Schematic Grothendieck Representation

A spectral Grothendieck representation Rep is said to be schematic if for every triple γ ≤ τ ≤ δ in Top(A), for every A in R^{^}(Ring) we have a commutative diagram in R^{^}:

 

If Rep is schematic, then, P : Top(A) → R^{^} is a presheaf with values in R^{^} over the lattice Top(A)^o, for every A in R.

The modality is to restrict attention to Tors(Rep(A)); that is, a lattice in the usual sense; and hence this should be viewed as the commutative shadow of a suitable noncommutative theory.

For obtaining the complete lattice Q(A), a duality is expressed by an order-reversing bijection: (−)⁻¹ : Q(A) → Q((Rep(A))^o). (Rep(A))^o is not a Grothendieck category. It is additive and has a projective generator; moreover, it is known to be a varietal category (also called triplable) in the sense that it has a projective regular generator P, it is co-complete and has kernel pairs with respect to the functor Hom(P, −), and moreover every equivalence relation in the category is a kernel pair. If a comparison functor is constructed via Hom(P, −) as a functor to the category of sets, it works well for the category of set-valued sheaves over a Grothendieck topology.

Now (−)⁻¹ is defined as an order-reversing bijection between idempotent radicals on Rep(A) and (Rep(A))^o, implying we write (Top(A))⁻¹ for the image of Top(A) in Q((Rep( A))^o). This is encoded in the exact sequence in Rep(A):

0 → ρ(M) → M → ρ⁻¹(M) → 0

(reversed in (Rep(A))^o). By restricting attention to hereditary torsion theories (kernel functors) when defining Tors(−), we introduce an asymmetry that breaks the duality because Top(A)⁻¹ is not in Tors((Rep(A))^op). If notationally, TT(G) is the complete lattice of torsion theories (not necessarily hereditary) of the category G; then (TT(G))⁻¹ ≅ TT(G^op). Hence we may view Tors(G)⁻¹ as a complete sublattice of TT(G^op).

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The Polity Today

The Polity Today…

Stifled and Suffocated Under Occupation,
Confined to a Life of Brazen Act of Annexation,
Subjected to a Toxicity by the Rest of the Confederacy,
Paying the Price for Some Cooked and Some Raw Conspiracy.

Counting Days and nights Cutoff from the Near and Dear Ones,
By the apathy and Atrophy of the Power that Runs,
Into a Corner and Forced Underground,
By the Despicable Vulgarity and Obscenity of the Dictatorial Sound.

Who’d Turn a Messiah to the Multiplying Affliction,
To Arrest the Basic Arithmetic of Division by Constriction,
To a Unity Imposed by Self-Rule and Determination,
By a Despot Shambolic and Ostentatious in Bringing Forth Malediction.

We are People With Indomitable Rights,
Sacrificed at the Altar of Rituals and Rites,
Chanted on by a Population that doesn’t seem to Care,
Legitimizing the Deeds of the Fundamentalists’ as an Internal Affair.

I, the Citizen, or You, the Citizen

Please cry for me India…
I, the citizen have lost my rights to be me,
I, the citizen have ceased to be,
I, the citizen have been stripped, raped, and burnt alive,
I, the citizen have been subject to justice denied,
I, the citizen have been alienated,
I, the citizen have been forcibly occupied,
I, the citizen have been living in constant fear,
I, the citizen have been subject to repeated smear,
I, the citizen have lost my right to express,
I, the citizen have my autonomy to suppress,
I, the citizen have my society divided,
I, the citizen have my politics lopsided,
I, the citizen are what you cast(e) me,
I, the citizen are what you class me,
Enough of this I, the citizen, for I shan’t be, what you prove me to be.
You, the citizen carve your own story,
You, the citizen, toy with my history,
You, the citizen decide what’s right,
You, the citizen have lost all foresight,
You, the citizen stifle my being,
You, the citizen compel me to fleeing,
You, the citizen regulate me with force,
You, the citizen admit, everything’s normal of course,
You, the citizen deny me free air,
You, the citizen snatch what’s my fair share,
You, the citizen dictate,
You, the citizen choose what’s my intimate,
You, the citizen force upon me your religion,
You, the citizen submit me to an unseen region,
Enough of this You, the Citizen, for WE the PEOPLE,
Will rise to dismantle,
What is yours to impose,
Won’t rest until you are deposed…

An Ode to a Silent Communicator…

With the setting and rising of the moon and the sun,
Shuffling back and forth in the derelict zone,
Mired in temporary infatuation and still on the run,
Waiting idly for the overwhelming attachment of what we don’t own.

Caught between the manic, the desirous and the anticipatory,
Playing by the gamble of this celestial cyclicity,
Stupefied in a hesitant, yet revelatory anxiety of the reconciliatory,
Calling upon to perpetually chasing along this periodicity.

Woken up to a reality of the ill-spraying wind,
To the lives torn asunder, and left to adrift,
Into the great chasm and the abyss of the chagrined,
Against the score of a deep silence, and the lyrically thrift…