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How can I find a list of open problems in Homotopy Type Theory and Univalent Foundations ?
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A list of open problrems for hpmotopy type theory is presented in HISTORIC.
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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 ?
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Hello, thank you for your reply.
The general idea is that all vertices are "essential" in the sense that removing a vertex will effect the semiconnectivity structure of the rest of the digraph. So I would say condition (1) captures this notion the best.
Thank you for the observation that there can be no source vertices.
There also can be no sinks (all indegrees).
An essentially interconnected graph need not be weakly connected (just take the disjoint union of cycles). Such examples are trivially also not strongly connected.
I think I just found an example of a weakly connected essentially interconnected graph which is not strongly connected:
Let the vertices of G be a,b,c,d,e,f and the directed edges be
(a,b),(b,c),(c,a) , (d,e),(e,f),(f,d) , (b,d)
It seems that G does satisfy condition (1) but is is not strongly connected as there is no oriented path from d to b.
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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).
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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.
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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.
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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:
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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.
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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!!!
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@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.
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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?
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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.
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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!
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 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) 
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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.
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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
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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?
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@ 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.
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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!
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You can also study this article 
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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)?
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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.
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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.
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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.
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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. 
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Thanks, Tahir. 
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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.
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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.
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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?
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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.
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If no - when it can and can not exist?
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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.
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For example Ivan Kolar worked on this issue but on functeurs with general fibers.
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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.
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I would like an explanation first in terms of physics (quantum field theory) and second in category theory (Fock functor).
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@Jared: Thank you very much, I'll study the paper.
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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.
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@Costas, oh YES! Here is a relevant paper on that connection by Albert Lautman: Dialectics in mathematics!
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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?
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I sometimes use Formal Concept Analysis to describe, how people use implicit ordering (sub/super-concepts) of their concepts. Disjunct and exhaustive "categories" (as a binary relation between subjects and attributes, S "has" A) are a prerequisite to detect graphical "scales" among concepts.