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Dec
18
comment Is a set closed under finite intersections? (about filters)
@KarlKronenfeld: Due funcoid magic $f_1\in\operatorname{GR} ( \Delta \times^{\mathsf{FCD}} \Delta) \Leftrightarrow f_2\in\operatorname{GR} ( \Delta \times^{\mathsf{FCD}} \Delta)$ and moreover $\operatorname{GR} ( \Delta \times^{\mathsf{FCD}} \Delta)$ is symmetric, see mathematics21.org/algebraic-general-topology.html
Dec
18
accepted Is a set closed under finite intersections? (about filters)
Dec
18
comment Is a set closed under finite intersections? (about filters)
@AlexRavsky: No! Your first comment was wrong. You provided a correct counter-example in your second comment. Please make an answer based on your second comment, not first!
Dec
18
comment Is a set closed under finite intersections? (about filters)
@AlexRavsky: I earlier wrote that your answer is not correct. That my comment was wrong and I deleted it. Really, your counter-example is correct. I suggest you to write it as an answer, so that I could accept it
Dec
17
comment Is a set closed under finite intersections? (about filters)
@AlexRavsky: It is not proved that there exists positive $\epsilon_1$ such that $f[\{x_1\}]\in\Delta$ for each $x_1\in(-\epsilon_1,\epsilon_1)$.
Dec
17
awarded  Yearling
Dec
12
comment Is a set closed under finite intersections? (about filters)
@KarlKronenfeld: Do you assume that $f$ is a function? $f$ is a binary relation, not necessarily a function
Dec
12
comment Is a set closed under finite intersections? (about filters)
@KarlKronenfeld: I added more parentheses to the formula. It seems you misunderstood me
Dec
12
revised Is a set closed under finite intersections? (about filters)
added more parentheses
Dec
12
awarded  Promoter
Dec
12
comment Is a set closed under finite intersections? (about filters)
The above comment is proved in portonmath.tiddlyspace.com/…
Dec
7
revised Is a set closed under finite intersections? (about filters)
added 55 characters in body
Nov
29
revised Name for a category
specified domain and codomain of the functions
Nov
27
comment Name for a category
For simplicity let limit both domains and codomains of all functions to some set $U$.
Nov
26
asked Name for a category
Nov
22
comment Category theory - where is my error?
@GiorgioMossa: It seems it is my error what you have pointed "It works just for generalized elements, not for object in general". But I do not 100% understand what is it exactly. Could you be so grateful to completely explain my last error?
Nov
22
comment Category theory - where is my error?
@GiorgioMossa: Isn't it true by definition of currying? Why do you mind it is wrong?
Nov
22
comment Category theory - where is my error?
For the category $\mathbf{Dig}$ exponential $\operatorname{MOR}(A;B)$ is the digraph whose vertexes are functions $\operatorname{Ob}A\rightarrow\operatorname{Ob}B$ and whose edges are such pairs $(f;g)$ of functions that $\forall (v;w) \in \operatorname{GR} A : ( f(v); g(w)) \in \operatorname{GR} B$ (where $\operatorname{Ob} X$ is the set of vertices of a digraph $X$ and $\operatorname{GR} X$ is the set of edges of a digraph $X$).
Nov
22
comment Category theory - where is my error?
$\operatorname{MOR}(A;B)$ (often designated $B^A$) is the exponential object as defined in en.wikipedia.org/wiki/Exponential_object or nlab.mathforge.org/nlab/show/exponential+object in our special case it is... (see the next comment)
Nov
22
comment Category theory - where is my error?
@GiorgioMossa: I mostly understood it, but one issue remains. Please see the updated question