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Stream: theory: category theory

Topic: The relative/map perspective/philosophy

Joe Moeller (Jul 29 2021 at 13:26):

In category theory, we have this perspective that we should focus attention on maps, on the relationships between objects, rather than on the objects themselves. What's your favorite examples of people giving a schpiel about this? Blog posts, snippets from books or papers, MO/SE answers, or even just giving your own take right now, are all welcome.

Simon Burton (Jul 29 2021 at 13:28):

Verbs over nouns.

Henry Story (Jul 29 2021 at 13:29):

I don't have a favorite one, but the The Yoneda Perspective is very well written.

Chad Nester (Jul 29 2021 at 13:51):

My take: The fact that there is a single-sorted presentation of the theory of categories, in which there is a sort of arrows but no sort of objects, tells us that objects are superfluous. What really matters are the source and target maps. The image of the source and target maps (they have the same image) in the single-sorted presentation corresponds to the collection of identity maps (and thus of objects) in the two-sorted presentation. Two arrows are composable in case the source of the first is the target of the second. It is convenient and, at least for me, conceptually simpler to imagine that the source and target of an arrow are "objects", but the only thing that's "really there" is the corresponding identity map.

Chad Nester (Jul 29 2021 at 13:56):

I've just noticed that in the single-sorted version the identity maps are precisely the fixed points of the source (equivalently, target) function. Neat!

Henry Story (Jul 29 2021 at 13:57):

I was going to mention those single sorted version of CT. It is not used much, but I think it gives one an interesting insight into Heraclitus: everything is change, "one never steps in the same river twice".

Henry Story (Jul 29 2021 at 13:58):

Chad Nester said:

I've just noticed that in the single-sorted version the identity maps are precisely the fixed points of the source (equivalently, target) function. Neat!

Where did you find that written out?

Chad Nester (Jul 29 2021 at 13:59):

I don't know if anyone's written it down.

Jon Awbrey (Jul 29 2021 at 14:04):

Joe Moeller said:

In category theory, we have this perspective that we should focus attention on maps, on the relationships between objects, rather than on the objects themselves. What's your favorite examples of people giving a schpiel about this? Blog posts, snippets from books or papers, MO/SE answers, or even just giving your own take right now, are all welcome.

My first “abstract algebra” course in college (U of Mich, 1970), the last project our instructor assigned us was to “do something creative”, a piece of creative writing, painting, sculpture, or other objet d'art, reflecting on one of the topics covered in the course.

I wrote a science fiction story about two species of creatures, the Sets and the Mappings.  No way I can remember all the details but I recall it explored a theme of duality between the two forms of life and the way ideas about “things in themselves” evolved over time into ideas about “that which changes into itself”.

Jon Awbrey (Jul 29 2021 at 14:20):

Buckminster Fuller's I Seem To Be A Verb was in the air about that time.  Don't know whether I had read it before I wrote my story or not, but I had been reading some category theory by then.

Spencer Breiner (Jul 29 2021 at 15:46):

For me, the easiest way to understand this perspective was to understand why arrows are generalized elements. An example that I find easy for most people to understand are elements $x:\mathbb{R}\to X$, viewed as time-varying elements $x=x(t)\in X$.

Morgan Rogers (he/him) (Jul 29 2021 at 15:59):

Chad Nester said:

I've just noticed that in the single-sorted version the identity maps are precisely the fixed points of the source (equivalently, target) function. Neat!

This is 1.13 in Freyd and Scedrov's Categories, Allegories, if you ever want a reference.

Jon Awbrey (Jul 29 2021 at 17:28):

Spencer Breiner said:

For me, the easiest way to understand this perspective was to understand why arrows are generalized elements. An example that I find easy for most people to understand are elements $x:\mathbb{R}\to X$, viewed as time-varying elements $x=x(t)\in X$.

The analogy between $\mathbb{R} \to X$ and $\mathbb{B} \to X$ where $\mathbb{B} = \{ 0, 1 \}$ is one of the things differential logic is all about.

See, for example, An Interlude on the Path.

Henry Story (Jul 31 2021 at 11:16):

I was just reading the famous Theorms for Free by Philip Wadler, as there is some trickery with having Sets be functors in Scala. Anyway, in on page 4 of Wadler's article, right after a short presentation of a naive model of programming languages as sets and functions, Wadler introduces an alternative where Types are relations. The Type Int would be the identity relation to itself. The type bool the identity relation to itself. I am not sure what the morphisms are then, but he refers to Freyd. Looking for references on ncatlab I found a long post by Mike Shulman Scones, Logical Relations and Parametricity that seems relevant, but I did not want to know quite that much right now.
In any case we have here relations made into primary objects to explain an important programming feature. I am not sure what to make of it yet.

Jon Awbrey (Jul 31 2021 at 12:16):

Henry Story said:

In any case we have here relations made into primary objects to explain an important programming feature. I am not sure what to make of it yet.

Dear Henry,

This is one of my oldest lines of inquiry, as Peirce naturally gets a person to thinking of relations as the primary things, and I made some progress with it in '84–'85 when I asked the question, “What Should a Relational Arrow Be?”

I know I wrote something about this on the web somewhere ...
but all I can find right at the moment is this bit from my files.

MOF -- Was sind und was sollen : relational arrows?

Question

I can remember asking myself this question sometime in the early (19)80's.
I worked on it a little more in the middle of that decade, as I was passing
through Champaign-Urbana, taking a course from John Gray on "Applications
of $\lambda$-calculus" or some such thing, and thinking a lot about the
mutual applications of category theory and computer science to each other.
I appear to be doomed to return to the question from time to time, at least
for the time being.

I started recording a few notes of what I can recall of my own approach at
the $n$-Lab wiki, so maybe that will serve as sufficient primer to get the
canon-ball moving:

• Relation Theory

Yes, I already tried walking toward the light, but the light is too fast for me to $c$.

http://mathoverflow.net/questions/6590/was-sind-und-was-sollen-relational-arrows

The “MOF” tells me it was a question or answer on MathOverFlow but it appears to be disapparated now.

Regards,

Jon

Jon Awbrey (Jul 31 2021 at 12:24):

Okay, the link to the $n$-lab wiki archive/history is still live in a zombie-ish sort of way, so here's what's there.

• Relation Theory

Jon

Jon Awbrey (Jul 31 2021 at 13:48):

Here's the first part of that …

Relations have types.
Types are functions.
Functions are relations.

Idea

To the best of my re*collection, it went a bit like this:

Was sind und was sollen —

• relational arrows?

In other words:

What Thing (1) would correspond to a Relation as an Arrow corresponds to a Function?

• Let’s call Thing 1 a Relational Arrow.

What Thing (2) would correspond to a Relational Arrow as a Category corresponds to an Arrow?

• Let’s call Thing 2 a Relational Category.

What Thing (3) would correspond to a Relational Category as a Metagraph corresponds to a Category?

• Let’s call Thing 3 a Relational Metagraph.

So it seems a relational metagraph ought to consist of objects $a, b, c, \ldots,$ relational arrows $f, g, h, \ldots,$ and a number of operations $\mathop{dom}_j$ assigning to each relational arrow $f$ an object $\mathop{dom}_j f$ for each $j$ in the arity of $f.$

Et sic deinceps …

Jérémie Koenig (Jul 31 2021 at 18:43):

Henry Story said:

I am not sure what the morphisms are then, but he refers to Freyd.

The usual thing to do would be to take functions between the underlying sets such that $x \mathrel{R} y \Rightarrow f(x) \mathrel{S} f(y)$. If you're using partial equivalence relations (symmetric and transitive), you can think of the self-related elements as "well-behaved", and think of the equivalence classes as your "points", and in particular you get a function space using the relation $f \mathrel{[R \rightarrow S]} g \Leftrightarrow \big(\forall x y \cdot x \mathrel{R} y \Rightarrow f(x) \mathrel{S} g(y) \big)$.

Jon Awbrey (Jul 31 2021 at 19:40):

You guys keep flashing me back to unfinished projects from the 80s !!!
Here's one I posted bits of to my blog a while back, copying ASCII graphics from some old file ...

Cf: Notes On Categories • 1

Continued from “Notes On Categories” (14 Jul 2003) • Inquiry List • Ontology List

NB. This page is a work in progress. I will have to dig up some still older notes from the days of pen and paper before I can remember how I left things last.

Here are some notes on a computational approach to category theory I started working on back in the 1980s, all of which work as yet remains in the “Schubert Category” of unfinished symphonies.

It helps me a little bit to write the names of categories in the plural, so as not to confuse them with individuals. It also helps if I treat the arrows of Arr(C) as the primary entities in the category C, recovering the objects of Obj(C) as secondary entities by collecting all the entities that appear in s(f) = Source(f) and t(f) = Target(f) as one ranges over all of the arrows f in Arr(C).

The last time that I tried to do “categories by computer”, I was using data structures that had the following shapes ...

I'll redo the graphics and get back to it ...

Jon

Keith Elliott Peterson (Aug 08 2021 at 03:15):

The arrow perspective tells us that we can't ignore variation through time and/or space, as opposed to the set-theoretic perspective where everything is seen as static entities.

As someone that lives in a dynamic universe where things can vary, I'd say that is a big deal.