RAPL (Right Adjoint Preserve Limits) Theorem. Part 8b/End Part.

Fix a small category I and a diagram D ∶ I → C of shape I. Then the limit of D, if it exists, consists of an object l ∈ C and a natural isomorphism

Cone ∶ HomC(−,l) ≅ HomCI((−)I,D) ∶ Uni

in the category SetCop.

This is intuitively plausible if we recall the definition of limits. Recall that a cone under D consists of an object l ∈ C and a natural transformation Λ ∶ lI ⇒ D. We say that the cone (l,Λ) is the the limit of D if, for any other cone Φ ∶ cI ⇒ D, there exists a unique arrow υ ∶ c → l making the following diagram in CI commute:


The map sending the cone Φ ∶ cI ⇒ D to the unique arrow υ ∶ c → l is the desired function HomCI (cI,D) → HomC(c,l). Furthermore, it’s clear that this function is a bijection since we can pull back any arrow α ∶ c → l to the cone Λ ○ αI ∶ cI ⇒ D. The main difficulty is to show that the data of naturality for these bijections is equivalent to the data of the canonical cone Λ ∶ lI ⇒ D.

Proof: First assume that the limit of D exists and is given by the cone (limID,Λ). In this case we want to define a family of bijections

Unic ∶HomcI(cI,D) →~ Hom (c,limID)

that is natural in c ∈ Cop. (Then the inverse Cone ∶= Uni−1 is automatically natural. So consider any element Φ ∈ HomCI(cI,D), i.e., any cone Φ ∶ cI ⇒ D. By the definition of limits we know that there exists a unique arrow υ ∶ c → limID making the following diagram commute:


Therefore the assignment Unic(Φ) ∶= υ defines an injective function (recall that the functor (−)I is faithful, so that υ1I = υ2I implies υ1 = υ2). To see that Unic is surjective, consider any arrow α ∶ c → limID in C. We want to define a cone Φα ∶ cI ⇒ D with the property that Unicα) = α. By definition of Unic this means that we must have Φα ∶= Λ ○ αI — in other words, we must have α)i ∶= Λi ○ α indices i ∈ I. And note that this does define a natural transformation Φα ∶ cI ⇒ D since for all arrows δ ∶ i ∈ j in I we have

D(δ) ○ (Φα)i =D(δ) ○ (Λi ○ α)

= (D(δ) ○ Λi) ○ α

= Λj ○ α (Naturality of Λ)

= (Φα)j

We conclude that Unic is a bijection. To see that Unic is natural in c ∈ Cop, consider any arrow γ ∶ c1 → c2 in C (i.e., any arrow γ ∶ c2 → c1 in C). We want to show that the following diagram commutes:


And to see this, consider any cone Φ ∶ cI1 ⇒ D. By composing with the natural transformation γI ∶ cI2 ⇒ cI1 we obtain the following commutative diagram in CI:


Since the diagonal embedding (−)I ∶ C → CI is a functor, the bottom arrow is given by

(Unic1 (Φ))I ○ γI = (Unic1 (Φ) ○ γ)I

But by the definition of the function Unic2 this arrow also equals (Unic2(Φ ○ γI))I

Then since (−)I is a faithful functor we conclude that

Unic2 (Φ ○ γI) = Unic1 (Φ) ○ γ

and hence the desired square commutes. Conversely, consider an object l ∈ C and suppose that we have a bijection

Conec ∶HomC(c,l) ←→ HomCI(cI,D) ∶ Unic

that is natural in c ∈ Cop. In other words, suppose that for each arrow γ ∶ c1 → c2 in Cop (i.e., for each arrow γ ∶ c2 → c1 in C) we have a commutative square:


We want to show that this determines a unique cone Λ ∶ lI ⇒ D such that (l, Λ) is the limit of D. The only possible choice is to define Λ ∶= Conel(idl). Now given any cone Φ ∶ cI ⇒ D we want to show that there exists a unique arrow υ ∶ c → l with the property Λ ○ υI = Φ.

So suppose that there exists some arrow υ ∶ c → l with the property Λ ○ υI = Φ. By substituting γ ∶= υ into the above diagram we obtain a commutative square:


Then following the arrow idl ∈ HomC(l, l) around the square in two different ways gives

idl ○ υ = Unic(Conel(idl) ○ υI)

υ = Unic(Λ ○ υI)

υ = Unic(Φ)

Thus there exists at most one such arrow υ. To show that there exists at least one such arrow, we must check that the arrow Unic(Φ) actually does satisfy Λ ○ (Unic(Φ))I = Φ. Indeed, by substituting υ ∶= Unic(Φ) into the above diagram we obtain a commutative square:


Then following the arrow idl ∈ HomC(l,l) around the

Conel(idl) ○ (Unic(Φ))I) = Conec(idl ○ Unic(Φ)) Λ ○ (Unic(Φ))I

= Conec(Unic(Φ))

Λ ○ (Unic(Φ))I = Φ

square in two ways gives as desired.

[Remark: We have proved that the limit of a diagram D ∶ I → C, if it exists, consists of an object limID ∈ C and a natural isomorphism

HomC(−, limID) ≅ HomCI((−)I,D) of functors Cop → Set. It turns out that if all limits of shape I exist in C then there is a unique way to extend this to a natural isomorphism

HomC(−,limI−) ≅ HomCI((−)I,−)

of functors Cop × CI → Set, and hence that we have an adjunction (−)I ∶ C ⇄ CI ∶ limI. However, we don’t need this result right now so we won’t prove it. Dually, the colimit of D, if it exists, consists of an object colimID ∈ C and a natural isomorphism HomC(colimID, −) ≅ HomCI(D, (−)I) of functors C → Set. If all colimits of shape I exist in C then this extends uniquely to an adjunction colimI ∶ CI ⇄ C ∶ (−)I. This explains the title of the previous lemma.]

Theorem (RAPL):

Let L ∶ C ⇄ D ∶ R be an adjunction and consider a diagram D ∶ I → D of shape I in D. If the diagram D ∶ I → D has a limit cone Λ ∶ lI ⇒ D then the composite diagram RI(D) ∶ I → C also has a limit cone, which is given by RI(Λ) ∶ R(l)I ⇒ RI(D).


In this proof we will write limID ∶= l ∈ D, and we will just assume that the limit object limI RI(D) ∈ C exists. Now we want to show that the following objects are isomorphic in C : R(limID) ≅ limI RI (D). (We will ignore the data of the limit cone.)

So assume that L ∶ C ⇄ D ∶ R is an adjunction. Then we have the following sequence of bijections, each of which is natural in c ∈ Cop:

Homc(c, R(limID)) →~ Homc(L(c), limID) (L ⊣ R)

~ HomDI(L(c),D) (Diagonal ⊣ Limit)

~ HomDI (LI (cI),D)

~ HomCI(c,RI(D))

~ Homc(c,limIRI(D)) (Diagonal ⊣ Limit)

By composing these we obtain a family of bijections

Homc(c,R(limID)) →~ Homc(c,limI RI(D))

that is natural in c ∈ Cop. In other words, we obtain an isomorphism of hom functors HR(limI(D)) ≅ HlimI RI(D) in the category SetCop. Then since the Yoneda embedding H(−) : C → SetCop is essentially injective (from the Embedding Lemma), we obtain an isomorphism of objects R(limID) ≅ limI RI(D) in the category C.



Donald Trump may be a challenge for AIIB and NDB. Is It and How Far?

Was reading this news on whether Trump would be a challenge for Chinese-led development banks AIIB and NDB. Here is the blurb hyperlinked to the whole story, and why I think that more Trump, is the outcome of upcoming elections in France and The Netherlands that could be the chief hurdle, if at all.


With the US president Donald Trump entering the stage, two new China-backed multilateral banks, the Asian Infrastructure Investment Bank (AIIB) and the New Development Bank (NDB) of Brics nations, may face a rough ride ahead, analysts say. 

China has benefited in diplomatic and political terms because these two banks have been sponsored by Beijing and are based in the country.

“China has gained in terms of ‘soft power’ because it could bring several European powers on the table through AIIB. At present, a lot of European countries are concerned about what they see in the US. This might increase potential cooperation between China and the European countries,” says Julian Evans-Pritchard, the China economist for Capital Economics……

Is Trump’s protectionist policy the real danger to Chinese-led banks? Is stepping out of the TPP the real threat? Or, is there some other hidden variable at work here? Holland and France go to Presidential polls this year and with BREXIT negotiations looming large, its the likelihood of referendum of the BREXIT kind in France and Holland once right of the centre parties are elected to power there, where there is a high probability of such happenings there.The rise of Alt-Right political parties in both countries such as the Front National (FN) led by Marine Le Pen in France and The Party for Freedom (PVV) led by Geert Wilders in the Netherlands could trigger the end of the European Union. AIIB, especially, since it has Europeans along its corridors. Moreover, US infrastructure is getting outmoded by the day and according to promises delivered by Trump during his electioneering, his focus would concentrate on domestic infrastructural projects, rather than get into the glamour created by the likes of multi-laterals and that too Chinese for the time being. Moreover, with his presidency the idea is to thaw the relationship between the US and Russia and stymying AIIB or NDB could be perplexing for that to happen unnecessarily. And would Russia really be subdued to these emanating pressures to put a period? As of now, it won’t look into the aspects of a cold relationship with China at US’s expense. While Trump labelled China a currency manipulator and threatened trade wars, he might have a more open ­attitude towards China-backed institutions and investment ­programmes. According to Jin Liquin, Chair of AIIB, “I was told that many in his (Trump’s) team have an opinion that Obama was not right not to join the AIIB, especially after Canada joined, which was a very loud endorsement of the bank.” So, the options are soft, but for reasons on domesticity, these might be the hard ones to steer clear for the US. What really should be taken into account is consensus on right-wing victories in the upcoming elections in Europe and a concomitant string of protectionist policies in their wake that could really be the derailing point for these Chinese-led development banks. As Josep Goded says, this year will be full of threats and challenges for global society.The potential disintegration of the European Union represents one of these, but no one knows how it will end. History is full of threats and challenge that are often happily resolved. The elections in The Netherlands and France will be all about choosing between tolerance or intolerance, war or peace, friendship or enemies, future or past…Their citizens will have an enormous advantage since they can see what Trump is doing in the U.S. and based on that they will make an important decision that will change the world for the better or worse.

von Neumann & Dis/belief in Hilbert Spaces

I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space any more.

— John von Neumann, letter to Garrett Birkhoff, 1935.


The mathematics: Let us consider the raison d’ˆetre for the Hilbert space formalism. So why would one need all this ‘Hilbert space stuff, i.e. the continuum structure, the field structure of complex numbers, a vector space over it, inner-product structure, etc. Why? According to von Neumann, he simply used it because it happened to be ‘available’. The use of linear algebra and complex numbers in so many different scientific areas, as well as results in model theory, clearly show that quite a bit of modeling can be done using Hilbert spaces. On the other hand, we can also model any movie by means of the data stream that runs through your cables when watching it. But does this mean that these data streams make up the stuff that makes a movie? Clearly not, we should rather turn our attention to the stuff that is being taught at drama schools and directing schools. Similarly, von Neumann turned his attention to the actual physical concepts behind quantum theory, more specifically, the notion of a physical property and the structure imposed on these by the peculiar nature of quantum observation. His quantum logic gave the resulting ‘algebra of physical properties’ a privileged role. All of this leads us to … the physics of it. Birkhoff and von Neumann crafted quantum logic in order to emphasize the notion of quantum superposition. In terms of states of a physical system and properties of that system, superposition means that the strongest property which is true for two distinct states is also true for states other than the two given ones. In order-theoretic terms this means, representing states by the atoms of a lattice of properties, that the join p ∨ q of two atoms p and q is also above other atoms. From this it easily follows that the distributive law breaks down: given atom r ≠ p, q with r < p ∨ q we have r ∧ (p ∨ q) = r while (r ∧ p) ∨ (r ∧ q) = 0 ∨ 0 = 0. Birkhoff and von Neumann as well as many others believed that understanding the deep structure of superposition is the key to obtaining a better understanding of quantum theory as a whole.

For Schrödinger, this is the behavior of compound quantum systems, described by the tensor product. While the quantum information endeavor is to a great extend the result of exploiting this important insight, the language of the field is still very much that of strings of complex numbers, which is akin to the strings of 0’s and 1’s in the early days of computer programming. If the manner in which we describe compound quantum systems captures so much of the essence of quantum theory, then it should be at the forefront of the presentation of the theory, and not preceded by continuum structure, field of complex numbers, vector space over the latter, etc, to only then pop up as some secondary construct. How much quantum phenomena can be derived from ‘compoundness + epsilon’. It turned out that epsilon can be taken to be ‘very little’, surely not involving anything like continuum, fields, vector spaces, but merely a ‘2D space’ of temporal composition and compoundness, together with some very natural purely operational assertion, including one which in a constructive manner asserts entanglement; among many other things, trace structure then follows.