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As it happened, one of Hall's contemporaries, Jacques Plante of the Montreal Canadiens, began getting the notion that being playing goal without a mask was a bad idea. Ever creative, even as a junior player, Plante began experimenting with the idea of a face protector.

https://www.nhl.com/news/jacques-plante-becomes-first-goalie-to-wear-mask/c-290431748




Now, all hockey goalies wear masks. Moreover, all college hockey players wear masks and the vast majority of professional players wear visors. How would we predict that Jacques Plante's idea would become the initial object? Also, is it a terminal object too? How should we construct a meaningful category to model that?
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It seems like Paul Romer makes a certain mistake in his approach to the theory of innovation. He assumes that characteristic function ("the recipe") is given, while it has to be discovered. In short, the set of elements is the initial object, not the terminal one.
Maybe we can think of the set of elements as the initial object, while the free group generated by it represents the terminal object.

E.g. printed book vs handwritten book (different terminal objects built using different initial objects.)
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The entire field of Law (A) and Economics (B) must have a topos structure f: BxA -> Ω, etc, with the efficiency criteria being the terminal object S->B; S->1; B->Ω. Since market mechanisms may not apply here, we can use the theory of Firm (e.g transaction costs) to establish a different criteria. Actually, such a topos would work with any mechanism for establishing an efficiency criteria. For now, we can assume that it's exogenous.

upd. I wonder whether it's even theoretically possible to have market-efficient healthcare laws. Singapore would probably make an interesting test case.
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I'm now stuck on this definition.




How can I construct a simple topos to work through the definition?

Let's start with (4) and say that B represents various kinds of sports games; S - a particular game of sports, e.g. football; 1 stands for the rules; Ω - truth table to determine the nature of the game.

Looking at (5), let PB be the score; that is, each game has a score according to the official rules. B -> PB.

Let A be a bet on the outcome of a game. g: A -> PB.

What's BxA? It looks like a matrix of games and related bets. Using f: BxA -> Ω, we can determine whether the bet was legit.

What's BxPB? It looks like a matrix of games and scores. Using epsilon: BxPB -> Ω we can determine whether the score was legit.

In this topos we can say that a unique function g maps all bets to legitimate scores.

Does this make sense?
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У человека возник хороший вопрос:
если информация истинная, то разве принятие публикой более информированных решений не даёт большее общественное благо?


С одной стороны, интуитивно очевидно, что если "истинная информация" односторонняя, то увеличение ее количества не влияет на качество решения. Поэтому, например, в суде есть принцип состязательности сторон.

С другой стороны, можно ли доказать эту интуитивную очевидность формально? Например, если мы добавляем в категорию объекты (subobjects), которые имеют тот же терминальный объект, что и все остальные объекты в категории, то это не может изменить whether the "topos square" commutes независимо от того, что находится "внутри" объекта.

upd. I think the confirmation bias has the same logical structure.
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Reading McLane & Moerdijk helped me understand that in CT objects don't matter. In most introductory CT books their authors go out of their way to say that CT can handle many useful objects, including sets, etc. In ML&M they cut through the chase in their Categorical Preliminaries



It's clear that object C works in the axioms as a placeholder, it is used briefly to denote any kind of object. What really matters is the structure of functions and rules to construct them.
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Another interesting quote from McLane&Moerdijk, 1999.



In short, we can create two contrasting categories: a poset P, which has one morphism and many objects, and a group G, which has one object and many morphisms.

Now, by thinking in topos terms, we can see that both categories have a terminal object ( "the arrow", "the object") and can be used in combination with a subobject classifier. The existence of a subobject classifier creates an opportunity for a simple decision making process. Then the question arises, "Decision making about what?"

One way to answer it would be to go back to Adam Smith and say, "Specialization!" That is, posets P enable us to specialize in arrows ("the arrow"), while groups G (including monoids) let us specialize in objects ("the object".) A combination (e.g. a disjoint union) of these two categories helps us build a broad range of large, highly specialized categories with relatively simple decision making rules. I bet most industrial systems can be modeled this way.
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Embedding from books doesn't work on dreamwidth, so here's a snapshot:

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Within the next 10-15 years the situation with Climate Change science will resolve itself through massive satellite data collection. Thanks to Elon Musk and others, people will be launching satellites and getting plenty of data to prove or disprove the hypothesis. Most likely, it'll be cheaper to get data from space than from a drone. A global science problem will be solved by a true global data data collection system plus DL.

I can probably describe the solution in terms of Category Theory through two categories: climate events C and data collection (D). Right now, implementation of the functor F: C->D is sufficient for localized weather-related conclusions, but not sufficient for climate ones.
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McLane, Moerdijk. Sheaves in geometry and logic. 1999. p. 31-32

* -> [0,1]

Jul. 5th, 2017 09:38 am
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Most popular American professional sports — football, basketball, baseball, hockey — don't have a draw. Until recently, hockey ( a Canadian game) had a draw but NHL rules got changed to a combination of overtime and penalty shots, so that the public would always have their winner.

This tells you something about the culture of certainty.
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A righteous person among all people (A) is an equalizer (R); therefore s/he can be used as a role model for another person (X).

R -> A -> Ω
R -> 1 ->  Ω

X -> R

The existence of an ordinary righteous person R simplifies life choices for X because it relieves X from thinking, doubt and anxiety.

Also see the discussion of religion vs personal moral quest ( terminology?) in Homo Deus, by Yuval Harari.
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Deadline is another important terminal object of the western civilization. It simplifies decision making by imposing a hard constraint, thus creating a powerful Ω.
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I think we can model a constraint as the terminal object within a category. Once a constraint is given, rationality boils down to * -> Ω.

This is a provisional note because I don't understand initial objects yet.
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It appears that as a result of the 2016 elections American social anthropologists discovered a major human tribe in the middle of a highly industrialized country. http://thenewpress.com/books/strangers-their-own-land
Strangers in Their Own Land goes beyond the commonplace liberal idea that many on the political right have been duped into voting against their interests. In the right-wing world she explores, Hochschild discovers powerful forces—fear of cultural eclipse, economic decline, perceived government betrayal—which override self-interest, as progressives see it, and help explain the emotional appeal of a candidate like Donald Trump. Hochschild draws on her expert knowledge of the sociology of emotion to help us understand what it feels like to live in “red” America.


As an illustration, a figure from Bostrom's book on Superintelligence. BTW, Bostrom is right that he doesn't use the term "artificial intelligence."

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After working on Subjobject Classifier diagrams, I think I understand the meaning of Make America Great Again. Essentially, it calls for the reestablishment of the old set of Social Norms as the terminal object.

Under the new terminal object created by the Elites, a typical Trump voter finds himself at a disadvantage, both culturally and economically. Furthermore, he can't transfer his values to his children because the values won't work under the new terminal object regime.

The same goes for the more traditional extractive American companies who struggle to cope with globalization and technological change. The steep drop in energy prices in 2014 hit them and their workers really hard. Obama's climate change rhetorics added insult to the injury. It's "us vs the world's elites" now.
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Speaking categorically of Zarathustra, he invented one of the most generic subobject classifiers - Good vs Bad.

The Greeks invented another one: mortal vs immortal.

Galileo Galilei postulated that mechanical systems with speed=0 (first derivative) as the final object are equivalent to systems with acceleration=0 (second derivative) as the final object.

Sir Isaac Newton postulated that gravitation has a final object - g (third derivative = 0).

etc. etc.

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