1. Today, in my Facebook, I wrote:
An example of "asymmetric bias" is where one group of people feel they are following a correct procedure and therefore, are immune from justice. In the new category theory of law and finance, this is the MISTAKE of confusing translational symmetry (eg procedure) for justice (eg rotational symmetry). An example of this bias can found in the ISDA Determinations Committee view of the "voluntary 70% nonpayment" of bonds as a "non-default." To understand why this determination is actually an asymmetric bias, you might ask yourself, "when has a banker ever treated you as being not in default when you failed to pay 70% of the debt due and payable?" The answer establishes a bilateral symmetry. Now imagine multiplying this question out to all members of society and receiving answers. If the answer is basically the same, then you will have a sense of the rotational symmetry. Biases form part of a transformational group that define the space. With category theory, we can "conceptually calculate" an instantiation of the space that is "presentation-invariant" of the theory. Sounds vague, but it is actually rigorously fun and exciting once you get the hang of it, like skiing. For the latest on the defensiveness of the ISDA determinations committee, see: http://www.zerohedge.com/news/isdas-take-lack-greek-cds-trigger-we-think-credit-eventdc-process-fair-transparent-and-well-tes
2. If you've bothered to follow the argument in this blog re a Category Theory of Law and Finance, you will note that I have tried to show how a few formalism coming from what is now called Category Theory may be applied to: (1) the Arrow-Debreu-Sharpe (ADS) Model of the Financial Contract (FC) can be thought of as the fundamental unit of the law and finance universe with the quadrangle sketch with the four objects t0 ("t zero") for initial conditions, infinite contingent states of the world (Inf-C), Pay, and t1 ("t one") as the "maturity condition" when payment is due and payable. Let's label the morphisms between the objects, f, g, h and i, so that:
f: t0, Inf-C
g: Inf-C, P
h: t1, P
i: t0, t1
It's a lot easier to see what's going on if you draw the morphisms, so it looks like a "square."
Inf-C -------->P
/\ /\
| |
| |
t0----------> t1
3. The above ADS Model is a brilliant reduction of the complex law and finance phenomena since it captures most of the important and significant features of the basic unit of behaviours in the financial markets, including primary (eg, prospectus disclosures under a 1933 Securities Act regime) and secondary (eg, ongoing reporting requirements under a 1934 Securities Exchange Act regime) behaviours. But there is one big issue that it tends to pass over and that is, the phenomenon of Default.
4. Default is a fact of life. In Aristotelian terms, default is at least a "potential-actuality." I would go so far as to assert that its primary substance is commingled with primary matter in the classical sense of undifferentiated matter, where really no distinctions can be made except for that it is an ingredient that makes up the world. This undifferentiated quality links it to Uncertainty, and Uncertainty is basically the same thing as saying "infinite contingent states of the world.". So,in terms of a Category Theory diagram (which uses equinal morphisms) we have:
Default --> Potential-Actuality --> Primary Substance --> The Undifferentiated --> Uncertainty --> Infinite States of the World
5. Default Invariance. I guess my contribution to the Category Theory of Law and Finance is that the ADS Model is insufficient to capture the legal and financial reality. To be even more precise, the ADS Model fails to show the presentation-invariance of the general structure in which financial contracts exist. This is an enormous claim! And it is an example of the philosophical importance of Category Theory in general. So, in order to correct the model so that it is ontologically and epistemologically coherent, we MUST add "not-pay" as a possible alternative object to "pay." Now this change may sound trivial, but it cleaves the universe of possibilities and gives us the "presentation-invariance" that we need for a fundamental unit of legal and financial space. In simpler terms, we can now do conceptual calculations that take account more of the space that constitute legal and financial reality. From this foundation, we can connect to:
(1) Akerloff (1970) where
(a) Part 1 of his paper where information asymmetry favouring the seller leads to market failure is simply the morphism from not-pay and where
(b) Part 2 of his paper where information asymmetry favouring the buyer never leads to market failure corresponds to the morphism from pay [sorry, this thought deserves a simple diagram and a couple thousand words of explanation];
(2) Hohfeld's Legal Conceptions where
(a) Jural Opposites and
(b) Jual Correlatives
form the Category of Legal Relations, which act as a sort for the various legal interpretations of Financial Contracts [again, apologies, this thought deserves an essay unto itself];
(3) Default Invariance implemented into
(a) the Great Cycle of Default which is the four phase mega-states of the world of law and finance (recall previous notes on the definition of Financial Innovation [I] and Bailout [B], where we have the ordered qaudtuple, [I][I], [I][B], [B][B] and [B][I], and where various regulatory proposals can be mapped as morphisms from one phase to another) eg,
(i) the Orderly Liquidation Authority of Title II of the Dodd-Frank Act is a morphism from [I][B] to [B][B], and
(ii) the Whistleblower Incentives Orotection of the Dodd-Frank Act is a morphism from [B][B] to [I][I].
These "regulatory highways" tend to accelerate transitions from one phase to another, AND there are
(4) Opetopes, that is, morphisms on morphisms between these regulatory highways which have a n-category characteristics--I.e., I suppose, at the crazy complex legal and financial level of decision-making, we have the miraculous opetopes (Many-to-One) morphisms.
Ok, got that?
6. To check that we are on the same page, you might try "sketching" all the items stated in paragraph 5 above. There's actually a whole school of Category Theory by C. Ehresmann which developed the concept of esquisse (sketch) that systematically developed sketches as the rigorous syntactics and semantics of the theory. [See, Marquis, Jean-Pierre (2009) p. 225.]
7. Now, we have been using a Naive form of Category Theory, that is, up to isomorphisms of objects where the diagrams commute to at least one "cone.". That is, where we have the objects, A, B and C, we have the following morphisms: f: A->B, g:B->C, and h:A->C, h = gf. But really, where Category Theory proper makes itself fantastically useful is when we can show that we have ADJUNCTIONS.
8. An adjunction, in one simple (but not exact) version, is merely an equivalence of functors between categories. If anyone can show that adjunctions exist in law and finance, then that person or group of people deserve at least one Nobel Prize. At this point in time, I have a couple ideas of why there is no objection to have adjunctions in law and finance systems. But to PROVE the positive of this statement....well, that would be really amazing!
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