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Aug
6
comment Notation about commutative diagrams and their vertices
@RobArthan But I need to explicitly refer to different vertices in my proof text. Well, maybe I should just say like "top right" node of the square diagram?
Aug
6
comment Notation about commutative diagrams and their vertices
Well, what's about $0[A]\overset{f}{\leftrightarrow} 1[A]$ or $A[0]\overset{f}{\leftrightarrow} A[1]$?
Aug
5
comment Help to write a proof (category theory diagram)
What about the similar question for a square graph?
Aug
5
comment Help to write a proof (category theory diagram)
@PVAL Also this proves only that cycles are identities, but we need to prove also that the diagram is commutative
Aug
5
comment Help to write a proof (category theory diagram)
@PVAL But how to show that every cycle is composed from cycles of the length no more than 3?
Aug
5
comment How do we prove commutativity of a diagram?
@RobArthan I haven't asked whether "a technique for dealing with the infinite" exists. I ask about a particular technique tailored with category theory diagrams, which I can use to prove particular results for a specific diagram
Aug
5
comment How do we prove commutativity of a diagram?
Isn't referring to a particular diagram as planar "proof by a picture"? How to show that a diagram is planar without writing actual X-Y equations of the curves which serve as graph edges?
Aug
5
comment How do we prove commutativity of a diagram?
@DanielFischer What is "cell"? Why is this enough?
Jun
28
comment Duals of filters, an explicit formula for meet?
After some thinking, I conclude that it seems that there is no explicit formula in this case
Jun
27
comment Duals of filters, an explicit formula for meet?
@AsafKaragila You've misunderstood. Ideal is a filter in dual order. But I take both dual order (that is replacing every element of the filter with its dual) and complement of the filter (considered as a set)
Jun
16
comment Characterization of monovalued functions
For every $x$ there are no more than one $y$ such that $(x,y)\in f$
Jun
16
comment Characterization of monovalued functions
$f=\varnothing$ is a function. I define function as a monovalued (including zero-valued) binary relation
Jun
16
comment Characterization of monovalued functions
I did a stupid thing: I got a 100 points bounty for this easy question. I've solved it myself soon after this. For a solution consider $G=\{\{(a;y)\};\{(b;y)\}\}$ for $a\ne b$
Jun
14
comment Labeled commutative diagram
Another question is how to paint such a diagram. For every node we need both an object and a label. Two symbols can't be in one place
Jun
14
comment Category theory: Enough that polygonal diagrams commute
@Unit "pentagonal" was a typo. Corrected
Jun
14
comment Prove that all cycles are identities
I want a more detailed proof
Jun
13
comment An alternate definition of ideals
It seems that we need an additional condition: $P$ contains dual poset of each element of $P$
Jun
13
comment An alternate definition of ideals
a key to this proof is decomposition $\theta = \operatorname{dual} \circ \omega$ where $\omega$ is an order isomorphism on $P$. It remains to fill proof details. It is easy to show that $\omega[\mathfrak{F}] = \mathfrak{F}$, but we need stronger formula $\omega[\mathfrak{F}\cap P] = \mathfrak{F}\cap P$
Jun
2
comment Categories of $n$-ary relations?
Relations (not only binary relations) form a "category with star morphisms" (over $\mathbf{Rel}$) as I define them in my book: mathematics21.org/algebraic-general-topology.html - I suspect that I am the first person who explicitly defined categories with star morphisms, as they are important for my research of products of morphisms.
May
23
comment About elements of a poset
Counterexample: mathematics21.org/binaries/star-comparison.pdf (suggested by sci.math people)