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.

 

Being Mediatized: How 3 Realms and 8 Dimensions Explain ‘Being’ by Peter Blank.

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Experience of Reflection: ‘Self itself is an empty word’
Leary – The neuroatomic winner: “In the province of the mind, what is believed true is true, or becomes true within limits to be learned by experience and experiment.” (Dr. John Lilly)

Media theory had noted the shoring up or even annihilation of the subject due to technologies that were used to reconfigure oneself and to see oneself as what one was: pictures, screens. Depersonalization was an often observed, reflective state of being that stood for the experience of anxiety dueto watching a ‘movie of one’s own life’ or experiencing a malfunction or anomaly in one’s self-awareness.

To look at one’s scaffolded media identity meant in some ways to look at the redactionary product of an extreme introspective process. Questioning what one interpreted oneself to be doing in shaping one’s media identities enhanced endogenous viewpoints and experience, similar to focusing on what made a car move instead of deciding whether it should stay on the paved road or drive across a field. This enabled the individual to see the formation of identity from the ‘engine perspective’.

Experience of the Hyperreal: ‘I am (my own) God’
Leary – The metaprogramming winner: “I make my own coincidences, synchronities, luck, and Destiny.”

Meta-analysis of distinctions – seeing a bird fly by, then seeing oneself seeing a bird fly by, then thinking the self that thought that – becomes routine in hyperreality. Media represent the opposite: a humongous distraction from Heidegger’s goal of the search for ‘Thinking’: capturing at present the most alarming of what occupies the mind. Hyperreal experiences could not be traced back to a person’s ‘real’ identities behind their aliases. The most questionable therefore related to dismantled privacy: a privacy that only existed because all aliases were constituting a false privacy realm. There was nothing personal about the conversations, no facts that led back to any person, no real change achieved, no political influence asserted.

From there it led to the difference between networked relations and other relations, call these other relations ‘single’ relations, or relations that remained solemnly silent. They were relations that could not be disclosed against their will because they were either too vague, absent, depressing, shifty, or dangerous to make the effort worthwhile to outsiders.

The privacy of hyperreal being became the ability to hide itself from being sensed by others through channels of information (sight, touch, hearing), but also to hide more private other selves, stored away in different, more private networks from others in more open social networks.

Choosing ‘true’ privacy, then, was throwing away distinctions one experienced between several identities. As identities were space the meaning of time became the capacity for introspection. The hyperreal being’s overall identity to the inside as lived history attained an extra meaning – indeed: as alter- or hyper-ego. With Nietzsche, the physical body within its materiality occasioned a performance that subjected its own subjectivity. Then and only then could it become its own freedom.

With Foucault one could say that the body was not so much subjected but still there functioning on its own premises. Therefore the sensitory systems lived the body’s life in connection with (not separated from) a language based in a mediated faraway from the body. If language and our sensitory systems were inseparable, beings and God may as well be.

Being Mediatized

Neo-Con Times are Non-Sociological Times

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This one is brute force.

Those who cut their eye-teeth on the likes of Austrian economics ten or fifteen years back have moved far beyond the analytical limits of the utilitarian framework and the facile solution to every social problem in terms of getting Big Government off our backs and leaving it to Smith’s invisible hand to automatically harmonize the private interests of individuals with the policy interests of the State and forge organic solidarities within and between Nations through the operation of the law of comparative advantage, while the State prudently limits itself to enforcing contracts and seeing to the military defense of the Nation. They are trying to perceive those aspects of life that lie outside the field of vision of the force-economics binocular- namely, Man’s specifically and irreducibly social being.  They have come to question the view of individuals as homogeneous, interchangeable, and isolate utility-maximizing machines endowed with rights derived from a fictive “state of Nature” in which all social relations are abstracted away as a methodological first principle, held to take shape only posterior to a putative “social contract” that binds the hitherto asocial individuals together, and only through the media of self-interested economic exchange and common subjection to the coercive power of positive law. In their rebellion against the poverty of these economic and juridical abstractions, they strive to piece together the concrete existence of the individual as a social animal compelled by its very Nature, and not just rewards and punishments, to seek out and affiliate with others….

Let’s contrast this inherently sociological current in politics to its nemesis. The latter strives to efface every aforementioned dimension of social belonging and personal identity and lump every individual into one big boundless mass, differentiated only in terms of technical specialization in the capitalist production process and by a proliferation of sexual and other “identities” shorn of their social substance and freely adopted and then discarded at will by consumers as though so many shifting vagaries of fashion. The social ties of shared descent, territory, memory, language, tradition, religion, and rule are progressively delegitimized by a relentless campaign of propaganda, homogenizing consumerist monoculture, and not least of all, coercive government action including military conquest (“regime change”). Meanwhile, asabiyyah is derided as so much ridiculously obsolete superstition, dull-witted provincialism, and mental pathology that stunts and “oppresses” the individual. Each particular society is progressively stripped of its boundary-maintaining capacity and slurred into the others- and since only particular societies exist, this means that society itself is becoming an endangered species.

Category-less Category Theory. Note Quote.

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Definition:

Let us axiomatically define a theory which we shall call an objectless or object free category theory. In this theory, the only primitive concepts (besides the usual logical concepts and the equality concept) are:

(I) α is a morphism,
(II) the composition αβ is defined and is equal to γ, The following axioms are assumed:

1. Associativity of compositions: Let α, β, γ be morphisms. If the compositions βα and γβ exist, then

• the compositions γ(βα) and (γβ)α exist and are equal;
• if γ(βα) exists, then γβ exists, and if (γβ)α exists then βα exists.

2. Existence of identities: For every morphism α there exist morphisms ι and ι′, called identities, such that

• βι = β whenever βι is defined (and analogously for ι′),

• ιγ = γ whenever ιγ is defined (and analogously for ι′).

• αι and ι′α are defined.

Lemma:

Identities ι and ι′ of axiom (2) are uniquely determined by the morphism α.

Proof:

Let us prove the uniqueness for ι (for ι′ the proof goes analogously). Let ι1 and ι2 be identities, and αι1 and αι2 exist. Then αι1 = α and (αι12 = αι2. From axiom (1) it follows that ι1ι2 is determined. But ι1ι2 exists if an only if ι1 = ι2. Indeed, let us assume that ι1ι2 exist then ι1 = ι1ι2 = ι2. And vice versa, assume that ι1 = ι2. Then from axiom (2) it follows that there exists an identity ι such that ιι1 exists, and hence is equal to ι (because ι1 is an identity). This, in turn, means that (ιι12 exists, because (ιι12 = ιι2 = ιι1 = ι. Therefore, ι1ι2 exists by Axiom 1.

Let us denote by d(α) and c(α) identities that are uniquely determined by a morphism α, i.e. such that the compositions αd(α) and c(α)α exist (letters d and c come from “domain” and “codomain”, respectively).

Lemma 2.2 The composition βα exists if and only if c(α) = d(β), and consequently,

d(βα) = d(α) and c(βα) = c(β).

Proof. Let c(α) = d(β) = ι, then βι and ια exist. From axiom (1) it follows that there exists the composition (βι)α = βα. Let us now assume that βα exists, and let us put ι = c(α). Then ια exists which implies that βα = β(ια) = (βι)α. Since βι exists then d(β) = ι.

Definition:

If for any two identities ι1 and ι2 the class ⟨ι12⟩ = {α : d(α) = ι1, c(α) = ι2},

is a set then objectless category theory is called small.

Definition:

Let us choose a class C of morphisms of the objectless category theory (i.e. C is a model of the objectless category theory), and let C0 denote the class of all identities of C. If ι123 ∈ C0, we define the composition

mC0ι123 : ⟨ι1, ι2⟩ × ⟨ι2, ι3⟩ → ⟨ι1, ι3

by mC0 (α, β) = βα. Class C is called objectless category.

Proposition:

The objectless category definition is equivalent to the standard definition of category.

Proof:

To prove the theorem it is enough to reformulate the standard category definition in the following way. A category C consists of

(I) a collection C0 of objects,
(II) for each A,B ∈ C0,
a collection ⟨A,B⟩ C0 of morphisms from A to B,

(III) for each A,B,C ∈ C0, if α ∈ ⟨A,B⟩ C0 and β ∈ ⟨B,C⟩ C0, the composition

mC0 : ⟨A,B⟩ C0 × ⟨B,C⟩ C0 → ⟨A,C⟩ C0

is defined by mC0A,B,C (α, β). The following axioms are assumed

1. Associativity: If α ∈ ⟨A,B⟩C0, β ∈ ⟨B,C⟩C0 , γ ∈ ⟨C,D⟩C0 then γ(βα) = (γβ)α.

2. Identities: For every B ∈ C0 there exists a morphism ιB ∈ ⟨B,B⟩C0 such that

A∈C0α∈⟨A,B⟩C0 ιBα = α, ∀C∈C0β∈⟨B,C⟩C0 βιB = β.

To see the equivalence of the two definitions it is enough to suitably replace in the above definition objects by their corresponding identities.

This theorem creates three possibilities to look at the category theory: (1) the standard way, in terms of objects and morphisms, (2) the objectless way, in terms of morphisms only, (3) the hybrid way in which we take into account the existence of objects but, if necessary or useful, we regard them as identity morphisms.

Marching From Galois Connections to Adjunctions. Part 4.

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

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

p ≤P R(q) ⇐⇒ L(p) ≤Q q

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

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

q1Qop q2 ⇐⇒ q2Q q1

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

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

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

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

L(p) ≤Q q ⇐⇒ p ≤P R(q) ⇐⇒ L′(p) ≤Q q

In particular, setting q = L(p) gives

L(p) ≤Q L(p) ⇒ L′(p) ≤Q L′(p)

and setting q = L′(p) gives

L′(p) ≤Q L(p) ⇒ L(p) ≤Q L′(p)

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

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

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

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

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

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

img_20170204_163208

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

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

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

L(S) ∶= ∪s∈S f(s) ∈ 2V,

R(T) ∶= {u∈U ∶ f(u) ⊆ T} ∈ 2U

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

S ⊆ Rf (T) ⇐⇒ ∀ s∈S, s ∈ R(T)

⇐⇒ ∀ s∈S, f(s) ⊆ T

⇐⇒ ∪s∈S f(s) ⊆ T

⇐⇒ L(S) ⊆ T

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

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

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

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

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

R f−1(S) = {v∈V ∶ f−1(v) ⊆ S}

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

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

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

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

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

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

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

φ ∶L(G) ⇄ L(H)∶ φ−1.

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

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

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

img_20170204_173156

…..