Unless I am mistaken, the categories of natural transformations that have been studied are categories with natural transformations as morphisms (hence the two compositions: horizontal and vertical, along with their Godement compatibility condition, among others). In thinking of natural transformation / commutative square as a generalization of one too many particular structure-respecting maps of various categories (in the category of categories), what added insight(s), often associated with generalization, can we expect to gain into: 'don't tear' (objectified as commutative square)?

I'd be truly grateful to you for any pointers / suggestions you may have.

Thanking you,
Yours respectfully,
posina

On a related note, there is a mathematical magazine Bhāvanā, which means composition / concept / bringing into being, among others, depending upon the context.  It may be of some interest to note that unlike the trouble ivory tower mathematicians had in recognizing the primacy of concepts and symbolic languages as secondary constructs used to present concepts for the purpose of calculations, a high school dropout / movie hero / politician (Pawan Kalyan) in my state of Andhra Pradesh is on a mission highlighting the significance of concepts and thinking above and beyond that of languages used to express concepts and ideas for the purpose of communication.

On a not too unrelated note, I couldn't help but browse through every issue of Bhavana hoping to find an article or two about Grothendieck, given that concepts figured prominently in the work of Grothendieck, but alas not one (or I missed them).

On a somewhat relatively distant note, I was amused to find a series of articles on beauty (and math, of course) by Langlands; it's mildly amusing to see many mathematicians lost in thought pondering beauty, but there are four definitions of BEAUTY, which are abstractions from everyday experience and planned perception (that appear to be missing in action ;)

Definition I. Beauty is mostly an average (evocative of pleasantness), with a sprinkling of outliers (needed to draw our attention).  This definition dates back to the perennial human pursuit of criminals before they commit crime; a clever police officer took a ton of pictures of known criminals and averaged them to identify the tell-tale features of criminals only to find the average criminal handsome (or so the story goes ;)

Definition II. Beauty is figure-sans-ground, with figure defined as that part of the image wherein small changes are readily perceived vis-a-vis the changes in the part of the image constituting background.

Definition III. Disappearing into the appearance of beauty, which must be all too palpable to those of us still in college.

Definition IV. Putting together those that fit together. 

I hope all is well.

As you all know, following Samuel's Universal Mappings, Mac Lane interpreted product (A x B of factors A and B) as a universal mapping property (pp. 489-490), i.e., 1-1 correspondence between (pairs of) maps to the factors T --> A x T --> B and maps to the (corresponding) products T --> A x B, which is what we use to calculate products (pp. 6-7) in a category, with T as basic shape(s; yes, there is more, but skipped to get to the main point).  Isn't this 1-1 correspondence actually a part of the isomorphism: A^T x B^T ~ (A x B)^T?  To begin with, map objects, unlike those in the case of sets, involve nontrivial calculations (e.g.,. A^A, where A is the basic shape of arrow (* --> *) in the category of graphs), leave alone establishing isomorphisms.  Be that as it may, does this mean algebraic concepts defined as universal mapping properties are as much unique as they are isomorphic?  Even if it all sound word salad, there is still the question of how the 'for every' and 'there exists' in universal mapping definitions relate to isomorphisms.  In this context, one is reminded of 'there exists' left adjoint of 'substitution / composition' left adjoint of 'for every' (p. 246), except for the uniqueness of existence as in 'there exists a unique map' in universal mapping definitions.  Once we acknowledge isomorphisms in the universal mapping property definitions of category theory, then aren't categorical constructs within the confines of the category of groupoids?

I look forward to your corrections!

Thanking you,

Yours respectfully,

posina

I hope all is well.

Upon reading Professor F. William Lawvere's discussion of Grothendieck's definition of subobject:

"It became clear in the early sixties that the definition of SUBOBJECT given by Grothendieck is not a pretense, circumlocution, or paraphrase, but the only correct definition.  Here 'correct' means in a foundational sense, i.e. the only definition universally and compatibly applicable across all the branches of mathematics:

a subobject is NOT an object, but a given inclusion map.

The intersection of two objects has no sense, for only maps (with common codomain) can overlap" (https://github.com/punkdit/categories/blob/master/www.mta.ca/cat-dist/archive/1996/96-3 lines 2756-; also attached, pp. 4-5).

I am very much enthusiastic about studying the original paper / book, where Grothendieck redefined subobject.  I'd be truly grateful to you for your insights into the context (problem / theory) that motivated Grothendieck's redefinition of subobject.

Thanking you,

Yours truly,

posina

Going by 'X is what X does' or, in terms of our universal mapping properties, X is what X is good for (which may be operationalized as: X is that which wouldn't be but for X), biology is, if any, not like physics, which appears hellbent on killing us without doing anything, so to speak.  Biology goes to great lengths spending enormous amounts of energy establishing the gradients (e.g., ionic gradients across neuronal membranes), but for which we would be brain dead.  Physics, uses the very gradients to dissipate the gradients (cf. water flows downhill) and biology fights back again and again until it gives up, and physics declares victory in equality embodied in our dead body.  There is more to the primal war of the worlds (physics vs. biology), which I can elaborate on in terms of pieces/connected components and souls (yes, I said soul; objects have souls, according to Professor F. William Lawvere), and with the express purpose of establishing the evil that is physics ;)  I must hasten to add that physics may not be doing anything out of the ordinary in the sense of converting everything into one of its kind (dead/pure matter) as if straightening the crooked timber of humanity.

Given that I can no longer resist the call of morning coffee, I am rushing to wind it up: unlike physics, with its shortest distance paths and minimum energy configurations, mathematics, more like me, takes the most convoluted route, which may have something to do with their nefarious actively concealed agenda of appearing to be more smart than they are (cf. https://conceptualmathematics.substack.com/p/crooked-timber-of-mathematics).

Summing up, we have two dualities

(i) Physics vs. Biology

and

(ii) Mathematics vs. Physics

within a trinity: Biology, Mathematics, and Physics.

Thanking you, yours truly, poison venkata rayudu

Refer to this article on my blog (written in French) 

https://www.entropologie.fr/2024/05/schema-de-presentation-au-cle-du-22/05/2024.html