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

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

I just thought of sharing with you all that Professor Johnstone, in his "Sketches of An Elephant", appears to be blind to the moral of the 'four blind men and the elephant' story (from which he borrowed the title of his book).  I have written to him, but didn't receive any reply.  I also shared my manuscript in which I spell-out his faulty reading of the story, but didn't receive any reply.  Sometime later, I published it: Gandhi's Satya: Truth entails Peace.  Having benefitted from his helpful answers to a number of my questions (e.g., subobject classifier, Yoneda lemma, universal element, natural transformation) over the years, I find it puzzling that it can be impossibly difficult to admit a minor misreading (or contest it).  This is all the more shocking in the context of Grothendieck insisting that Professor F. William Lawvere keep all the mistakes that Grothendieck made in his feverish writing (see pp. 7-8 in FAREWELL TO AURELIO).  Also, Professor F. William Lawvere wrote to me quite a few times that he made a mistake in one of his earlier emails during our quarter-century correspondence.  But, Professor Johnstone is in the good company of Nobel laureates like Kahneman.

I could be wrong, but given the prevailing vigilante-mode of moderation and editorial decisions, one way to document dissenting voices is to submit to a major journal and as soon as you get a manuscript number submit to an archive (we can't find fault with moderators/editors, but give them enough reason for approving for which they have to answer).  I remember reading John Maddox's editorial, accompanying Hoyle-Narlikar steady-state universe paper, in which he openly stated that he, as the Editor-in-Chief of Nature, decided to publish the steady-state universe paper without external review because he felt, given the popularity of Einstein, it would be next to impossible to get a fair review.  Now it all seems to have gotten worse, with editors begging pardon for removing, without informing the authors, published unpalatable online comments in the name of publishers' decision to reformat the journal (e.g., https://disqus.com/home/discussion/cell-press/dialogue_across_chasm_are_psychology_and_neurophysiology_incompatible_neuron/, https://disqus.com/home/discussion/pnascomments/opinion_compound_risks_and_complex_emergencies_require_new_approaches_to_preparedness/).  On a related note, one of my coauthors' paper questioning one of the gazillion mistakes of Einstein wasn't allowed to be uploaded to the physics preprint archive.

Summing it all, dissent is difficult (and not invariably correct ;)

Thanking you, yours truly, posina

p.s. i thought of suing Cambridge University for being party to cultural misappropriation (but i remembered losing a silly lawsuit i filled against my university, so I had to let go off the millions I could have bagged in compensation for the emotional trauma induced by Professor Johnstone's misreading ;)

I would like to share with you a paper 'The Constituents of Sets, Numbers, and Other Mathematical Objects: Part One' of my good friend The Prophet of Köningsberg in which he constructed a category of constituent structures (sans cardinality).  As you may know, Cantor, in abstracting his Kardinalen of lauter Einsen, seems to have intended it as a first step towards getting at qualities (see Footnote 4 on p. 11 in Cantor's iauter Einsen).  Be that as it may, Bourbaki's deduction of structures (with structures as geometric objectifications of conceptualizations of qualities) can be considered as inaugurating the scientific study of qualities (Lawvere & Schanuel, Sets for Mathematics, p. 240).  Also, note that Professor F. William Lawvere in his seminal paper Axiomatic Cohesion explicitly states that he looks forward to the development of "categories of constitutive relations" for the express purpose of studying the constituent relations of cohesively varying bodies all around us directly in terms of their qualities as a refinement of the contemporary quantitative approximations of the objective qualities (https://conceptualmathematics.wordpress.com/2012/06/04/perugia-notes-prof-f-w-lawvere/; https://zenodo.org/records/7164047. p. 2; https://conceptualmathematics.wordpress.com/wp-content/uploads/2013/02/spacesanddatatypes.pdf, p. 105).

In the light of all of the above, am I mistaken in thinking that mathematics is about qualities (along with their conceptualizations), while most mathematicians along with the rest of the humanity think mathematics is all about quantities (and calculations)?  If so, is there a need for a math outreach (a' la Galileo's physics outreach)?

Thanking you, yours truly, posina

In the early 1960s there was a mathematical advance, an advance on par with Newtonian mechanism in physics and Darwinian evolution in biology. A mathematical theory, prior to F. William Lawvere’s Functorial Semantics of Algebraic Theories (Lawvere, 1963/2004/2013), was a list of statements, which together determined whether a given object is this or that. So, a theory of a universe of discourse, say, the category of graphs (consisting of dots and arrows), had no choice but to leave the given universe for one, with no readily discernible kinship with graphs, of arbitrary symbols, words, and sentences, i.e., language. Following Lawvere’s functorial semantics, a theory of a given category of objects is a category with their basic properties as objects and mutual determinations of properties as morphisms (Lawvere, 2003; see also Posina, Ghista, and Roy, 2017). Along with the functorial semantics of Lawvere, sketches of Bastiani and Ehresmann (1972), and Grothendieck’s descent (see Clementino and Picado, 2007/2008, p. 15) contributed to the monumental development of our mathematical understanding of mathematics, wherein the relationship between particulars, theory, models, presentations, and doctrine is spelled out in a spellbinding display of science: ever-proper alignment of reason with experience.