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Stream: learning: questions

Topic: Partial sets and negatives

Keith Elliott Peterson (Aug 17 2023 at 22:05):

It is well known that $Set$ doesn't have so-call 'negative sets' since it contains a unique up-to-iso initial object.

However, because of the existence of a zero object, this is not the case for $Set_{Part}$, the category of sets and partial functions. Couldn't one consider the overcategory $Set_{Part}/\varnothing$ as a kind of overcategory of negative partial sets? Ironically, such an overcategory is equivalent (at least classically) to the category $Set$.

This doesn't actually seem all that odd to me, seeing as $Set_{Part}$ is an additive category.

I only ask because I'm curious as to how one might model the concept of owing someone a set of things that, when traded, are kind of "deleted" or "negated" from their possession.
(Perhaps such a "set" should be called a "deletive set" instead of a "negative set" to avoid assumptions these are categorified negative numbers.)

John Baez (Aug 18 2023 at 15:28):

If the slice category $\mathsf{Set}_{\rm part}/\empty$ is equivalent to $\mathsf{Set}$ (and that looks right to me, at least classically) then how will it let you do anything that you can't already do with $\mathsf{Set}$?

John Baez (Aug 18 2023 at 15:29):

Or are you just trying to say it's a category of "negative sets", equivalent to the usual category of "positive sets". But that would seem to be just a matter of attitude.

John Baez (Aug 18 2023 at 15:30):

What's really interesting is to get a category that has both negative and positive sets! And people have tried to do this in various ways, with three listed here.

Keith Elliott Peterson (Aug 19 2023 at 20:57):

John Baez said:

What's really interesting is to get a category that has both negative and positive sets! And people have tried to do this in various ways, with three listed here.

Hmm. I don't exactly find the solutions for a category that has both "negative" and "positive" sets to be all that appealing... especially for modeling exchange with liabilities restricted to a single person's so-called "possession space." But I'm always open to correction.

John Baez said:

If the slice category $\mathsf{Set}_{\rm part}/\empty$ is equivalent to $\mathsf{Set}$ (and that looks right to me, at least classically) then how will it let you do anything that you can't already do with $\mathsf{Set}$?

Arguably this is more a change in perspective than a change in the underlying math (I think at least).

In $\mathsf{Set}_{\rm part}/\empty$ (or equivalently in the overcategory $\mathsf{Set}_\bot/\empty_\bot$), maps into $\empty$ (resp. $\empty_\bot := \empty\amalg\{\bot\}$) are reconceptualized as maps into "nothing," even though (classically) the underlying category $\mathsf{Set}$ would have it be a plain old mapping into the terminal singleton $\{\bot\}$. In a sense, $\mathsf{Set}$ is (in a classical setting) acting like a low-language assembly language that just so happens to also model the "deletive" part within the higher order language $\mathsf{Set}_{\rm part}$ (resp. $\mathsf{Set}_\bot$). From the perspective of $\mathsf{Set}_{\rm part}$, $\mathsf{Set}$ cleans up after itself.

John Baez (Aug 19 2023 at 21:26):

Keith Elliott Peterson said:

John Baez said:

What's really interesting is to get a category that has both negative and positive sets! And people have tried to do this in various ways, with three listed here.

Hmm. I don't exactly find the solutions for a category that has both "negative" and "positive" sets to be all that appealing...

Finding an intuitively appealing category of set-like things whose cardinalities can be any integer is a famous hard problem. Loeb's solution is straightforward but somehow not fully satisfying, while Schanuel's solution is profound, exciting but also a bit shocking. Yet another answer is the sphere spectrum, but this is an $\infty$-groupoid, not a category.

Keith Elliott Peterson (Aug 21 2023 at 23:56):

John Baez said:

Keith Elliott Peterson said:

John Baez said:

What's really interesting is to get a category that has both negative and positive sets! And people have tried to do this in various ways, with three listed here.

Hmm. I don't exactly find the solutions for a category that has both "negative" and "positive" sets to be all that appealing...

Finding an intuitively appealing category of set-like things whose cardinalities can be any integer is a famous hard problem. Loeb's solution is straightforward but somehow not fully satisfying, while Schanuel's solution is profound, exciting but also a bit shocking. Yet another answer is the sphere spectrum, but this is an $\infty$-groupoid, not a category.

I read Schanuel's solution in Kock's 'Note's on Polynomial Functors,' and though it is indeed very interesting, I'm not sure how using this topological approach, I would apply it to modeling owing and then giving up to someone apples and oranges, so to speak.

John Baez (Aug 22 2023 at 09:40):

It's hard to see how Schanuel's approach to negative cardinalities works for apples and oranges. It makes more sense for islands and bridges. If you have two islands separated by water, adding a bridge between them is like adding a negative island.

Since negative numbers were first introduced by Venetian bankers for purely 'formal' reasons - to keep track of debts - it's possible that for many purposes, negative numbers really are just formal entities, i.e. obtained by starting with the rig of natural numbers and taking equivalence classes of formal differences n - m.

Keith Elliott Peterson (Aug 29 2023 at 07:48):

John Baez said:

Since negative numbers were first introduced by Venetian bankers for purely 'formal' reasons - to keep track of debts - it's possible that for many purposes, negative numbers really are just formal entities, i.e. obtained by starting with the rig of natural numbers and taking equivalence classes of formal differences n - m.

Mathematical phantoms as you call them.

Though, now that you bring it up, the group the Ventician banks came up with, what Ellerman calls the Pacioli group (which, in accordance with Stigler's law of eponymy, is named after Benedetto Cotrugli, who was unforunately not Luca Pacioli, a collaborator with the famous Leonardo da Vinci), and the Grothendicke group of the naturals aren't the same entity. Interestingly, this Pacioli group behaves very much analogously to the rationals: one being additive, the other multiplicative. Ellerman even gives nice tables showing the analogy.

See: *On Double-Entry Bookkeeping: The Mathematical Treatment* or even just the blog post *The Math of Double-Entry Bookkeeping*

We can describe vectors of entries then much like how one would describe a multiset:

One usually defines multisets as a set $X$ and a function $\mu_X: X\to\mathbb{N}$,

$$\langle X, \mu_X \rangle,$$

(though, as an English speaker, I prefer the symmetric $\langle \mu_X, X \rangle$, which I shall stick to)

a typical example being,

$$\{(4\text{Apples}), (6,\text{Oranges}, (70, \text{miles per hour})\},$$

or if we want to annoy our elementary teachers, $(4,\text{Apples})+(6,\text{Oranges})+(70, \text{miles per hour})$. :upside_down:

one could also define a "Double-Entry Bookkeeping Set" or "DEB-set" as a set $D$ and a function into the Pacioli group $\delta_X: X\to\mathrm{P}(\mathbb{N})$,

$$\langle \delta_X\rangle, D \rangle,$$

an example might be,

$$\{([2 // 5], \text{Apples}),([3 // 0], \text{Oranges}),([0 // 65], \text{miles per hour})\}$$

or in sum notation,

$$\begin{align*} && ([2 // 5], \text{Apples}) \\ + && ([3 // 0], \text{Oranges}) \\ + && ([0 // 65], \text{miles per hour}) \end{align*}$$

But I'm digressing.

As Ellerman points out, what is special about numbers in the Pacioli group of $\mathbb{N}$ is that they are unsigned numbers within a double-sided account, rather than double-sided numbers with a single account (aka integer). Since there is no such thing as far as humanity truly knows of a double-sided (discrete) set, perhaps we should take up the spirit of a 15th-century Venetian merchant.

$$\_ \\$$

Let us consider a "Pacioli Set":

$$[\![X // Y ]\!],$$

and we simply bite the bullet and declare $[\![X // Y ]\!]$ is in reduced form when $X\not\subseteq Y$ or $X\not\supseteq Y$ and $X\cap Y\cong\varnothing$.

Otherwise if $X\subseteq Y$, then $[\![X // Y ]\!] \cong [\![ \varnothing // (Y\setminus X)]\!]$.

If $X\supseteq Y$, then $[\![X // Y ]\!] \cong [\![(X\setminus Y) // \varnothing] \!]$.

Or if $X\cap Y\not\cong\varnothing$, then $[\![(X\setminus(X\cap Y)) // (Y\setminus(X\cap Y)) ]\!]$.

(The first two of course derived from the third, but I put them in for illustrative purposes)

$$\_ \\$$

Squinking a bit, the "negative" part $[\![X // \varnothing]\!]$ does look and act a bit like $\hom_{\text{Set}_\text{Part}}[X,\varnothing]$.

And $[\![\varnothing // Y]\!]$, being the "positive" part, is just the set $Y$ (or, classically, in hom notation: $\hom_{\text{Set}_\text{Part}}[\varnothing, Y]$).

$$\_ \\$$

Does the analogy hold further?

Can we define a coproduct where $[\![X // Y]\!] + [\![Z // W]\!] \cong [\![X+Z // Y+W]\!]$ with unit $[\![\varnothing // \varnothing]\!]$?

Can we define a notion of isomorphism where $[\![X // Y]\!] \cong [\![W // Z]\!] \text{ if } [\![X+Z // Y+W]\!]$?

Is it the case that we have inverses $-[\![X // Y]\!] \cong [\![Y // X]\!]$?

Is it the case that $[\![X // Y]\!] \cong \hom_{\text{Set}_\text{Part}}[X,Y]$ in general? (This one I'm more doubtful of.)

Do all these components play nicely with each other?