07/12/2022, 09:01 AM
I have that Conway... tbh honest I have a shit-ton of books... just in case... you know... xD
Anyways... let's be sure I understand. I guess that we are not on the same page...
What do you exactly mean when you say
Question: Assume \(S:A\to A\),\(f:X\to X\).... the two case you are describing can be formalized as...
I don't think that it is what you mean... or it is...? Maybe It is what you mean because when you say
I read it as follows: Assume \(S:A\to A\),\(f:X\to X\) and \(U_0\) and \(U_1\) subsets of \(A\), we know how to compute two functions \(\chi_0:U_0\to X\) and \(\chi_1:U_1\to X\) s.t.
\[\chi_0 S|_{U_0}=f \chi_0\quad and\quad \chi_1 S|_{U_1}=f \chi_1
\]
saying that they do not agree, to me can be expressed as... assume \(U_0\cap U_1\) is non empty, the restrictions \((\chi_0)|_{U_0\cap U_1}\) is not the same as \((\chi_1)|_{U_0\cap U_1}\). In other words, exists an \(a\in U_0\cap U_1\) s.t. \[\chi_0(a)\neq \chi_1(a)\]
This seems good but in my mind it is not consistent with your use of the phrase "superfunction about something (e.g a fixed point)".
. In my mind this read as the superfunction \(\chi:A\to X\) is about a neighborhood of a fixed point \(p\in X\)... so locality isn't about a small set in the domain of the superfunction but is about a small set in the codomain, i.e. \(X\), where the fixed points of \(f:X\to X\) live.
Pleas help me make this clear. My mind is breaking around this terminology.
Anyways... let's be sure I understand. I guess that we are not on the same page...
What do you exactly mean when you say
Quote:super function either works for a tiny domain, or it works almost everywhere
Question: Assume \(S:A\to A\),\(f:X\to X\).... the two case you are describing can be formalized as...
- "works for a tiny domain"=the f.eq \(\chi S=f \chi\) has a solutions only on some tiny sets \(U\subseteq A\). I.e. we can only find functions \(\chi:U\to Y\) satisfying \(\chi S=f \chi\) only for "some small sets" \(U\subseteq A\)...
- "or it works almost eveywhere"=the f.eq \(\chi S=f \chi\) has an ugly solution on some set \(A'=A\setminus D\), where \(D\) is very small. I.e. we can only find functions \(\chi:A'\to Y\) satisfying \(\chi S=f \chi\) except on a small set of points \(D\), where \(A'=A\setminus D\)...
I don't think that it is what you mean... or it is...? Maybe It is what you mean because when you say
Quote:different super functions do not agree
I read it as follows: Assume \(S:A\to A\),\(f:X\to X\) and \(U_0\) and \(U_1\) subsets of \(A\), we know how to compute two functions \(\chi_0:U_0\to X\) and \(\chi_1:U_1\to X\) s.t.
\[\chi_0 S|_{U_0}=f \chi_0\quad and\quad \chi_1 S|_{U_1}=f \chi_1
\]
saying that they do not agree, to me can be expressed as... assume \(U_0\cap U_1\) is non empty, the restrictions \((\chi_0)|_{U_0\cap U_1}\) is not the same as \((\chi_1)|_{U_0\cap U_1}\). In other words, exists an \(a\in U_0\cap U_1\) s.t. \[\chi_0(a)\neq \chi_1(a)\]
This seems good but in my mind it is not consistent with your use of the phrase "superfunction about something (e.g a fixed point)".
. In my mind this read as the superfunction \(\chi:A\to X\) is about a neighborhood of a fixed point \(p\in X\)... so locality isn't about a small set in the domain of the superfunction but is about a small set in the codomain, i.e. \(X\), where the fixed points of \(f:X\to X\) live.
Pleas help me make this clear. My mind is breaking around this terminology.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)