , v ∈ V , φ ∈ V. For any morphism f : V → W , let f: W → V be the morphism dual to f. For any V in Rep(H), we define the duality morphisms dV : V ⊗ V → Ht, bV : Ht → V ⊗ V∗ as follows. For ∑j φj ⊗ vj ∈ V* ⊗ V, set

dV(∑j φj ⊗ vj)  = ∑j φj (1(1) · vj) 1(2) —– (3)

Let {fi}i and {ξi}i be bases of V and V, respectively, dual to each other. The element ∑i fi ⊗ ξi does not depend on choice of these bases; moreover, ∀ v ∈ V, φ ∈ V one has φ = ∑i φ(fi) ξi and v = ∑i fi ξi (v). Set

bV(z) = z · (∑i fi ⊗ ξi) —– (4)

The category Rep(H) is a monoidal category with duality. We know already that Rep(H) is monoidal, it remains to prove that dV and bV are H-linear and satisfy the identities

(idV ⊗ dV)(bV ⊗ idV) = idV, (dV ⊗ idV)(idV ⊗ bV) = idV.

Take ∑j φj ⊗ vj ∈ V ⊗ V, z ∈ Ht, h ∈ H. Using the axioms of a quantum groupoid, we have

h · dV(∑j φj ⊗ vj) = ((∑j φj (1(1) · vj) εt(h1(2))

= (∑j φj ⊗ εs(1(1)h) · vj 1(2)j φj S(h(1))1(1)h(2) · vj 1(2)

= (∑j h(1) · φj )(1(1) · (h(2) · vj))1(2)

= (∑j dV(h(1) · φj  ⊗ h(2) · vj) = dV(h · ∑j φj ⊗ vj)

therefore, dV is H-linear. To check the H-linearity of bV we have to show that h · bV(z) =

bV (h · z), i.e., that

i h(1)z · fi ⊗ h(2) · ξi = ∑i 1(1) εt(hz) · fi ⊗ 1(2) · ξi

Since both sides of the above equality are elements of V ⊗k V, evaluating the second factor on v ∈ V, we get the equivalent condition

h(1)zS(h(2)) · v = 1(1)εt (hz)S(1(2)) · v, which is easy to check. Thus, bV is H-linear.

Using the isomorphisms lV and rV identifying Ht ⊗ V, V ⊗ Ht, and V, ∀ v ∈ V and φ ∈ V we have:

(idV ⊗ dV)(bV ⊗ idV)(v)

=(idV ⊗ dV)bV(1(1) · 1) ⊗ 1(2) · v

= (idV ⊗ dV)bV(1(2)) ⊗ S−1(1(1)) · v

= ∑i (idV ⊗ dV) 1(2) · fi ⊗ 1(3) · ξi ⊗ S−1 (1(1)) · v

= ∑1(2) · fi ⊗ 1(3) · ξi (1′(1)S-1 (1(1)) · v) 1′(2)

= 1(2) S(1(3)) 1′(1) S-1 (1(1)) · v ⊗ 1′(2) = v

(dV ⊗ idV)(idV ⊗ bV)(φ)

= (dV ⊗ idV) 1(1) · φ ⊗ bV(1(2))

= ∑i (dV ⊗ idV)(1(1) · φ ⊗ 1(2) · 1(2) · 1(3) · ξi )

= ∑i (1(1) · φ (1′(1)1(2) · fi)1′(2) ⊗ 1(3) · ξi

= 1′(2) ⊗ 1(3)1(1) S(1′(1)1(2)) · φ = φ,

QED.

# Quantum Groupoid A (finite) quantum groupoid over k is a finite-dimensional k-vector space H with the structures of an associative algebra (H, m, 1) with multiplication m : H ⊗k H → H and unit 1 ∈ H and a coassociative coalgebra (H, ∆, ε) with comultiplication ∆ : H → H ⊗k H and counit ε : H → k such that:

1. The comultiplication ∆ is a (not necessarily unit-preserving) homomorphism of algebras such that

(∆ ⊗ id)∆(1) = (∆(1) ⊗ 1) (1 ⊗ ∆(1)) = (1 ⊗ ∆(1)) (∆(1) ⊗ 1) —– (1)

2.  The counit is a k-linear map satisfying the identity:

ε(fgh) = ε(fg(1))ε(g(2)h) = ε(fg(2))ε(g(1)h), (2) ∀ f, g, h ∈ H —– (2)

3.   There is an algebra and coalgebra anti-homomorphism S : H → H, called an antipode, such that, ∀ h ∈ H ,

m(id ⊗ S) ∆(h) = (ε ⊗ id) ∆(1)(h ⊗ 1) —– (3)

m(S ⊗ id) ∆(h) = (id ⊗ ε)(1 ⊗ h) ∆(1) —– (4)

A quantum groupoid is a Hopf algebra iff one of the following equivalent conditions holds: (i) the comultiplication is unit preserving or (ii) the counit is a homomorphism of algebras.

A morphism of quantum groupoids is a map between them which is both an algebra and a coalgebra morphism preserving unit and counit and commuting with the antipode. The image of such a morphism is clearly a quantum groupoid. The tensor product of two quantum groupoids is defined in an obvious way.

The set of axioms is self-dual. This allows to define a natural quantum groupoid  structure on the dual vector space H’ = Homk (H, k) by “reversing the arrows”:

⟨h,φ ψ⟩ = ∆(h), φ ⊗ ψ —– (5)

⟨g ⊗ h, ∆'(φ)⟩ = ⟨gh, φ⟩ —– (6)

⟨h, S'(φ)⟩ = ⟨S(h), φ⟩ —– (7)

∀ φ, ψ ∈ H’, g, h ∈ H. The unit 1ˆ ∈ H’ is ε and counit ε’ is φ → ⟨φ,1⟩. The linear endomorphisms of H defined by

h → m(id ⊗ S) ∆(h), h → m(S ⊗ id) ∆(h) —– (8)

are called the target and source counital maps and denoted εt and εs, respectively.

From axioms (3) and (4),

εt(h) = (ε ⊗ id) ∆(1)(h ⊗ 1), εs(h) = (id ⊗ ε) (1 ⊗ h)∆(1) . (9)

In the Hopf algebra case εt(h) = εs(h) = ε(h)1.

We have S ◦ εs = εt ◦ S and εs ◦ S = S ◦ εt. The images of these maps εt and εs

Ht = εt (H) = {h ∈ H | ∆(h) =∆(1)(h ⊗ 1)} —– (10)

Hs = εs (H) = {h ∈ H | ∆(h) = (1⊗h) ∆(1)} —– (11)

are subalgebras of H, called the target (respectively source) counital subalgebras. They play the role of ground algebras for H. They commute with each other and

Ht = {(φ ⊗ id) ∆(1)|φ ∈ H’,

Hs = (id ⊗ φ) ∆(1)| φ ∈ H’,

i.e., Ht (respectively Hs) is generated by the right (respectively left) tensorands of ∆(1). The restriction of S defines an algebra anti-isomorphism between Ht and Hs. Any morphism H → K of quantum groupoids preserves counital subalgebras, i.e., Ht ≅ Kt and Hs ≅ Ks.

In what follows we will use the Sweedler arrows, writing ∀ h ∈ H , φ ∈ H’:

h ⇀ φ = φ(1)⟨h, φ(2)⟩,

φ ↼ h = ⟨h, φ(1)⟩φ(2) —– (12)

∀ h ∈ H, φ ∈ H’. Then the map z → (z ⇀ ε) is an algebra isomorphism between Ht and H. Similarly, the map y → (ε ↼ y) is an algebra isomorphism between H and H’t. Thus, the counital subalgebras of H’ are canonically anti-isomorphic to those of H. A quantum groupoid H is called connected if Hs ∩ Z(H) = k, or, equivalently, Ht ∩ Z(H ) = k, where Z(H) denotes the center of H. A k-algebra A is separable if the multiplication epimorphism m : A ⊗k A → A has a right inverse as an A − A bimodule homomorphism. When the characteristic of k is 0, this is equivalent to the existence of a separability element e ∈ A ⊗k A such that m(e) = 1 and (a ⊗ 1)e = e(1 ⊗ a), (1 ⊗ a)e = e(a ⊗ 1) ∀ a ∈ A. The counital subalgebras Ht and Hs are separable, with separability elements et = (S ⊗ id)∆(1) and es = (id ⊗S)∆(1), respectively. Observe that the adjoint actions of 1 ∈ H give rise to non-trivial maps

H → H : h → 1(1)hS(1(2)) = Adl1(h), h → S(1(1))h1(2) = Adr1(h), h ∈ H —– (13) …….

# Why Shouldn’t Philosophers Worry that the Detector Criterion is too Operationalist? Scattering Theory to the Rescue. Note Quote. It is not true that a representation (K,π) of U must be a Fock representation in order for states in the Hilbert space K to have an interpretation as particle states. Indeed, one of the central tasks of “scattering theory,” is to provide criteria – in the absence of full Fock space structure – for defining particle states. These criteria are needed in order to describe scattering experiments which cannot be described in a Fock representation, but which need particle states to describe the input and output states.

Haag and Swieca propose to pick out the n-particle states by means of localized detectors; we call this the detector criterion: A state with at least n-particles is a state that would trigger n detectors that are far separated in space. Philosophers might worry that the detector criterion is too operationalist. Indeed, some might claim that detectors themselves are made out of particles, and so defining a particle in terms of a detector would be viciously circular.

If we were trying to give an analysis of the concept of a particle, then we would need to address such worries. However, scattering theory does not end with the detector criterion. Indeed, the goal is to tie the detector criterion back to some other more intrinsic definition of particle states. The traditional intrinsic definition of particle states is in terms of Wigner’s symmetry criterion:

A state of n particles (of spins si and masses mi) is a state in the tensor product of the corresponding representations of the Poincaré group.

Thus, scattering theory – as originally conceived – needs to show that the states satisfying the detector criterion correspond to an appropriate representation of the Poincaré group. In particular, the goal is to show that there are isometries Ωin, Ωout that embed Fock space F(H) into K, and that intertwine the given representations of the Poincaré group on F(H) and K.

Based on these ideas, detailed models have been worked out for the case where there is a mass gap. Unfortunately, as of yet, there is no model in which Hin = Hout, which is a necessary condition for the theory to have an S-matrix, and to define transition probabilities between incoming and outgoing states. (Here Hin is the image of Fock space in K under the isometry Ωin, and similarly for Hout.)

Buchholz and collaborators have claimed that Wigner’s symmetry criterion is too stringent – i.e. there is a more general definition of particle states. They claim that it is only by means of this more general criterion that we can solve the “infraparticles” problem, where massive particles carry a cloud of photons.

The “measurement problem” of nonrelativistic QM shows that the standard approach to the theory is impaled on the horns of a dilemma: either

(i) one must make ad hoc adjustments to the dynamics (“collapse”) when needed to explain the results of measurements, or

(ii) measurements do not, contrary to appearances, have outcomes.

There are two main responses to the dilemma: On the one hand, some suggest that we abandon the unitary dynamics of QM in favor of stochastic dynamics that accurately predicts our experience of measurement outcomes. On the other hand, some suggest that we maintain the unitary dynamics of the quantum state, but that certain quantities (e.g. position of particles) have values even though these values are not specified by the quantum state.

Both approaches – the approach that alters the dynamics, and the approach with additional values – are completely successful as responses to the measurement problem in nonrelativistic QM. But both approaches run into obstacles when it comes to synthesizing quantum mechanics with relativity. In particular, the additional values approach (e.g. the de Broglie–Bohm pilot-wave theory) appears to require a preferred frame of reference to define the dynamics of the additional values, and in this case it would fail the test of Lorentz invariance.

The “modal” interpretation of quantum mechanics is similar in spirit to the de Broglie–Bohm theory, but begins from a more abstract perspective on the question of assigning definite values to some observables. Rather than making an intuitively physically motivated choice of the determinate values (e.g. particle positions), the modal interpretation makes the mathematically motivated choice of the spectral decomposition of the quantum state (i.e. the density operator) as determinate.

Unlike the de Broglie–Bohm theory, it is not obvious that the modal interpretation must violate the spirit or letter of relativistic constraints, e.g. Lorentz invariance. So, it seems that there should be some hope of developing a modal interpretation within the framework of Algebraic Quantum Field Theory…..

# 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…..