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Stream: learning: questions

Topic: being a good community member

Daniel Geisler (Dec 25 2023 at 11:41):

Maybe the most important question I can ask is, "how to be a good community member?" I have been reviewing my earlier posting here and I must say I'm disappointed with myself. Being rather autistic, I have problems with being appropriate, communicating well and picking up signals of people being frustrated with me. So my apologies. I appreciate this community's support of inclusion, but I need to support the inclusion from my side also.

There is a second type of inclusion worth mentioning. While many here have begun their studies in category theory, I am ending my studies with it. I fit in the group of multi-disciplinary people needing category theory to complete their work. As a social activist I am interested in the applications of my work with ACT. Folks like me could use an onramp to ACT. For example, instead of expecting people to see how my work fits into the compositionality movement, I need to understand the work in compositionality and then see if my work can benefit what is already there.

I expect I have a significant credibility problem. I have no formal education, no publications and yet I am making claims about dynamics that likely seem unbelievable. But I never ask anyone to support my work. I do ask folks support the process. Paraphrasing something said about Galios, incredible claims require incredible communication and proof. My game plan is to make my work so trivial in category theory that it isn't worth disputing.

I struggle with the yoga of being a good person, having a nice balance of being a good person to work with and promoting my own work. I suspect I can't afford to benefit the world at the expense of the individuals I deal with.

Todd Trimble (Dec 25 2023 at 16:30):

Well, I'd just say: don't berate yourself too much. All of us struggle in one way or another. And from what I have seen, your posts here aren't going to seriously annoy people -- it's not like you're blathering on and on.

Todd Trimble (Dec 25 2023 at 16:30):

To make good use of category theory, you have to be hard-nosed about definitions. If you can exactly define the categories and functors and natural transformations you're working with, then your chances of making yourself understood are excellent. For example, in the "Our work" stream under your name, I see the string of symbols

$\mathcal{P}[n] \cong \frac{d^n}{dx^n} f(g(x))$

and I don't know what to make of that, because an isomorphism is a certain type of morphism in a category, and I don't know what category this is, that has objects $\mathcal{P}[n]$ and $\frac{d^n}{dx^n} f(g(x))$. I'm not even sure I know what $\mathcal{P}[n]$ is: number of partitions of the natural number $n$? Set of partitions of $\{1, \ldots, n\}$? Also you don't say what $f$ and $g$ are, although I would guess they denote formal power series or something. But most of all: what category are we talking about, that these expressions denote certain objects in the category?

Todd Trimble (Dec 25 2023 at 16:30):

So keep plugging away, but if you want to use category theory to express yourself, answer some basic questions like this. Even if no one else asks them (possibly out of politeness).

Todd Trimble (Dec 25 2023 at 16:30):

Also -- fanatic though I am about category theory -- I would say: use category theory if and only if you have an organic need to. Euler managed to make do without any category theory.

(Some people starting out in CT have preconceived ideas about how CT is going to help them as a research tool -- and I encourage them to explore whatever notions they have -- but it frequently happens that the application of CT that renders the greatest service comes from an unanticipated direction. So just let the mathematics and your personal quest for understanding and mastering it guide the way, without necessarily shoehorning in CT, until the way how reveals itself.)

Daniel Geisler (Dec 25 2023 at 19:17):

Thanks @Todd Trimble , your feedback is very helpful. I wasn't sure whether my questions were to simple to be worthy of answers or if I had failed to communicate with people. I do appreciate the importance of rigorous definitions and systematically learning the fundamental theorems. I needed to learn enough CT to understand its relevance to my work, but now that I do I am working my way through some books in a linear fashion.

The term $\mathcal{P}[n]$ comes from Analytic Combinatorics although I likely butchered the notation. I find the book so accessible that I thought others might be familiar with it. In From Finite Sets to Feynman Diagrams Baez and Dolan discuss structure types, where I guess $P_n$ for the integer partitions would be used instead of the following $F_n$.

The idea is simple. Let $F$ be any type of structure that we can put on
finite sets, and let $F_n$ be the set of all structures of this type that can be
put on your favorite $n$-element set (which we will simply call ‘$n$’). Define the
generating function of $F$ to be the formal power series

$$|F|(x) = \sum_{n=0}^∞ \frac{|F_n|}{n!} x^n.$$

Using this definition, operations on structure types correspond to operations
on their generating functions.

I will ask another basic question in learning:questions in the simplest terms that I can.

Most of my research progress comes through the abstraction and generalization of previous results. I suspect anyone that goes through this cycle enough times arrives at a need for CT. I need CT just to straighten my own head out, but beyond that I have been horribly ineffective at communicating my research. I believe being able to communicate effectively using CT will fix that problem.

Todd Trimble (Dec 25 2023 at 20:54):

I am familiar with the Baez-Dolan paper and also with the background of species.

But you haven't answered the question of what $\mathcal{P}[n]$ (as opposed to $P_n$?) is; you just refer to Analytic Combinatorics. I guess I could hunt down Analytic Combinatorics to see what it says, but it might be quicker if you just quoted it directly.

(Also, when you wrote down that isomorphism, did you in fact have a category in mind where $\cong$ denotes some invertible morphism? That's what an isomorphism is.)

Daniel Geisler (Dec 25 2023 at 21:38):

Sorry Todd, I was thinking out a longer response.

From Analytic Combinatorics

$$\mathcal{I}=(1,2,3,\cdots)$$

$$\mathcal{P}=MSET(\mathcal{I})$$

$\mathcal{P}$ is referred to as the class of all (integer) partitions. My understanding is a bit dicey because I understand structure types are 2-categories of rigs, but I certainly don't have a clear idea of what that means. My use of $\mathcal{P}[n]$ was meant to be a category of $P_n$ objects where $P_n$ is the partition number. I understand isomorphisms are defined by strict criteria and I tried to do that at Learning:question > discussing isomorphisms where I try and prove the isomorphism. I will try and rephrase my isomorphism question in a clearer manner.

Todd Trimble (Dec 25 2023 at 21:53):

That's quotation from the book is not answering the question, unfortunately: it's just more notation, and not even the notation I asked about. Is that $\mathcal{I}$ really an infinite tuple as written, or is it supposed to be a set? And what does $MSET(\mathcal{I})$ mean?

There are other questions I have in response to your last comment, but all this is not germane to "being a good community member", so I'll have to end here, unless this gets moved elsewhere.

John Baez (Dec 26 2023 at 15:09):

I understand structure types are 2-categories of rigs, but I certainly don't have a clear idea of what that means.

Where did you learn that definition? The paper you mentioned, From finite sets to Feynman diagrams, defines structure types to be functors from the groupoid of finite sets to the category of sets. These are the same as what people often call 'species' or 'combinatorial species'. They are not 2-categories of rigs - indeed I don't even know what a 2-category of rigs would be.

Daniel Geisler (Dec 26 2023 at 16:35):

@John Baez , my bad, I thought the quote came from you. Thank you for the correction and the article on combinatorial species.