Category Theory - Science topic

Further support:

"This particularly elegant theorem shows the inverse function relationship of the derivative and the integral and serves as the backbone of the physical sciences"(Brittanica 2023).

Image belongs to Brittanica(I added the highlight)

Britannica, The Editors of Encyclopaedia. "fundamental theorem of calculus". Encyclopedia Britannica, 29 Jul. 2023, https://www.britannica.com/science/fundamental-theorem-of-calculus. Accessed 20 September 2023.

Thank you, Andrew for your contribution. I see categories ass a sort of refined logic, where you distinguish several ways of getting from A to B. In so far logic is concerned you also would like some features like products (conjunction) co-products (disjunction) and cartesian closedness for implication. Well this is quite general and is certainly important if you speak of dialectics in logic.

To state it in a different way: Logic is about 0-categories and ordinary categories are 1-categories. Going further, and as far as one is interested in n-categories the notion of dialectics gets a new flavor, reminiscent of algebraic topology. To formulate it in an approximate way, one player would have an n-simplex to defend and the other(s ?) would try to detect the weakness.

I believe your sentiment is that Category Theory is so general that it might if applied to physics provide an insight into the overarching principles of the science. But Category Theory is not going to give you any insight that you do not already possess. It might provide a convenient notation for expressing that insight. Using Category Theory to navigate physics without a physical understanding would be like sailing a yacht without a keel.

The idea of diagonalization in operad theory has been studied in the literature, although it is typically formulated in terms of "partial compositions" rather than "variable identification" as in your proposed Dij operation.

One approach to diagonalization in operads is to define a "partial composition" operation that takes two elements of an operad P and produces a new element by composing them along a diagonal. More precisely, given elements f ∈ P(m) and g ∈ P(n), we define their diagonal composition f ∘g ∈ P(m+n-1) as follows:

(f ∘g)(x1,...,xm+n-1) = f(x1,...,xi,...,xm,g(i-m+1),...,g(n))

where i is the unique index such that i-m+1 ≤ j < i for all j ∈ {1,...,m+n-1}.

This partial composition operation satisfies some important algebraic properties and has been extensively studied in the context of operad theory. However, it may not be directly applicable to your specific problem of identifying variables in an operad.

Another approach to diagonalization in operad theory involves the use of "modular operads", which are operads that allow for the composition of operations in a non-symmetric fashion. Modular operads provide a powerful framework for studying algebraic structures that arise in geometry, topology, and mathematical physics, and they have been used to study a wide range of phenomena, including Feynman diagrams, string field theory, and knot invariants.

In summary, while the specific operation you propose (Dij : P(n) -> P(n-1)) may not have been studied in the literature, there are related concepts in operad theory that may be useful for your purposes, such as diagonal composition and modular operads. I would recommend exploring these ideas further to see if they can be adapted to your specific problem.

A list of open problrems for hpmotopy type theory is presented in HISTORIC.

Hello Clarence! I believe that your question is answered in the paper 'Minimal strong digraphs' by Garcia-Lopez and Marijuan. Here is a link:

There is a view that if mathematical categories are kinds of mathematical structure, then what is important mathematically are the functors from one category to another, because they provide a means of find a neat way of discovering a new property in a category by translating proofs in another category. This is a way of formalising reasoning by "analogy". Personally I find reasoning about categories as abstract algebras difficult and unintuitive, and find it much easier to look at a concrete realisation of a category than considering a category with a list of pre-defined desirable properties; but I recognise that that is a matter of learning preferences.

hope this can help u a lil bit.

Of course it is well known that the category of R-modules is suitable for this setting. Let me add one more reference

"Continuous Geometry" by J von Neumann, Oxford University Press, 1960.

@Romeo P.G

For sure this is true. There are many examples from algebraic geometry. I often work with kinds of locally ringed spaces - think of just generalising the sheaf of smooth functions on a manifold -  and these are not `set theoretical objects' in the sense they are not simply a set of points (together with a topology or something like that). 

But of course, as ringed spaces morphisms can be defined as so can their category. The point is that the objects are not sets, though in the cases I deal with, the Hom sets are genuine sets.

This is good question.

A complete answer to your question is given in

The category of matroids

An overview of the categorical nature of matroids (Fig. 1 in the article) is shown in the attached image.

 The question contradicts the fundamental principle of thought process and one of the basic axioms of ontology, something either "is" or " it is not", but not both. Based on the definition of a set being finite or infinite, if a set is finite then it can not be infinite and if it is infinite then it can not be finite. No finite set that is equivalent to any of its proper subsets and hence can not be infinite. Besides, the set of natural numbers do not end or finished as you have said (if that is what you meant) 

Dear Professor Waldemar Koczkodaj, thank you for your ideas from your work.

The "math nails": when facing “infinite things” in present classical infinite related science theory system, no one is sure whether they are “potential infinite things” or “actual infinite things” and are not sure how to treat them scientifically------- more and more suspended infinite related paradox families are produced by the “confusing of potential infinite and actual infinite” such as infinitesimal relating paradoxes in analysis, infinity relating paradoxes in set theory, both infinitesimal and infinity relating paradoxes in the ideas and skills of “Cantor’s diagonal-contradictory proofs and the conclusion on “Real Number Set has more elements than Natural Number Set (infinite elements in Real Number Set is more infinite than that in Natural Number Set-------Infinite R is more infinite than Infinite N)”, and these paradox family members are surely unsolvable in present classical infinite related mathematics where they were produced and nourished.

Yours,

Geng

@ Daniel Baldomir

You put a nice question. We have no real evidence of the infinity in physics, as far as I know. But this is mainly because our observation possibilities are limited. This is true both for big distances as for small scales. Physically, the infinities remain potential. Also, our definition for some categories, like temperature, does not really make sense outside a given interval. We cannot speak about negative  absolute temperature, for example.

Shortly, I am not aware about direct physical applications of this Cantorian construction. But there are some indirect applications. Take the notion of volume of a body.  There is a geometric construction called Banach-Tarski paradox. One cand decompose a full spherical body in finitely many parts, move them rigidly in the space, and build with them two full spherical bodies, both of them equal with the body you started with. The parts have infinitely small details. This cannot be done physically, and so we have an indirect proof of the existence of atoms, which do not allow us any construction with infinitely small details.

Another interesting question is if the existing frequencies of light and electromagnetic waves do really create a continuum or not. (The question is if really for every positive real number in some interval there is a wave of this frequency or not. In mathematics, the cardinality of the real interval is bigger than then the countable cardinality...)

@ Genk: As I told also Daniel Baldomir above, almost nobody doubts about the fact that there are more real numbers (geometric points on a line) then natural numbers, although both sets are infinite.

You can also study this article 

This is a good question.

The classic overvew of categories is given in

J. Adamek, H. Herrlich, G.E. Strecker, Abstract and Concrete Categories.   The Joy of Cats, 2004:

See Section VI.21 on the topological categories, which is both detailed and helpful.  This book is also written in a witty way, which is also helpful for those who  are brushing up against this subject for the first time.   For example, the notion of lifting of a source in a topological category is represented by the attached drawling.

The "or" is quite important. It is not just a union datatype. Either[A, B] = Left[A, B] + Right[A,B]. In the case when M is Set it allows to build a Set whose members are Left[E] or Right[SetTree[E]]. This is more general than Set[E] + Set[Set[E]] + ...

In the case when M is a simpler monad like Option/Maybe you are right that you just get a chain of Some/Just wrappers of arbitrary length, and that's not very interesting, but it is a monad. I haven't worked out an intuition of what this means if M is the State monad.

Thanks, Tahir. 

If I understood the Iaroslav Karkunov's question correctly, when he speaks about isomorphism K/N1 = N2 he means the isomorphism of canonical form that maps an element of N2 to its coset \in K/N1. In Geoffrey Mckenna's example, the isomorphism does not look this way.

This is a good question without an obvious answer.

An almost mathematical approach to answering this question is the whimsical proposal of a category Reality introduced in

R.E. Lauro, Beyond the colonization of human imagining and everyday life: Crafting mythopoeic lifeworlds as a theological response to hyperreality, Ph.D. thesis, University of St. Andrews, Scotland, 2012:

file:///Users/jfpeters/Downloads/RenoLauroPhDThesis.pdf

See page 76, based on Baudrillard 1972 book.

On RG, see the works of S.A. de Groot:

In particular, see

S.A. de Groot, In search of beauty.  Developing beautiful organizations, Ph.D. thesis, Technische Universiteit Endhoven, 2014.

Splitting problem is an important problem in group theory. The answer of your question can be a paper!!

As a simple example, consider the subgroup Z (integer numbers)  of Q (rationales numbers). Since Q is divisible and Z reduced, so Q is not isomorphic to Z product Q/Z.

In the category of abelian groups, if N is a divisible subgroup of G, then the answer of your question is yes.

As a first iteration, I would try this http://dml.cz/bitstream/handle/10338.dmlcz/107982/ArchMathRetro_042-2006-1_7.pdf ... It took me to this one http://journals.impan.pl/cgi-bin/doi?ap82-3-6 where is a reference [6]. This might be what you are looking for. However, the fiber preserving only.  When more general, this should be mentioned in [6] (Kolar, Mikulski DGA 1999. I guess the biggest group working in this area contains Doupovec, Kurka, Miukulski and Kolar (they should be active in this field...as far as I know). Of course, it depends in waht extent functors you want to classify ... and how fine and explicit is the classification-- often thse are done using the algebraic (Weil algebra)  or vector space data.

@Jared: Thank you very much, I'll study the paper.

@Costas, oh YES! Here is a relevant paper on that connection by Albert Lautman: Dialectics in mathematics!

I think categorization is a mental process. Our cognition guide us to categorize any event based on our past experience, values, environment and most importantly the state of mind at that point of time. Richard Thaler in 1980 while defining mental accounting highlighted how people categorize incomes based on the source. Similar classification can be done from different dimensions. There can be any unique rule for categorization which will be acceptable for all.