….ad infinitum trajectory…..

The rationale  behind the 10-odd sentences was to delve into “Realism”, and economize philosophy from the clutches of “Idealism”. The complex weaving of phrases denoted chaosmos of thought structures that have become too very fluid, and tendentially getting attracted towards a basin, where, rest assured, the dynamics of such structures do not turn themselves in on statics, but are further disturbed, and then follow through to an ad infinitum trajectory. Meaning becomes a mean concept, and is always betrayed by an ally, it thought of as such. Mind you, this ain’t just about philosophy, but even the rubric of economizing does not escape the fate. I could generically cover the whole of disciplinary sets under a sheet, with the caveat that such coverings would not help my, or anybody else’s forays into such studies cut the mustard by any chance.

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Ejecting and Injecting Pronouns in Capitalism. Drunken Risibility.

I want to de-agentify myself. It is not ‘you’, or ‘I’, or ‘me’. It is the questioning of such pronouns. And why are they required at all. As I have always maintained, agencies would disappear in capitalism. And when we have capitalism everywhere, we would cease to be “we”. “We” would become part of the “IT”. And this is emancipation of the highest order. Teleology would be replaced by Eschatology. Alienation would be replaced with emancipation. Teleology is alienating, whereas eschatology is emancipating. Agency would become un-agency. An emancipation from alienation, from being, into the arms of becoming, for the former is a mere snapshot of the illusory order, whereas the latter is a continuum of fluidity, the fluid dynamics of deracinated from the illusory order. The “IT” is pure and brute materialism, the cosmic unfoldings beyond our understanding and importantly mirrored in on the terrestrial. “IT” is not to be realized. “It” is what engulfs us, kills us, and in the process emancipates us from alienation. “IT” is “Realism”, a philosophy without “we”, Capitalism’s excessive power. “IT” enslaves “us” to the point of us losing any identification. Capital is the fluidity of encampment. Ideologizing it is capitalism. This fluid is hotter than molten lava and colder than ice-caps. This fluidity does not believe in layers, and to that extent is democratic. Capital is the flow of desires/passions/reasons that criss-cross with the material that go on to build them. Capital is abstract only as far as speculations are concrete. Both morph into each other.

From Vector Spaces to Categories. Part 6.

We began by thinking of categories as “posets with extra arrows”. This analogy gives excellent intuition for the general facts about adjoint functors. However, our intuition from posets is insufficient to actually prove anything about adjoint functors.

To complete the proofs we will switch to a new analogy between categories and vector spaces. Let V be a vector space over a field K and let V ∗ be the dual space consisting of K-linear functions V → K. Now consider any K-bilinear function ⟨−,−⟩ ∶ V × V → K. We say that the function ⟨−,−⟩ is non-degenerate in both coordinates if we have

⟨u₁,v⟩ = ⟨u₂,v⟩ ∀ v ∈ V ⇒ u₁ = u₂, ⟨u,v₁⟩ = ⟨u,v₂⟩ ∀ u ∈ V ⇒ v₁ = v₂

We say that two K-linear operators L ∶ V ⇄ V ∶ R define an adjunction with respect to ⟨−, −⟩ if, ∀ vectors u,v ∈ V, we have

⟨u, R(v)⟩ = ⟨L(u), v⟩

Uniqueness of Adjoint Operators. Let L ⊣ R be an adjoint pair of operators with respect to a non-degenerate bilinear function ⟨−, −⟩ ∶ V × V → K. Then each of L and R determines

the other uniquely.

Proof: To show that R determines L, suppose that L′ ⊣ R is another adjoint pair. Thus, ∀ vectors u,v ∈ V we have

⟨L(u), v⟩ = ⟨u, R(v)⟩ = ⟨L′(u), v⟩

Now consider any vector u ∈ V. The non-degeneracy of ⟨−, −⟩ tells us that

⟨L(u), v⟩ = ⟨L′(u), v⟩ ∀ v ∈ V ⇒ L(u) = L′(u)

and since this is true ∀ u ∈ V we conclude that L = L′

RAPL for Operators:

Suppose that the function ⟨−, −⟩ ∶ V × V → K is non-degenerate and continuous. Now let T ∶ V → V be any linear operator. If T has a left or a right adjoint, then T is continuous.

Proof:

Suppose that T ∶ V → V has a left adjoint L ⊣ T, and suppose that the sequence of vectors v_i ∈ V has a limit lim_iv_i ∈ V. Furthermore, suppose that the limit lim_iT(v_i) ∈ V exists. Then for each u ∈ V, the continuity of ⟨−, −⟩ in the second coordinate tells us that

⟨u, T (lim_iv_i)⟩ = ⟨L(u), lim_iv_i⟩

= lim_i⟨L(u), v_i⟩

= lim_i⟨u,T(v_i)⟩

= ⟨u, lim_iT (v_i)⟩

Since this is true for all u ∈ V, non-degeneracy gives

T (lim_iv_i) = lim_iT (v_i)

The theorem can be made rigorous if we work with topological vector spaces. If (V, ∥ − ∥) is a normed (real or complex) vector space, then an operator T ∶ V → V is bounded if and only if it is continuous. Furthermore, if (V,⟨−,−⟩) is a Hilbert space then an operator T ∶ V → V having an adjoint is necessarily bounded, hence continuous. Many theorems of category have direct analogues in functional analysis. After all, Grothendieck began as a functional analyst.

We can summarize these two results as follows. Let ⟨−,−⟩ ∶ V ×V → K be a K-bilinear function. Then for each vector v ∈ V we have two elements of the dual space H^v, H_v ∈ V^∗ defined by

H^v ∶= ⟨v,−⟩ ∶ V → K,

H_v ∶= ⟨−,v⟩ ∶ V → K

The mappings v ↦ H^v and v ↦ H_v thus define two K-linear functions from V to V^∗ : H(−) ∶V → V^∗ and H(−) ∶ V → V^∗

Furthermore, if the function is ⟨−,−⟩ is non-degenerate and continuous then the functions H(−), H(−) ∶ V → V^∗ are both injective and continuous.

the hom bifunctor

Hom_C(−,−) ∶ C^op × C → Set behaves like a “non-degenerate and continuous bilinear function”……