## Friday, 19 July 2013

Default Invariance:  The Three Approximations of Legal and Financial Reality

I took a break from blogging to work on a theory paper entitled, "Default Invariance, A Naive Category Theory of Law and Finance."  You can see its abstract on www.ssrn.org.   I think the idea of taking a simple logical structure implied by the simplest form of a legal-financial phenomenon, namely, a financial contract with a one-period payment, and looking to the topological space implied by its terminal object will forever change the way we do law and finance.  In the paper there are three approximations of law and finance that correspond to the structure implied by the terminal objects, Pay, Not-Pay and Pay & Not-Pay.  In the simplest rough and ready terms, these terminal objects imply a point, a line ("risk homological chain complex") and a cyclic matrix ("a ring structure").  Each approximation defines the context-environment of the legal financial structures.  And I'm happy to say that we can may make explicit conceptual calculations which improve on ("correct") the works of three Nobel laureates in economics, Arrow, Debreu and Sharpe.  Arrow-Debreu-Sharpe basically set out the contingent claim model (Debreu, by the way applied abstract algebraic topological methods in his seminal work, A Theory of Value, which got rid of probability for one-period claims) which underlies everything we know and do in risk management, corporate governance, portfolio theory and practically, everything else we call "modern finance theory."  So, one way to read my Default Invariance paper is that it puts Arrow-Debreu-Sharpe into the perspective of a naive category theory, and shows our syntactical structures pre-dispose our conceptual calculations.  Anyway, you can read the paper to find out for yourself.  It's got 35 original diagrams that are meant to help "re-wire" one's own internal mapping of how the law and finance world works.

Fourth Approximation:  Taking Parts and Partitions Seriously

If we wanted to study law and finance as a physical process, we might find that there is a Darwinian-light version to the selection of laws and financial products that appears to apply.  Recall Darwin used the principle random selection for the adaptation of certain macro-features appearing to differentiate species according to external environments.

Consider Lancelot Law Whyte (1965) Internal Factors in Evolution, cited by John Bonner (20 July 2013) "Evolution, by chance?" New Scientist, 26-27, 26.  As Bonner states, "His [Whyte's] thesis was straightforward. Not only is there selection of organisms in the environment--Darwinian natural selection, which is eternal--but there is also continuous internal selection during development.  Maybe the idea was too simple and straightforward to have taken root." Bonner then goes on to state his own thesis, "This fits in neatly with my contention that the shape of microorganisms is more affected by randomness than for large, complex organisms.  Being small means very few development steps, with little or internal selection.  The effect of a mutation is likely to be immediately evident in the eternal morphology, so adult variants are produced with large numbers of different shapes and there is an increased chance that some of these will be untouched by natural selection. Compare this with what happens in a big, complex organism--a mammal, say. Only those mutations that occur at a late stage of development are likely to be viable--eye or hair colour in humans are obvious examples.  Any unfavourable mutation will likely be eliminated by internal selection."  He points out the evidence that the shapes of microorganisms are "less likely to be culled by natural selection" by citing Radiolaria (50,000 species) and diatoms (100,000 species) and Foraminifera (270,000 species).   Then he states, "If you are a strict adaptionist, you have to find a separate explanation for each shape. If you favour my suggestion that their shapes arose through random mutation and there is little or no selection, the problem vanishes." [p. 27]

What structure is implied by Bonner's internal versus external environment selection thesis?  I find his terminology a bit confusing.  For what is the external environment of a micro-organism?  Isn't everything outside it in a sense a micro-structure and therefore, could be in it, as well?  Perhaps we can clarify the thesis by translating the situation into a morphism f: A-->2.  Imagine the object A population with lots and lots of elements but having one partition such that you can maximise or minimise either part.  The f-morphism are injections to either of the two elements in 2.  So long as the partition exists, the 2 separate values will exist in 2.  So, we don't need the internal versus external division.  In category theory, there is a theorem which just gets rid of all "internal diagrams" so that anything and everything that can be possibly expressed can be done with external diagrams only.  I think the same can be said about Whyte's and Bonner's thesis above.  In other words, Darwin's random selection to adaption is preserved in the structure of parts and partition via a morphism f: A-->2.

The Fourth Approximation is taking the Third Approximation of Pay and Not-Pay as parts with a partition.  The structured implied from this terminal object is "Continuous Contingencies" (CC) to "Infinitely Discrete Randomness" (IDR).  This sounds extraordinarily vague, but what it means is that which is undifferentiable can be made into discrete unit choice.  In syntactic form:  g: CC-->IDR.  This is similar to the conceptual step of moving from "God as ubiquitous being" to "eating a properly cooked vegetarian meal is a morally correct choice of being."

## Monday, 14 May 2012

### On the Unity of Science: Law, Finance and the Philosophy of Category Theory

1.  We should measure our progress as a species of knowledge by how well individual disciplines meet the criterion of the unity of science.  This criterion was first stated as an almost urgent request by Edward O. Wilson with his concept of consilience.

2.  There are so many analogies between one subject and another, the vocabularies (as objects may be different), but the way in which these vocabularies are used (the morphisms) are so similar that they might as well be said to be the same.  What do we mean by "same" or better, if we capture the water colourist wash, and call it by its technical name, "weakened equivalence"?

3.  Mazur (June 12, 2007) in "When is one thing equal to some other thing," htt://www.math.harvard.edu/%7Emazur/preprints/when_is_one.pdf writes in tribute to one of the founders of category theory, Mac Lane, sets out an approachable essay on the question of the meaning of equivalence.  This is the deep point where all our equations and assertions in science sink to.  He sets out three approaches of how we have answered this question.

4.   The first is the "bureau of standards" where by convention we can point to something in a designated office that is an equivalent exemplar. [Id @ 4-5.]  The second is a type of universal quantification as in Frege's definition of cardinality. [Id @ 5.]  And the third is a compromise where "we indicate what we do rather than what we say we do when quizzed about our foundations." [Ibid.]  I call this third method a promiscuous stitching, using the same needle and thread or glue may be all we need to make appropriate connections between subjects, disciplines and fields of knowledge.

5.  In mathematics, you can "package" entire mathematical theories either as (1) formal systems a la the David Hilbert programme or as (2) categories. [Id @6.]  On the one hand, the formal systems go all the way back to Euclid and are much admired under the rubric of axiomatization.  On the other hand, categories are a relatively recent invention (1945 with a paper by Samuel Eilenberg and Saunders Mac Lane was more an announcement of new technique than a new view of mathematics) and its method of a sparse vocabulary and sketches of arrows betrays its deep goal which is to reveal structure.  Mathematicians, like others engaged in doing or performing in their particular discourse, don't really "axiomize" but rather "play games with conviction."

6.  Somewhere in "Tool and Object, A History and Philosophy of Category Theory," (2007), Kromer quotes Bill Lawvere (my preferred radical category theorist) for saying something to the effect, "the point is not to achieve maximal abstraction, but an optimal abstraction, a just-right abstraction that works appropriately at the level where it is most needed and used."  Of course, I am attributing a certain line of argument to Lawvere which I do not think he would disagree with.  Lawvere was motivated to find a theory of physics, to explain how things worked, but his work on the philosophy of category theory takes him on exoduses into Hegel.  I believe it is this urge to find "synthesis" with simple tools that motivates him.  He has been accused of being both revolutionary and idiosyncratic.  Revolutionary for advocating that all of mathematics can be thought of as a category of category theory.  And idiosyncratic because for such a great mathematician, he and Schanuel wrote a best selling book entitled, Conceptual Mathematics, A First Introduction to Category Theory (2009 2nd Edition) wherein you don't need any university level mathematics to understand.  In fact, I recommend this book to all my law and finance students who are interested in pursuing the application of category theory to the field of law and finance.

7.  The main points about the test for the unity of science (consilience) is that the most appropriate method for pursuing a rigorous apprehension of science (i.e., the three approaches stated by Mazur: bureaucratic standard, universal axiomization or balanced compromse (I call "promiscuous stitching") may be something so natural and simple that even our high school students can be engaged in this entreprise.

8.  For theories of law and finance, we see that there has been an influence of the latest trend or fashion from other fields that have filtered into the vocabulary of the legal theorist or financial theorist.  For example in the last 5 to 10 years, in both fields there is an emphasis at least in the titles to papers on the concept of "complexity" and "behaviours."  This is not to say these concepts are  red-herrings.  From my view, they are just another batch of ideas that come from a few equations.  Another example, what would Hart's programme of primary and secondary laws be without the notions of first order and second order logics emanating from the Cambridge logicians in the early part of the 20th century?  Not that Hart genuinely meant to implement the same programme, but the inspiration for an orderly resolution of the definition of the meaning of law was certainly intended to take the script from the philosophy department--and these were the ideas pre-Wittengstein.  In finance, the initial idea of covariance goes back to Bachelier's PhD dissertation (1905) and then developed as various methods for "curve fitting" against time horizons.  Very little work has been done on how the various theories of law and finance might be approached in a unified way.  But here the stumbling block may have been the limited view on the number of approaches to reach unification.  I do not mean by "unification" a form of axiomatiion or foundational premises evolved and expanded in a universalistic sense that may have endeared Spinoza.  Rather we have a very powerful alternative which is a kind of Kantian insight that the intellectual revolution begins with a recognition that

"There are only two possible ways in which synthetic representations and their objects...can meet one another.  Either the object (Genenstand) alone must make the representation possible, or the representation alone must make the object possible." [quoted from Mazur supra @ 20.]

9.  In law and finance theory, I would (and will) argue that one of the significant leaps in our imagination of how law and finance work together is to recognize the structure of something called "default invariance."  This is captured by or very conveniently set out with a category theory approach.  Default invariance permeates all financial contracts, and all states of the financial-regulatory-political system.

## Friday, 13 April 2012

### HFT FLASH CRASHES AND REGULATION: AN INCHOATE BIBLIOGRAPHY

( ) Research question: Why is it important to understand the physical limits of financial trading?
( ) Application: possible training course for supervisors and regulators around the world

I shall be adding to this list of references from time to time and reorganising its entirety when appropriate:

( ) IOSCO'S Recommendation for Standard Regulation on HFTs
Regulatory Issues Raised by the Impact of Technological Changes on Market Integrity and Efficiency - Consultative Report
OISU - ISOCO (July 2011)
http://www.iosco.org/library/pubdocs/pdf/IOSCOPD354.pdf

( ) SEC Report on May 6, 2010 Flash Crash (100930)
http://www.sec.gov/news/studies/2010/marketevents-report.pdf

( ) Zerohedge (100623): http://www.zerohedge.com/article/how-hft-quote-stuffing-caused-market-crash-may-6-and-threatens-destroy-entire-market-any-mom

( ) Nanex Analysis of the Flash Crash (100618): http://www.nanex.net/20100506/FlashCrashAnalysis_CompleteText.html

[ ] pay particular attention to "quote stuffing" which Nanex says should be banned: http://www.nanex.net/20100506/FlashCrashAnalysis_Part4-1.html

( ) Nanex Analysis of the SEC's Flash Crash (May 6, 2010) Report of Sept 30, 2010 (120412):

( ) Email exchange on the definition of LIQUIDITY: http://www.nanex.net/aqck/2977.html

( ) Nanex updated analysis on April 11, 2011--check out the 2nd chart especially when Wadell and Clarke whom the SEC has blamed for the Flash Crash and the W&C has accepted such blame, could NOT PHYSICALLY be the one who was involved in the perpetration of the flash crash!  http://www.nanex.net/FlashCrashFinal/FlashCrashAnalysis_WR_Update.html

[ ] http://www.zerohedge.com/article/guest-post-deconstructing-algos-part-1

[ ] http://www.zerohedge.com/article/guest-post-deconstructing-algos-part-2-leveraging-chaos-high-frequency-arbitrage-opportuniti

[ ] http://www.zerohedge.com/article/guest-post-deconstructing-algos-3-quote-stuffing-means-restoring-arbitrageable-latency-or-cq

HFT REVIEW White papers, academic and industry reports

INVESTIGATIONS
[ ]  SEC Probes Ties to High Speed Traders
http://online.wsj.com/article/SB10001424052702303816504577321864050711038.html

NY TIMES ARTICLES
135 articles on high frequency trading as of April 4, 2012

EUROPEAN REGULATIONS
[ ] MiFID 2 Barnier

REUTERS THOMSON HFT NEWS
http://hft.thomsonreuters.com/

WHARTON article
http://knowledge.wharton.upenn.edu/article.cfm?articleid=2345

INSTITUTIONAL INVESTOR
10 articles

[ ] Conflicting Codes and Codings
How Algorithmic Trading Is Reshaping Financial Regulation
Marc Lenglet
Abstract

Contemporary financial markets have recently witnessed a sea change with the ‘algorithmic revolution’, as trading automats are used to ease the execution sequences and reduce market impact. Being constantly monitored, they take an active part in the shaping of markets, and sometimes generate crises when ‘they mess up’ or when they entail situations where traders cannot go backwards. Algorithms are software codes coding practices in an IT significant ‘textual’ device, designed to replicate trading patterns. To be accepted, however, they need to comply with regulatory texts, which are nothing else but codes of conduct coding accepted practices in the markets. In this article, I draw on ethnographic fieldwork in order to open these black boxes, while trying to describe their existence as devices encapsulating several points of views. I address the question of a possible misalignment between those visions, and more specifically try to draw the consequences raised by such discrepancies as regards the future of financial regulation.
algorithmic trading codes of conduct codings financial markets practices regulation
Codes and Codings in Crisis: Signification, Performativity and Excess
Theory, Culture & Society November 1, 2011 28: 3-23
AbstractFull Text (PDF)

[ ] Investigating Financial Fraud in High Frequency Trading
Afroditi Katika
University of Manchester
Babis Theodoulidis
University of Manchester - Manchester Business School
David Diaz
University of Manchester - Manchester Business School; Universidad de Chile - Escuela de Economia y Negocios

December 20, 2011

Abstract:
Market monitoring is a very important process to the stock market. It is necessary in order to verify that all trades comply with the existing rules, as well as to detect any act of manipulation. Recently, a new kind of trading has emerged, High Frequency Trading, which allows traders to place and execute orders within milliseconds via a program running in a computer. It is doubtful whether existent market systems are capable of detecting inconsistencies in trades at this speed. This project has attempted to propose a design of a detection engine that could be incorporated to a monitoring framework so as to accommodate High Frequency trades. Understanding the field of stock markets and High Frequency Trading was a crucial part of this project. Since it is such a recent phenomenon with no confirmed cases (to the best of our knowledge) of market manipulation, we attempted to answer whether there are certain conditions that could benefit market manipulators. We used business intelligence techniques to analyse historical data and discover what sort of indications our detection engine should look for. Our results show that there have been violations of regulations that were not blocked in real-time which proves the inefficiency of current market monitoring systems. We also prove that an extreme number of orders within seconds can delay an exchange’s processes and systems. In the end we propose a design that takes those results into consideration.

Keywords: high frequency trading, flash crash, quote stuffing, financial markets, market manipulation, fraud detection, market monitoring, market surveillance

Working Paper Series
Date posted: December 20, 2011
Suggested Citation
Katika, Afroditi, Theodoulidis, Babis and Diaz, David, Investigating Financial Fraud in High Frequency Trading (December 20, 2011). Available at SSRN: http://ssrn.com/abstract=1974911

[ ] On the Dark Side of the Market: Identifying and Analyzing Hidden Order Placements
Nikolaus Hautsch
Humboldt-Universität zu Berlin; CASE - Center for Applied Statistics and Economics; CFS
Ruihong Huang
Humboldt University of Berlin

February 8, 2012

Abstract:

Number of Pages in PDF File: 43

Keywords: limit order market, hidden liquidity, high-frequency trading, non-display order, iceberg orders

JEL Classifications: G14, C24, C25, G17

Working Paper Series
Date posted: February 13, 2012
Suggested Citation
Hautsch, Nikolaus and Huang, Ruihong, On the Dark Side of the Market: Identifying and Analyzing Hidden Order Placements (February 8, 2012). Available at SSRN: http://ssrn.com/abstract=2004231 or http://dx.doi.org/10.2139/ssrn.2004231

CASE STUDIES

[ ]  BATS Take-down
http://www.zerohedge.com/news/skynet-wars-how-nasdaq-algo-destroyed-bats

[ ] http://www.zerohedge.com/news/presenting-todays-21-least-mini-flash-crashes

[ ] http://www.zerohedge.com/article/fractal-limit-order-buster-latest-market-manipulation-algo-gimmick

[ ]  Man Vs Machine: How Each Sees The Stock Market
Submitted by Tyler Durden on 04/03/2012
available at: http://www.zerohedge.com/news/man-vs-machine-how-each-sees-stock-market

[ ]  http://www.zerohedge.com/news/fractal-algo-strikes-again-time-impacts-popular-bond-bear-etf-tbt

[ ] http://www.zerohedge.com/news/stop-loss-terminator-algo-reemerges-picks-national-bank-greece-todays-victim

[ ] http://www.zerohedge.com/news/step-right-its-hft-whack-mole-time

[ ] http://www.zerohedge.com/news/whack-mole-algo-back

[ ] http://www.zerohedge.com/article/are-hft-algos-taking-aim-dominating-and-manipulating-wonderful-world-etfs-next

[ ] http://www.zerohedge.com/article/visualizing-market-topology-video-real-time-exchange-routing

[ ] http://www.zerohedge.com/article/caught-act-hft-option-algos-observed-frontrunning-todays-pmi-release

[ ] http://www.zerohedge.com/article/some-observations-spy-vwap-and-block-manipulation-fsa-launches-probe-front-running-block-tra

[ ] http://www.zerohedge.com/article/no-its-not-nat-gas-fractal-algo-nanex-discloses-very-ominous-implications-todays-berserk-cru

[ ] http://www.zerohedge.com/article/another-algo-gone-wild

RANDOM QUOTES:

[ ] Guest Post: Deconstructing Algos, Part 4 -Phase Space Reconstructions Of CNTY Busted Trades Suggests High Speed Gang-Bangs In The Market
Submitted by The World Complex Deconstructing Algos, Part 4: Phase Space ... Volatility Submitted by The World ComplexDeconstructing Algos, Part 4: Phase Space Reconstructions ... the algos picked up by Nanex on June 21, 2011 in CNTY using the data appended here . ...

Story - Tyler Durden - 07/31/2011 - 17:43 - 39 comments - 0 attachments

[ ] Guest Post: Deconstructing Algos, Part 2: Leveraging Chaos Into High-Frequency Arbitrage Opportunities
Submitted by The World Complex Deconstructing Algos, Part 2: Leveraging Chaos Into High-Frequency Arbitrage Opportunities The recent elegant explanation for the activities of the HFT algos ... Natural Gas Submitted by The World ComplexDeconstructing Algos, Part 2: Leveraging Chaos ...

[ ] Story - Tyler Durden - 07/01/2011 - 00:21 - 32 comments - 0 attachments

Guest Post: Deconstructing Algos 3: Quote Stuffing As A Means Of Restoring Arbitrageable Latency; Or Is The CQS TRYING To Crash The Market?
Submitted by The World Complex Deconstructing Algos 3: Quote Stuffing As A Means ... ComplexDeconstructing Algos 3: Quote Stuffing As A Means Of Restoring Arbitrageable LatencyIn a recent article Nanex has ... at least 33% too low. 3. An algo is testing how much more quote noise it needs to generate to cause ...

[ ] Story - Tyler Durden - 07/08/2011 - 22:08 - 63 comments - 0 attachments

Guest Post: Deconstructing Algos, Part 1
Submitted by The World Complex Deconstructing algos, part 1 The third part ... Natural Gas Submitted by The World ComplexDeconstructing algos, part 1 The third part of the series ... are critical to the decision-making module of the algo. The reconstructed phase space The difficulties ...

[ ] Story - Tyler Durden - 06/24/2011 - 21:16 - 82 comments - 0 attachments

Gold Tumbles More Than \$100 As \$1700 Stops Triggered
to algo driven liquidations following the earlier described shift in sentiment, or has some assistance ... whispers?  Felt Any MFG style whispers?  Felt like a bit more than algos.  More like a company iquidation ...

## Thursday, 12 April 2012

### Default Invariance: Product & Sum Modules - Sans Esquisses

Default Invariance:
Product and Sum Category Theory Models of Financial Contracts
Product Model:  Pay or Not-Pay
Sum Model: Financial Legal Remedies

1.  Recall the Arrow-Debreu-Sharpe model of t0 to t1 corresponding to the state of initial financial contract in a world of infinite contingent states and the state of pay, respectively.

2.  We improved the ADS Model to a Default Invariance Model where the t1 state is now bi-valued to include (a) Pay and (b) Not-Pay, i.e. P and -P.

3.  If t1 results in -P then the infinite contingent states continues at t1.  This is equivalent (or isomorphic) to the infinite-contingent-states being multiplied by 1.

4. If t1 results in P then the infinite contingent states is annihilated and the certainty of payment makes the financial contract certain and therefore, immediately disengages from the infinitely contingent states of world.  This is equivalent (or isomorphic) to the inifinite-contingent-states being multiplied by 0.

5.  The 2-state at t1 default invariance model can be further specified in terms of Product and Sum.  [This is going to get a bit technical, I warn you.]

Definition of Product
An object P together with a pair of maps P1:P->B1, and P2:P->B2 is called a product of B1 and B2 if for each object X and each pair of maps f1:X->B1, f2:X->B2, there is exactly one map f:X->P for which both f1=P1f and f2=P2f. [See Lawvere & Schanuel, 2009, p. 217.]

[I'll insert a diagram later. Hint: it looks like a chevron with X on the left and an arrow from X to P which is , another arrow from X to B1 labelled f1, an arrow P1 from P to B1, an arrow from X to B2 labelled f2, and an arrow P2 from P to B2.]

6.  Given this definition of Product, we can now apply what we stated in an earlier blog that payment of a financial contract makes it certain and therefore, takes it out of the realm of uncertainty and is no longer part of ("resides in") a world of infinite contingent states.  Thus, an occurrence of payment is equivalent to the value of 0 in the infinite contingent world.  In our Product Diagram B1 = (Infinite-Contigency) x (0).  Another way of saying this is B1 = Uncertainty x 0, which means, no more uncertainty.  This valuation is not what Sharpe and others had supposed, and had in fact given payment the value of 1, which leads to inconsistent and contradictory results.

7.  Also, we can see that non-payment or not-paying at t1 means that the infinite-contingent-states of the world continues at t1.  This is equivalent to:  (infinite-contingent-states) x (1).  So, non-payment is actually an identity morphism.  In our Default Invariance Product Diagram, B2 = 1.  Strangely, f2 will have to be equivalent to infinite contingency divided by infinite contingency.

8.  As a sum, we have the following definition:

A pair j1: B1->S, j2; B2->S, of maps in a category makes S a sum of B1 and B2 if for ach object Y and each pair g1:B1-Y, g2:B2-Y, there is exactly one map g:S->Y for which both g1 = gj1 and g2 = gj2. [See Lawvere & Schanuel (2009) Conceptual Mathematics, p. 222]

It is my contention that a legal remedy to a financial contract has the form of a sum as above, where Y is a legal remedy and B1 and B2 to breach and not-breach situations of the contract.  This is of course a first approximation of a legal risk theory.

9.  The virtue of a Product and a Sum Model for the fundamental and universal unit of law and finance is that by putting them together THROUGH TIME (that is, from t0, t1, t2,...tn) we begin to have a view of how law and finance may be seen under one perspective that allows for both (1) simplification and anticipation of direct results -- i.e., "rough and ready" calculations that border on immediate insight through very complex legal and financial phenomena; and (2) a very detailed and rigorous, bookkeeping or tracking methodology to ensure that our predictions make sense and are grounded on facts.

## Tuesday, 3 April 2012

### Platonic Signage: From Discrete Algebraic Individual Complaints to Continuous Geometric Social Solutions

Or, Infinite Smoothness: From The Discrete Complaints of Individuality to The Ironic Continuum of Small Rules

1. I think it's in book 2 or 3 of The Republic that Plato makes a rhetorical analogy that is engrained in the way we argue and complain about ourselves and our relations to society.  He asks to transfer the argument about whether a person's happiness depends on power and money  rather than virtue (that is, knowledge of the good) to a much larger body that can be dissected objectively, namely, the body of society.  The move from examining the personal body and its object of happiness (what we might now call psychology) to the social body and its aim of good government, may seem quite weak, for how can we trust in an answer to a problem which is at a level different from the problem itself. The problem is more particular and local than the solution which is general and indeed, allegedly universal.

2. This move has bothered me for 42 years.  Now, I think I have an answer to the quandary both at the substantive and procedural levels.  And both technically can be viewed from what we call a category theory.  We can also immediately understand Plato's overall strategy as a movement from a local composition in algebra to a general conception of geometry.  (There is an apocryphal saying that the signage above Plato's Academy said something to the effect, "Let no one without Geometry enter [this space]."

3. I would also temper this interpretation with a couple caveats.  First, Plato though genius of the first degree did not have the devices necessary to found category theory.  Second, when I call out the term of art, category theory, I mean in the first and foremost instance, the conceptualisation of certain data about objects, arrows (morphisms), the associativity axiom [(a*b)*c = a*(b*c)], and the identity axiom [a*I = a = 1*a].   In its most austere form and substance, namely, in diagrams or sketches ("esquisses") very very complex structures that would not be very comprehensible in a grammatical English are very evidently laid out like Dr. Seuss drawings.  One can for example re-wire quantum mechanical expressions rather naturally in category theory and understand how so-complex financial contracts work in an infinitely contingent world.

4. Coming back to the Platonic move from small (individual) to big (society), we recall Socrates' reason for this move was because he was trapped. The two nephews of Plato, x and y, [i can't remember their names but will look them up later--maybe Glaucon and Adimentaus?] had argued so eloquently and convincingly that at the PERSONAL level only power and money counted for anything in a real person's real happiness.  This was an argument based on personal observation ("what is evidently and evidentially true").  So, Platonic philosophy based whole heartedly on one's personal knowledge and education could hardly win against someone whose argument was based on personally observed facts.

5. Socrates had to go a level up.  Plato through Socrates was making an argument about the  SPACE around the individual, the interconnections, the networks, the various specialised objects ("specialists are better for the good of the whole society than the inefficient workings of generalists"), and how ultimately, this led to an analysis of governance, from benevolent dictator to timocracy, oligarchy, democracy and tyrant.  All of this was rounded by the question asked of Socrates, "And what if none of what you have imagined comes about?" Socrates' answer again is of a SPACE, "Well then, we have not wasted time, because forever and a day, people will now know where they should be heading."  In other words, Socrates had literally given future generations a map to travel, with signposts and monuments.  Now, just as no one on the tube in London can say that the tube maps are what the tubes and tube lines are, we can't say that Plato's Republic describes exactly what is front of us.  Much depends on our understanding of how to make the correspondences between the map and our own presence or location.  We use the map as a guide to help make sure we steer ourselves to a particular destination.

6. The more general meaning and thus, philosophical import of the move from individual psyche to the social space is in the form of generalisation.  Note that the rules defining the linkages in the local space, i.e., the relations between individuals, which in a strict sense might be defined as "rules" establish a structure which is the definition of the entire space in which any and all individuals exist.  This movement from particular individual mind to the structures which link us together may appear to be a type of algebraic expression of composition, that is, combinations of bilateral transactions. But the point is that these groups defined by rules when applied generally literally create a geometry of social space.  It is a space that is topologically continuous.

7. As we say in the old lingo, "from the many to the one."  We can now understand this as a movement from the particular algebraic rules that appear discrete in form but are actually, when taken together, define the infinite smoothness of the skin of reality.  The slogan here is, "small rules define big space."

8. As a meditation one might take the slogan to its limit in breath, balance and humour.