Category of a Quantum Groupoid

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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.

 

Hypercoverings, or Fibrant Homotopies

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Given that a Grothendieck topology is essentially about abstracting a notion of ‘covering’, it is not surprising that modified Čech methods can be applied. Artin and Mazur used Verdier’s idea of a hypercovering to get, for each Grothendieck topos, E, a pro-object in Ho(S) (i.e. an inverse system of simplicial sets), which they call the étale homotopy type of the topos E (which for them is ‘sheaves for the étale topology on a variety’). Applying homotopy group functors gives pro-groups πi(E) such that π1(E) is essentially the same as Grothendieck’s π1(E).

Grothendieck’s nice π1 has thus an interpretation as a limit of a Čech type, or shape theoretic, system of π1s of ‘hypercoverings’. Can shape theory be useful for studying ́etale homotopy type? Not without extra work, since the Artin-Mazur-Verdier approach leads one to look at inverse systems in proHo(S), i.e. inverse systems in a homotopy category not a homotopy category of inverse systems as in Strong Shape Theory.

One of the difficulties with this hypercovering approach is that ‘hypercovering’ is a difficult concept and to the ‘non-expert’ seem non-geometric and lacking in intuition. As the Grothendieck topos E ‘pretends to be’ the category of Sets, but with a strange logic, we can ‘do’ simplicial set theory in Simp(E) as long as we take care of the arguments we use. To see a bit of this in action we can note that the object [0] in Simp(E) will be the constant simplicial sheaf with value the ordinary [0], “constant” here taking on two meanings at the same time, (a) constant sheaf, i.e. not varying ‘over X’ if E is thought of as Sh(X), and (b) constant simplicial object, i.e. each Kn is the same and all face and degeneracy maps are identities. Thus [0] interpreted as an étale space is the identity map X → X as a space over X. Of course not all simplicial objects are constant and so Simp(E) can store a lot of information about the space (or site) X. One can look at the homotopy structure of Simp(E). Ken Brown showed it had a fibration category structure (i.e. more or less dual to the axioms) and if we look at those fibrant objects K in which the natural map

p : K → [0]

is a weak equivalence, we find that these K are exactly the hypercoverings. Global sections of p give a simplicial set, Γ(K) and varying K amongst the hypercoverings gives a pro-simplicial set (still in proHo(S) not in Hopro(S) unfortunately) which determines the Artin-Mazur pro-homotopy type of E.

This makes the link between shape theoretic methods and derived category theory more explicit. In the first, the ‘space’ is resolved using ‘coverings’ and these, in a sheaf theoretic setting, lead to simplicial objects in Sh(X) that are weakly equivalent to [0]; in the second, to evaluate the derived functor of some functor F : C → A, say, on an object C, one takes the ‘average’ of the values of F on objects weakly equivalent to G, i.e. one works with the functor

F′ : W(C) → A

(where W(C) has objects, α : C → C′, α a weak equivalence, and maps, the commuting ‘triangles’, and this has a ‘domain’ functor δ : W(C) → C, δ(α) = C′ and F′ is the composite Fδ). This is in many cases a pro-object in A – unfortunately standard derived functor theory interprets ‘commuting triangles’ in too weak a sense and thus corresponds to shape rather than strong shape theory – one thus, in some sense, arrives in proHo(A) instead of in Ho(proA).

Grothendieck’s Abstract Homotopy Theory

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Let E be a Grothendieck topos (think of E as the category, Sh(X), of set valued sheaves on a space X). Within E, we can pick out a subcategory, C, of locally finite, locally constant objects in E. (If X is a space with E = Sh(X), C corresponds to those sheaves whose espace étale is a finite covering space of X.) Picking a base point in X generalises to picking a ‘fibre functor’ F : C → Setsfin, a functor satisfying various conditions implying that it is pro-representable. (If x0 ∈ X is a base point {x0} → X induces a ‘fibre functor’ Sh(X) → Sh{x0} ≅ Sets, by pullback.)

If F is pro-representable by P, then π1(E, F) is defined to be Aut(P), which is a profinite group. Grothendieck proves there is an equivalence of categories C ≃ π1(E) − Setsfin, the category of finite π1(E)-sets. If X is a locally nicely behaved space such as a CW-complex and E = Sh(X), then π1(E) is the profinite completion of π1(X). This profinite completion occurs only because Grothendieck considers locally finite objects. Without this restriction, a covering space Y of X would correspond to a π1(X) – set, Y′, but if Y is a finite covering of X then the homomorphism from π1(X) to the finite group of transformations of Y factors through the profinite completion of π1(X). This is defined by : if G is a group, Gˆ = lim(G/H : H ◅ G, H of finite index) is its profinite completion. This idea of using covering spaces or their analogue in E raises several important points:

a) These are homotopy theoretic results, but no paths are used. The argument involving sheaf theory, the theory of (pro)representable functors, etc., is of a purely categorical nature. This means it is applicable to spaces where the use of paths, and other homotopies is impossible because of bad (or unknown) local properties. Such spaces have been studied within Shape Theory and Strong Shape Theory, although not by using Grothendieck’s fundamental group, nor using sheaf theory.

b) As no paths are used, these methods can also be applied to non-spaces, e.g. locales and possibly to their non-commutative analogues, quantales. For instance, classically one could consider a field k and an algebraic closure K of k and then choose C to be a category of étale algebras over k, in such a way that π1(E) ≅ Gal(K/k), the Galois group of k. It, in fact, leads to a classification theorem for Grothendieck toposes. From this viewpoint, low dimensional homotopy theory is ssen as being part of Galois theory, or vice versa.

c) This underlines the fact that π1(X) classifies covering spaces – but for i > 1, πi(X) does not seem to classify anything other than maps from Si into X!

This is abstract homotopy theory par excellence.

Superalgebras

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Let k be an algebraically closed field. Given a superalgebra A we will denote with A0 the even part, with A1 the odd part and with IAodd the ideal generated by the odd part.

A superalgebra is said to be commutative (or supercommutative) if

xy = (−1)p(x)p(y)yx, ∀ homogeneous x, y

where p denotes the parity of an homogeneous element (p(x) = 0 if x ∈ A0, p(x) = 1 if x ∈ A1).

Let’s denote with A the category of affine superalgebras that is commutative superalgebras such that, modulo the ideal generated by their odd part, they are affine algebras (an affine algebra is a finitely generated reduced commutative algebra).

Define affine algebraic supervariety over k a representable functor V from the category A of affine superalgebras to the category S of sets. Let’s call k[V] the commutative k-superalgebra representing the functor V,

V (A) = Homk−superalg(k[V], A), A ∈ A

We will call V (A) the A-points of the variety V. A morphism of affine supervarieties is identified with a morphism between the representing objects, that is a morphism of affine superalgebras.

We also define the functor Vred associated to V from the category Ac of affine k-algebras to the category of sets:

Vred(Ac)= Homk−alg(k[V]/Ik[V]odd, Ac), Ac ∈ Ac

Vred is an affine algebraic variety and it is called the reduced variety associated to V. If the algebra k[V] representing the functor V has the additional structure of a commutative Hopf superalgebra, we say that V is an affine algebraic supergroup.

Let G be an affine algebraic supergroup. As in the classical setting, the condition k[G] being a commutative Hopf superalgebra makes the functor group valued, that is the product of two morphisms is still a morphism. In fact let A be a commutative superalgebra and let x, y ∈ Homk−superalg(k[G], A) be two points of G(A). The product of x and y is defined as:

x · y = defmA · x ⊗ y · ∆

where mA is the multiplication in A and ∆ the comultiplication in k[G]. One can find that x · y ∈ Homk−superalg(k[G], A), that is:

(x · y)(ab) = (x · y)(a)(x · y)(b)

The non commutativity of the Hopf algebra in the quantum setting does not allow to multiply morphisms(=points). In fact in the quantum (super)group setting the product of two morphisms is not in general a morphism.

Let V be an affine algebraic supervariety. Let k0 ⊂ k be a subfield of k. We say that V is a k0-variety if there exists a k0-superalgebra k0[V] such that k[V] ≅ k0[V] ⊗k0 k and

V(A) = Homk0 − superalg(k0[V], A) = Homk−superalg(k[V], A), A ∈ A.

We obtain a functor that we still denote by V from the category Ak0 of affine k0-superalgebras to the category of sets:

V(Ak0) = Homk0−superalg(k0[V], Ak0), A ∈ Ak0

thus opening up for consideration of rationality on supervariety.

Derivability from Relational Logic of Charles Sanders Peirce to Essential Laws of Quantum Mechanics

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Charles Sanders Peirce made important contributions in logic, where he invented and elaborated novel system of logical syntax and fundamental logical concepts. The starting point is the binary relation SiRSj between the two ‘individual terms’ (subjects) Sj and Si. In a short hand notation we represent this relation by Rij. Relations may be composed: whenever we have relations of the form Rij, Rjl, a third transitive relation Ril emerges following the rule

RijRkl = δjkRil —– (1)

In ordinary logic the individual subject is the starting point and it is defined as a member of a set. Peirce considered the individual as the aggregate of all its relations

Si = ∑j Rij —– (2)

The individual Si thus defined is an eigenstate of the Rii relation

RiiSi = Si —– (3)

The relations Rii are idempotent

R2ii = Rii —– (4)

and they span the identity

i Rii = 1 —– (5)

The Peircean logical structure bears resemblance to category theory. In categories the concept of transformation (transition, map, morphism or arrow) enjoys an autonomous, primary and irreducible role. A category consists of objects A, B, C,… and arrows (morphisms) f, g, h,… . Each arrow f is assigned an object A as domain and an object B as codomain, indicated by writing f : A → B. If g is an arrow g : B → C with domain B, the codomain of f, then f and g can be “composed” to give an arrow gof : A → C. The composition obeys the associative law ho(gof) = (hog)of. For each object A there is an arrow 1A : A → A called the identity arrow of A. The analogy with the relational logic of Peirce is evident, Rij stands as an arrow, the composition rule is manifested in equation (1) and the identity arrow for A ≡ Si is Rii.

Rij may receive multiple interpretations: as a transition from the j state to the i state, as a measurement process that rejects all impinging systems except those in the state j and permits only systems in the state i to emerge from the apparatus, as a transformation replacing the j state by the i state. We proceed to a representation of Rij

Rij = |ri⟩⟨rj| —– (6)

where state ⟨ri | is the dual of the state |ri⟩ and they obey the orthonormal condition

⟨ri |rj⟩ = δij —– (7)

It is immediately seen that our representation satisfies the composition rule equation (1). The completeness, equation (5), takes the form

n|ri⟩⟨ri|=1 —– (8)

All relations remain satisfied if we replace the state |ri⟩ by |ξi⟩ where

i⟩ = 1/√N ∑n |ri⟩⟨rn| —– (9)

with N the number of states. Thus we verify Peirce’s suggestion, equation (2), and the state |ri⟩ is derived as the sum of all its interactions with the other states. Rij acts as a projection, transferring from one r state to another r state

Rij |rk⟩ = δjk |ri⟩ —– (10)

We may think also of another property characterizing our states and define a corresponding operator

Qij = |qi⟩⟨qj | —– (11)

with

Qij |qk⟩ = δjk |qi⟩ —– (12)

and

n |qi⟩⟨qi| = 1 —– (13)

Successive measurements of the q-ness and r-ness of the states is provided by the operator

RijQkl = |ri⟩⟨rj |qk⟩⟨ql | = ⟨rj |qk⟩ Sil —– (14)

with

Sil = |ri⟩⟨ql | —– (15)

Considering the matrix elements of an operator A as Anm = ⟨rn |A |rm⟩ we find for the trace

Tr(Sil) = ∑n ⟨rn |Sil |rn⟩ = ⟨ql |ri⟩ —– (16)

From the above relation we deduce

Tr(Rij) = δij —– (17)

Any operator can be expressed as a linear superposition of the Rij

A = ∑i,j AijRij —– (18)

with

Aij =Tr(ARji) —– (19)

The individual states could be redefined

|ri⟩ → ei |ri⟩ —– (20)

|qi⟩ → ei |qi⟩ —– (21)

without affecting the corresponding composition laws. However the overlap number ⟨ri |qj⟩ changes and therefore we need an invariant formulation for the transition |ri⟩ → |qj⟩. This is provided by the trace of the closed operation RiiQjjRii

Tr(RiiQjjRii) ≡ p(qj, ri) = |⟨ri |qj⟩|2 —– (22)

The completeness relation, equation (13), guarantees that p(qj, ri) may assume the role of a probability since

j p(qj, ri) = 1 —– (23)

We discover that starting from the relational logic of Peirce we obtain all the essential laws of Quantum Mechanics. Our derivation underlines the outmost relational nature of Quantum Mechanics and goes in parallel with the analysis of the quantum algebra of microscopic measurement.

Categorial Functorial Monads

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Algebraic constructs (A,U), such as Vec, Grp, Mon, and Lat, can be fully described by the following data, called the monad associated with (A,U):

1. the functor T : Set → Set, where T = U ◦ F and F : Set → A is the associated free functor,

2. the natural transformation η : idSet → T formed by universal arrows, and

3. the natural transformation μ : T ◦ T → T given by the unique homomorphism μX : T(TX) → TX that extends idTX.

In these cases, there is a canonical concrete isomorphism K between (A,U) and the full concrete subcategory of Alg(T) consisting of those T-algebras TX →x X that satisfy the equations x ◦ ηX = idX and x ◦ Tx = x ◦ μX. The latter subcategory is called the Eilenberg-Moore category of the monad (T, η, μ). The above observation makes it possible, in the following four steps, to express the “degree of algebraic character” of arbitrary concrete categories that have free objects:

Step 1: With every concrete category (A,U) over X that has free objects (or, more generally, with every adjoint functor A →U X) one can associate, in an essentially unique way, an adjoint situation (η, ε) : F -|U : A → X.

Step 2: With every adjoint situation (η, ε) : F -|U : A → X one can associate a monad T = (T, η, μ) on X, where T = U ◦ F : X → X.

Step 3: With every monad T = (T, η, μ) on X one can associate a concrete subcategory of Alg(T) denoted by (XT, UT) and called the category of T-algebras.

Step 4:  With every concrete category (A,U) over X that has free objects one can associate a distinguished concrete functor (A,U) →K (XT , UT) into the associated category of T-algebras called the comparison functor for (A, U).

Concrete categories that are concretely isomorphic to a category of T-algebras for some monad T have a distinct “algebraic flavor”. Such categories (A,U) and their forgetful functors U are called monadic. It turns out that a concrete category (A, U ) is monadic iff it has free objects and its associated comparison functor (A,U) →K (XT , UT) is an isomorphism. Thus, for concrete categories (A,U) that have free objects, the associated comparison functor can be considered as a means of measuring the “algebraic character” of (A,U); and the associated category of T-algebras can be considered to be the “algebraic part” of (A,U). In particular,

(a) every finitary variety is monadic,

(b) the category TopGrp, considered as a concrete category

  1. over Top, is monadic,
  2. over Set, is not monadic; the associated comparison functor is the forgetful functor TopGrp → Grp, so that the construct Grp may be considered as the “algebraic part” of the construct TopGrp,

(c) the construct Top is not monadic; the associated comparison functor is the forgetful functor Top → Set itself, so that the construct Set may be considered as the “algebraic part” of the construct Top; hence the construct Top may be considered as having a trivial “algebraic part”.

Among constructs, monadicity captures the idea of “algebraicness” rather well. Unfortunately, however, the behavior of monadic categories in general is far from satisfactory. Monadic functors can fail badly to reflect properties of the base category (e.g., the existence of colimits or of suitable factorization structures), and they are not closed under composition.

Category Theory of a Sketch. Thought of the Day 50.0

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

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

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

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