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How can I find a list of open problems in Homotopy Type Theory and Univalent Foundations ?
I call a digraph G= (V,E) essentially interconnected if whenever any vertex $a$ is removed from $G$ there is always at least one pair of distinct vertices $v$ and $w$ which can no longer be joined by an oriented path in $G$.
Are there essentially interconnected graphs ?
Example: The cycle: (a,b), (b,c),(c,a)
If I remove $b$ then I cannot connect $a$ to $c$ (although I can connected $c$ to $a$) and analogusly for $a$ and $c$.
Can we characterise such digraphs in general ?
1) There is some tradition in philosophy of mathematics starting at the late 19th century and culminating in the crisis of foundations at the beginning of the 20th century. Names here are Zermelo, Frege, Whitehead and Russel, Cantor, Brouwer, Hilbert, Gödel, Cavaillès, and some more. At that time mathematics was already focused on itself, separated from general rationalist philosophy and epistemology, from a philosophy of the cosmos and the spirit.
2) Stepping backwards in time we have the great “rationalist” philosophers of the 17th, 18th, 19th century: Descartes, Leibniz, Malebranche, Spinoza, Hegel proposing a global view of the universe in which the subject, trying to understand his situation, is immersed.
3) Still making a big step backwards in time, we have the philosophers of the late antiquity and the beginning of our era (Greek philosophy, Neoplatonist schools, oriental philosophies). These should not be left out from our considerations.
4) Returning to the late 20th century we see inside mathematics appears the foundation (Eilenberg, Lavwere, Grothendieck, Maclane,…) of Category theory, which is in some sense a transversal theory inside mathematics. Among its basic principles are the notions of object, arrow, functor, on which then are founded adjunctions, (co-)limits, monads, and more evolved concepts.
Do you think these principles have their signification a) for science b) the rationalist philosophies we described before, and ultimately c) for more general philosophies of the cosmos?
Examples: The existence of an adjunction of two functors could have a meaning in physics e.g.. The existence of a natural numbers - object known from topos theory could have philosophical consequences. (cf. Immanuel Kant, Antinomien der reinen Vernunft).
Fix a category D, a site (C,J), an object d of D. Assume that:
a1) C is a subcategory of D;
a2) d is not an object of C;
a3) both D and C have all limits and colimits;
a4) is given a collection K of arrows of D with codomain d and domain in C.
I am looking for references about the construction of a site (C', J') with:
1) any object of C is also an object of C';
2) J(c) is contained in J'(c) for any object c of C;
3) d is an object of C';
4) any arrow of the collection K belongs to a sieve of J'(d);
5) C' has all limits and colimits;
6) if (C'',J'') is another site fulfilling 1-5 then there is one and only one functor F:C'--->C'' which commutes with inclusion of C in C' and C'', F(d)=d, F(f)=f for any arrow f in K.
Since J. von Neumann physicists stick to categories of Hilbert spaces to modelize quantum phenomena. Categories of modules over a ring might represent an alternative if we add axioms (e.g. the existence of particular limits or co-limits) that would respond to the experimental requirements.
A very general setting for the purpose would be abelian categories. Have there been attempts to make use of them?
References:
Most of the modern texts use category theory for algebraic topology rather than set theory. What are the pros and cons of both the set theory and the category theory in this formulation.
Moreover, is it necessary to use Category theory or set theory suffices for all the concepts.
Thanks in advance!!!
Can anybody give me a tip of a text that treats about Matroids and maps between, I mean Category of Matroids, which relates to the Universal Property definition in the sense of categories?
Also, has anyone heard about the Category of Matroids over Ring, either have a clue of how to define its structure?
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.
It is true that during the “one-to-one correspondence” operation between Real Number Set and Natural Number Set, after the elements in Natural Number Set have been finished up, the elements in Real Number Set are still a lot (infinite) remained:
1, The elements in real number set are never-to-be-finished, endless, limitless------Real Number Set is really infinite!
2, The elements in natural number set are sure-to-be-finished, ended, limited-------- Natural Number Set is actually finite?!
There are still some other proofs of “one-to-one correspondence” operation between the two sets telling us a fact that the elements in many infinite sets are sure-to-be-finished, ended, limited and they are actually finite！
A typical tool and technique is Cantor's Power Set Theorem: all the elements in any infinite set can be prove “sure-to-be-finished, ended, limited and they are actually finite” in front of its own Power Set-------because during the “one-to-one correspondence” operation between the original set and its power set, after the elements in the original set have been finished up （finite）, the elements in its own Power Set are still a lot (infinite) remained!
Set theory and analysis have brought us human tremendous achievements in our science, but so many suspended “infinite related paradox family members” strongly proved that there have been fatal defects in our “ontology-form” cognitions to "infinite" at least since Zeno’ time 2500 years ago ---------the fatal mistakes in the foundation of present classical infinite related science theory system.
2500-year is like a filliping in the long history of human beings, it is the right time for us human to integrate our works（achievements and faults）and look for a way out.
Our studies have proved that discard the contradictory, confusing, mistaken contents of “potential infinite and actual infinite” in the foundation of present classical science system and study and develop systematically all kinds of “infinite carriers” is the only way out; this is a step we should take sooner or later beyond our will. A newly constructed "infinite theory system" with newly discovered but having been long existing virgin land--------“the field of infinite carriers”, is waiting to be pioneered and developed.
If someone says: “Set A has limitless (boundless, endless) elements, so we say Set A is an infinite set; but if Infinite Set B has more elements than Infinite Set A, we say Infinite Set B is more infinite (more limitless, more boundless, more endless) than Infinite Set A”, what do we think?
If someone says “apple is fruit, banana is more (much) fruit than apple, pineapple is more more (much much) fruit than apple, pear is more more more (much much much) fruit than apple,… in our daily life, what do we think?
Can we really have quantitative cognizing idea of “mixing up ‘potential infinite’ and ‘actual infinite’ to create many different definitions for 'infinite' in our mathematics”?
Cantor is really a great mathematician who opens up a new field of quantitative cognitions to “infinite things” within set theory--------it is a must for us human to have quantitative cognitions to “infinite things” mathematically, but is it necessary and possible to improve and develop his work?
There is a more detailed post on this project here: https://www.researchgate.net/post/What_are_the_proper_techniques_for_analyzing_reliability_of_categorical_variables_with_a_large_number_of_categories, but I wanted post a new, more specific question in a new thread.
For those of you used to looking at I&O code reliabilities (or using I&O codes in substantive research), what levels of agreement/reliability are acceptable to you? Or, what levels have you commonly seen? I'm using Census, NAICS, and SOC codes, and seeing ARs and kappas between mid-high 0.70s and mid-high 0.80s. Do those seem high, low, or average? This is my first time working with I&O data.
Thanks in advance!
This question is for specialists in Category Theory. Can anybody find some kind of index (or atlas) of so-called topological categories in the literature (or internet)?
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.
For a particular application area, I found a need for a "monadic tree", defined essentially as follows:
Let M be a monad (as in functional programming).
Let an MTree[E] be the least fixed point (in N) of N[E] = M[E or N[E]], where "or" is shorthand for an Either bifunctor.
For example, if M is Set, then SetTree = Set[E or SetTree[E]] is a model of a tree of arbitrary, but finite, depth and arbitrary (possibly infinite) breadth where there is no order among branches, and all the leaves are of type E.
I have not found any mention of this monad among the references I am familiar with, or through googling. I needed to be sure it was a monad, so I wrote a proof of the monad laws for myself[1] and also tested the monad laws through some implementations [2]. But I find it hard to believe that no one has created this monad before. Is there a publication already in the literature proving that this recursive definition does generate a monad from any other monad?
Also, my proof is only applicable to M-trees of finite depth, and I am curious about the infinite depth trees that could be generated from this recursion. But I am not familiar with the category-theoretical methods that might be used to investigate it.
The specific context is industry and occupation codes. These are 6-digit codes that are hierarchical and categorize a person's industry and occupation based on what they report to an open-ended survey question, and there are hundreds of codes total (i.e., hundreds of categories). We are comparing a set of human-coded data against an electronic/automatic coding system. I've Googled and found some good articles on reliability, but nothing (at least nothing recent) addressing a method for checking the reliability of these codes or variables with many codes in general. I'm going to get Fliess's book from the library since I recall that being a helpful one in the past.
Here's the problem as I see it, and some suggested approaches. All comments welcome. Thanks!
1) Since we have hundreds of categories, and particularly because the codes have a hierarchical structure, a simple agreement rate or kappa will be low simply because of the number of categories and they way they are applied. Minor mis-codings or unreliability in the latter digits will throw of the overall agreement/reliability. For example, the code "lawyer" might be reliable, but the code for the type of lawyer may not be. An overall analysis would result in lower reliability, even if the first digits of the codes are reliable.
2) In addition, some of the categories are likely used frequently and some infrequently if at all. Something tells me this will be a problem that will attenuate overall reliability, but I can't express it any better than that at this point.
3) My first thought is to check the reliability on each digit of the codes (or combination of digits depending on how the codes are applied by coders, eg., two- or three-digit chunks). I have to learn what their structure is but don't have the data right now. I believe they are NIOCCS codes (http://www.cdc.gov/niosh/topics/coding/overview.html).
4) My second thought was to split up the data by "job type" (i.e,. the first level of coding) and look at reliability within job type. Similarly I could look at reliability for "professional" v. "trade" jobs if I can find a key for coding the NIOCCS codes into broad classes of jobs.
5) Finally, my colleagues are talking about doing a "concordance" analysis (must be a public health term). From what I can tell this is just an agreement rate. I'm familiar with kappa and weighted kappa, but not with techniques where there are so many categories. I found the irr R package (http://cran.r-project.org/web/packages/irr/irr.pdf) and read the description of each technique it has but didn't see any specifically for large numbers of categories.
Thanks for your thoughts and leads.
Let K - group with normal subgroups N1 and N2. Intersection of which is trivial N1 \cap N2 = {e_k} and quotient groups of K by those normal groups are isomorphic to: K/N1 = N2, K/N2 = N1.
Is it true that in this case K = N1*N2? Where '*' gives us all posible products of N1 and N2 elements.
I am investigating the use of a mathematical Category Theory to explore a deductive model of the emergence and evolution of cooperative structures in human organizing. I am aware of work by Ehresmann & Vanbremeersch. Is there other related work or some alternative formulations?
If no - when it can and can not exist?
For example Ivan Kolar worked on this issue but on functeurs with general fibers.
I would like an explanation first in terms of physics (quantum field theory) and second in category theory (Fock functor).
Considering classical two-valued logic, it seems that there is place for dialectics in mathematics. In order to have a chance for that we should consider Many-valued Logic. However even without introducing Many-valued Logic, we may apply dialectics in some mathematical constructions.
Beyond this there is Category theory, a kind of theory of “variable elements”, where we may consider dialectics, in the sense of Lawvere. In the attached paper by J. Lambek, you can find a description of the categorical concept of dialectics. There are some six examples of concepts (e.g. convex hull, etc.) which they result as a synthesis of a dialectical scheme.
I am interested in categorisation, category formation and how we use categories in our daily lives: both positively and negatively. Why are other people interested in categories?
