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So far the only *examples* of posets I've talked about in the lectures are the real numbers \(\mathbb{R}\) and the natural numbers \(\mathbb{N}\) with their usual order \(\le\). Of course every natural number is a real number, so there's a function

$$ i : \mathbb{N} \to \mathbb{R} $$ sending any natural number \(x \in \mathbb{N}\) to the exact same number regarded as a real number. This function is monotone, so you now know instinctively to ask this question:

**Puzzle 21.** Does the monotone function \(i : \mathbb{N} \to \mathbb{R}\) have a left adjoint? Does it have a right adjoint? If so, what are they?

This is nice, but we need to look at other examples to appreciate the diversity of posets. Both \(\mathbb{N}\) and \(\mathbb{Z}\) have a very special property. They are **totally ordered sets**: posets such that

$$ \textrm{ for all } x, y, \textrm{ either } x \le y \textrm{ or } y \le x . $$
If you want to show off, you can call totally ordered sets **tosets**. They're also called **linearly ordered**, because you can imagine them as lines:

Totally ordered sets are limiting. Suppose you're trying to order foods on a restaurant menu based on how much you like them. What's better: a cheese sandwich or a pancake? There may be no answer, because you like them in *different ways*. To get a totally ordered set you have to ignore this and arrange all the foods in a line.

In standard economics we *do* try to arrange everything in a line. We measure the worth of everything in real numbers: numbers of *dollars*. There's even a theorem to justify this, proved by von Neumann and Morgenstern. But the assumptions of this theorem don't hold in real life. It's mainly just *convenient* to measure value, or "utility", in real numbers. With computer technologies we could set up cryptocurrencies based on other posets. But will we?

Luckily, human thought as a whole is not limited to total orders. A good example is logic. Logic, in its simplest form, is about statements \(P, Q, R, \dots \) and whether one statement implies another. If \(P\) implies \(Q\) we often write \(P \implies Q\). There are many kinds of logic, but every kind I know, this relation \(\implies\) makes statements into a preorder, since we have

1) reflexivity: \(P \implies P\)

2) transitivity: if \(P \implies Q\) and \( Q \implies R \) then \(P \implies R\).

Often people make this preorder into a poset by imposing this rule:

3) antisymmetry: if \(P \implies Q\) and \(Q \implies P\) then \(P = Q \).

This amounts to decreeing that we count two statements as "the same" if they both imply each other. We may not always want to do this. And we certainly don't want a linear order: it's easy to find examples of statements such that neither \( P \implies Q\) nor \(Q \implies P\), like "I am a millionaire" and "I am happy", or "I like this food for breakfast" and "I like this food for lunch".

So, to continue our study of preorders, posets, monotone functions and Galois connections, we'll turn to logic! Category-theoretic logic is an enormous wonderful field, but we'll just do a bit of logic based on the poset of subsets of a set, followed by a bit of logic based on the poset of partitions of a set. The latter underlies Fong and Spivak's discussion of "generative effects" in Chapter 1.

## Comments

Regarding Puzzle 21, if we consider \(\mathbb{N} = \{0, 1, 2, \ldots\}\), then we can construct a left-adjoint function \(f : \mathbb{R}\to\mathbb{N}\) to our \(i : \mathbb{N}\to\mathbb{R}\) such that

$$f = \begin{cases} 0 & \text{if } x\le0\\ \lceil x \rceil & \text{if } x>0 \end{cases}$$ We see that our function \(f\) is monotone since if \(a \le b\) then \(f(a) \le f(b)\). Then, we can check that if \(x \le i(y)\), then \(f(x) \le y\), which is the definition of a left-adjoint.

`Regarding Puzzle 21, if we consider \\(\mathbb{N} = \\{0, 1, 2, \ldots\\}\\), then we can construct a left-adjoint function \\(f : \mathbb{R}\to\mathbb{N}\\) to our \\(i : \mathbb{N}\to\mathbb{R}\\) such that $$f = \begin{cases} 0 & \text{if } x\le0\\\\ \lceil x \rceil & \text{if } x>0 \end{cases}$$ We see that our function \\(f\\) is monotone since if \\(a \le b\\) then \\(f(a) \le f(b)\\). Then, we can check that if \\(x \le i(y)\\), then \\(f(x) \le y\\), which is the definition of a left-adjoint.`

An aside on the topic of partial orders in currency: perhaps we can take inspiration to what humans did in the past when (totally-ordered) bullion money was scarce but still needed to do commerce: we often relied on recording transactions with others through tally sticks and other forms of credit. You could even make it effectively tamper-proof by recording transactions in which two halves of a stick, forging the exact locations of notches natural grain of the split stick would have been practically impossible. Debt in this form could then be traded as currency and someone to try to collect on that debt.

This is partially ordered because a mark on one set of sticks is not necessarily fungible or comparable with another: one person's debt might not be perceived as being able to pay back their debt, so in exchange, it might not be perceived worth the nominal value.

`An aside on the topic of partial orders in currency: perhaps we can take inspiration to what humans did in the past when (totally-ordered) bullion money was scarce but still needed to do commerce: we often relied on recording transactions with others through [tally sticks](https://en.wikipedia.org/wiki/Tally_stick) and other forms of credit. You could even make it effectively tamper-proof by recording transactions in which two halves of a stick, forging the exact locations of notches natural grain of the split stick would have been practically impossible. Debt in this form [could then be traded as currency](http://www.bbc.com/news/business-40189959) and someone to try to collect on that debt. This is partially ordered because a mark on one set of sticks is not necessarily fungible or comparable with another: one person's debt might not be perceived as being able to pay back their debt, so in exchange, it might not be perceived worth the nominal value.`

Continuing with Yakov's comment.

We can construct a left-adjoint function \( L : \mathbb{R}\to\mathbb{N} \) to our \( I : \mathbb{N}\to\mathbb{R}\) such that

$$L = \begin{cases} 0 & \text{if } x\le0\\ \lceil x \rceil & \text{if } x>0 \end{cases}$$ Can we construct a right-adjoint function \( R : \mathbb{R}\to\mathbb{N} \) to our \( I : \mathbb{N}\to\mathbb{R}\) such that

$$R = \begin{cases} ? & \text{if } x\lt0\\ \lfloor x \rfloor & \text{if } x \ge0 \end{cases}$$ ...what do we do when \( x \lt0 \)?

`Continuing with Yakov's comment. We can construct a left-adjoint function \\( L : \mathbb{R}\to\mathbb{N} \\) to our \\( I : \mathbb{N}\to\mathbb{R}\\) such that $$L = \begin{cases} 0 & \text{if } x\le0\\\\ \lceil x \rceil & \text{if } x>0 \end{cases}$$ Can we construct a right-adjoint function \\( R : \mathbb{R}\to\mathbb{N} \\) to our \\( I : \mathbb{N}\to\mathbb{R}\\) such that $$R = \begin{cases} ? & \text{if } x\lt0\\\\ \lfloor x \rfloor & \text{if } x \ge0 \end{cases}$$ ...what do we do when \\( x \lt0 \\)? ![Adjoints](https://docs.google.com/drawings/d/e/2PACX-1vRD1FFfwQ4qGDkT8XVX4tjcQx3XlPewnc1_UxMpHJIQCXdzv8lneYvt5YToniHrKnD2tIMhfwQfdcCY/pub?w=754&h=188)`

Brian Cohen wrote:

What you're describing is Double-Entry Bookkeeping. David Ellerman (the same author from the Partition Logic paper) also made a paper giving a mathematical treatment of double-entry bookkeeping: On Double-Entry Bookkeeping: The Mathematical Treatment.

Fredrick Eisele wrote:

Why not simply map \(x \in \mathbb{R} \) to \( 0 \in \mathbb{N} \)?

`Brian Cohen wrote: >An aside on the topic of partial orders in currency: perhaps we can take inspiration to what humans did in the past when (totally-ordered) bullion money was scarce but still needed to do commerce: we often relied on recording transactions with others through tally sticks and other forms of credit. You could even make it effectively tamper-proof by recording transactions in which two halves of a stick, forging the exact locations of notches natural grain of the split stick would have been practically impossible. Debt in this form could then be traded as currency and someone to try to collect on that debt. >This is partially ordered because a mark on one set of sticks is not necessarily fungible or comparable with another: one person's debt might not be perceived as being able to pay back their debt, so in exchange, it might not be perceived worth the nominal value. What you're describing is Double-Entry Bookkeeping. David Ellerman (the same author from the [Partition Logic](https://arxiv.org/abs/0902.1950) paper) also made a paper giving a mathematical treatment of double-entry bookkeeping: [On Double-Entry Bookkeeping: The Mathematical Treatment](https://arxiv.org/abs/1407.1898). Fredrick Eisele wrote: >...what do we do when x<0? Why not simply map \\(x \in \mathbb{R} \\) to \\( 0 \in \mathbb{N} \\)?`

Fredrick wrote:

We have a formula for the right adjoint if it exists: we saw it near the end of Lecture 6. So, we can use this to figure out what \(R(x)\) must be if the right adjoint exists... and if the formula gives an undefined result, we know the right adjoint cannot exist.

Another approach is to use Proposition 1.81 in

Seven Sketches. Applied to our example, this implies that if \(R : \mathbb{R} \to \mathbb{N}\) is a right adjoint to \(I : \mathbb{N} \to \mathbb{R} \), we must have$$ I(R(x)) \le x $$ for all \(x \in \mathbb{R}\). See what this means?

`Fredrick wrote: > ...what do we do when \\( x \lt 0 \\)? We have a formula for the right adjoint if it exists: we saw it near the end of [Lecture 6](https://forum.azimuthproject.org/discussion/1901/lecture-6-chapter-1-computing-adjoints/p1). So, we can use this to figure out what \\(R(x)\\) must be if the right adjoint exists... and if the formula gives an undefined result, we know the right adjoint cannot exist. Another approach is to use Proposition 1.81 in _[Seven Sketches](http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf)_. Applied to our example, this implies that if \\(R : \mathbb{R} \to \mathbb{N}\\) is a right adjoint to \\(I : \mathbb{N} \to \mathbb{R} \\), we must have $$ I(R(x)) \le x $$ for all \\(x \in \mathbb{R}\\). See what this means?`

\(R = 0 \text{ if } x \lt0 \) gives \( I(R(-0.2)) = 0.0 \nleq -0.2 \) meaning there is no right adjoint. Simply mapping negative values to 0 is a good mapping, it is called the left adjoint. There is a right adjoint for \( I : \mathbb{N} \rightarrow \mathbb{R}^+ \) though. Another, visual, way to think about \( I(R(x)) \le x \) is that red arrows cannot bend to the right [what I was trying to say with the picture].

`\\(R = 0 \text{ if } x \lt0 \\) gives \\( I(R(-0.2)) = 0.0 \nleq -0.2 \\) meaning there is no right adjoint. Simply mapping negative values to 0 is a good mapping, it is called the left adjoint. There is a right adjoint for \\( I : \mathbb{N} \rightarrow \mathbb{R}^+ \\) though. Another, visual, way to think about \\( I(R(x)) \le x \\) is that red arrows cannot bend to the right [what I was trying to say with the picture].`

Fredrick - yes!

`Fredrick - yes!`

John, if you ever have time to expand on this, I'd love to hear what you mean by

I wonder both what you mean by

currencies based on other posets(whether they are cryptocurrencies or other) and, as your "But will we?" part suggests), why you think that would be a good idea? I have no opposition, since I have no clue what this is even about.Minor quibbles: von Neumann and Morgenstern didn't prove a theorem to justify measuring everything in dollars in general -- the theorem has bite

onlywhen you have uncertainty or risk, and in that case you can apply it. I say that because if you don't have uncertainty, you can't get going, which I always found quite interesting (well, you can get going by simply slapping numbers on things, but "10 dollars more" will not mean the same thing in all choices, whereas with risky choices "10 dollars more" will always mean the same gain in utility). Even then the theorem doesn't say you can measure everything in dollars, but that you can measure everything in some measure (usually called utility). For dollars to work as utility measurement, it has to be true that, for every thing that exists, there is an amount of money that you like more -- which needn't be true if you cannot buy everything in the market (say kidneys).And just to be clear, these really are quibbles.

`John, if you ever have time to expand on this, I'd love to hear what you mean by > With computer technologies we could set up cryptocurrencies based on other posets. But will we? I wonder both what you mean by *currencies based on other posets* (whether they are cryptocurrencies or other) and, as your "But will we?" part suggests), why you think that would be a good idea? I have no opposition, since I have no clue what this is even about. Minor quibbles: von Neumann and Morgenstern didn't prove a theorem to justify measuring everything in dollars in general -- the theorem has bite *only* when you have uncertainty or risk, and in that case you can apply it. I say that because if you don't have uncertainty, you can't get going, which I always found quite interesting (well, you can get going by simply slapping numbers on things, but "10 dollars more" will not mean the same thing in all choices, whereas with risky choices "10 dollars more" will always mean the same gain in utility). Even then the theorem doesn't say you can measure everything in dollars, but that you can measure everything in some measure (usually called utility). For dollars to work as utility measurement, it has to be true that, for every thing that exists, there is an amount of money that you like more -- which needn't be true if you cannot buy everything in the market (say kidneys). And just to be clear, these really are quibbles.`

Well, putting numbers on preferences is very useful even without uncertainty and at the cost of losing cardinality for merely ordinal representations ;)

Btw, “10 dollars more” are not always worth the same gain in utility with risky choices unless the decision maker is risk neutral.

`Well, putting numbers on preferences is very useful even without uncertainty and at the cost of losing cardinality for merely ordinal representations ;) Btw, “10 dollars more” are not always worth the same gain in utility with risky choices unless the decision maker is risk neutral.`

Yeah, I should have been clearer about the 10 dollars more. It's really about "probability that you final life-time wealth is 100,000 dollars" - at least in the way economists usually think about it (and if you always have savings, etc etc).

The point though is that von Neumann-Morgenstern that John mentioned is about uncertainty, and that it is the uncertainty that makes it special -- the cardinality vs ordinality you mention, which is or can be a pretty big deal, depending on what you want. I am sure you know that Valter, but since it took a while before I realized how important that distinction was, I figured it worth mentioning, even on the off-chance of being nitpicky.

`Yeah, I should have been clearer about the 10 dollars more. It's really about "probability that you final life-time wealth is 100,000 dollars" - at least in the way economists usually think about it (and if you always have savings, etc etc). The point though is that von Neumann-Morgenstern that John mentioned is about uncertainty, and that it is the uncertainty that makes it special -- the cardinality vs ordinality you mention, which is or can be a pretty big deal, depending on what you want. I am sure you know that Valter, but since it took a while before I realized how important that distinction was, I figured it worth mentioning, even on the off-chance of being nitpicky.`