1. Pullback Square. Let P, A, B and C be Objects (Categories) in a category such that we have the following morphisms (functors) P->A, P->B, A->C, B->C. Visually, this is a pullback square where P, A, B and C are corner vertices. We can write this as
P = A xC B
2. Pullback Pasting Lemma. Assume the facts of 1 above and A->A1, C->C1, and A1->C1. So it looks like we have two adjacent squares that form a rectangle. The pullback pasting lemma states that given this "two adjacent squares," where the right square is a pullback, then the left square is a pullback if and only if the entire rectangle of two adjacent squares has a pullback.
3. Applying the Pullback Pasting Lemma (PPL) to the Structure of Laws. PPL applied to the interpretation of laws, we have the following three stages.
(1) Interpretation as a Matter of Conceptual Binary Operation
Here, Hohfeldian symmetries apply setting out the jurisprudential algebra for the calculation of legal results under the framework assumptions of Anglo-American common law. Substantive interpretation of the laws applied to particular facts follows a calculus of reifying non-legalised facts to the quanta of Hohfeldian legal relations. Slogan: what is the most real in the practice of the laws, in the very action of the laws is in the phenomenology of human actors driven by the professional discourse which is underlined by a clarifying code.
(2) Enactment and Interpretation of the Laws is Surfacetial Forming Arbitrary Series and Sequences of Conjunctions called Objects.
Between objects (legal events), we have morphisms of Time (t) and Space (s). Time is universal linear (Euclidean and pre-Reimann) and space is a minimum 3 degrees of freedom and up to and including an infinite (omega) fractal dimension of 4. To illustrate, a mere 3-dimensional location is stationary rigid body in a Newtonian-Kantian cosmogony. A 3+ fractal dimensional legal and financial object will "spread" itself over more and more possibilities of conventional space. The limit is reached at a convergent infinity fractal 3.999... which covers the entirety of all possible 3-dimensional space configurations. The 4-fractal dimensional object is for all intents and purposes the adjunction of Aristotle's unmoved mover found in Book XII of his Metaphysics.
(3) PPL Allows Decomposition of Surfacetial Structure to (a) First Pasting to Transformation of Potentially Similar Legal Objects to Categories of Isomorphically Conjoined Categories and (b)Second Pasting where Maximal Compression of all of (a) results in a singular omega object which is the generalised category of isotropic infinitude or simply, emptiness.
4. To illustrate what's going on in 3, imagine square tilings with an arbitrary square having its four corners labelled P, A, B and C, such that we have the four sides converted into morphisms, P->A, P->B, A->C and B->C. Now imagine another square where the B->C is the same as P1->A1, and we have the other three sides of the second square labelled as P1->B1, A1->C1 and B1->C1. Now imagine "pulling back" C so that it appears above the plane formed by P, A and B. Now visualise how the second square is merged to the first square. Note how C and C1 are separate points. The PPL can be read so that if we begin with a pullback in the second square, a pullback will occur in the first square if there is a pullback for the entire rectangle. This means that C and C1 will be into the same point (!) that is, that the two squares can share the same conceptual result although starting from different objects. Achieving precisely the same result as a matter of structural indulgence is extremely suggestive for a theory of law and finance.
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