Completed proof of reflection
This commit is contained in:
parent
93d565a4c7
commit
5c75b2b701
GPG Key ID:
0220AC4A3D6A328B
@ -456,11 +456,11 @@
|
||||
\subsubsection{Proof of H1}
|
||||
|
||||
\todo{Relire + réeexpliquer pourquoi ça prouve}
|
||||
|
||||
\label{sec:coproductConstr}
|
||||
We will define the sums of the form $X \oplus_i L_0^i Y$ in $\BB_i$.
|
||||
|
||||
\[
|
||||
X \oplus_i L_0^i Y := \left(R_{i-1}^i \oplus_{i-1} L_0^{i-1} Y, (R_0^{i-1} \inj_1^{i-1})_\UU \circ \Cstr_i^X \circ (\inj_1^{i-1} \circ \dash)^{-1}\right)
|
||||
X \oplus_i L_0^i Y := \left(R_{i-1}^i X \oplus_{i-1} L_0^{i-1} Y, (R_0^{i-1} \inj_1^{i-1})_\UU \circ \Cstr_i^X \circ (H_iF_{i-1}\inj_1^{i-1})^{-1}\right)
|
||||
\]
|
||||
|
||||
Here, $(\inj_1^{i-1} \circ \dash)^{-1}$ is the inverse of the isomorphism of hypothesis H3', and
|
||||
@ -490,12 +490,12 @@
|
||||
|
||||
The second injector is defined as follows:
|
||||
\[
|
||||
inj_2^i := (\varepsilon_i \oplus_i \id_{L_0^i Y}) \circ L_{i-1}^i \inj_2^{i-1}
|
||||
\inj_2^i := (\varepsilon_i \oplus_i \id_{L_0^i Y}) \circ L_{i-1}^i \inj_2^{i-1}
|
||||
\]
|
||||
|
||||
Where $\varepsilon_i$ is the counit of the adjunction $R_{i-1}^i \vdash L_{i-1}^i$, going from $L_{i-1}^i R_{i-1}^i X$ to $X$.
|
||||
|
||||
This goes from $L_0^i Y = L_{i-1}^i L_0^{i-1} Y$ to $L_{i-1}^i(R_{i-1}^i X \oplus_{i-1} L_0^{i-1} Y)$, which is equivalent to $L_{i-1}^i R_{i-1}^i X \oplus_i L_0^i Y)$ as $L_{i-1}^i$ is a left-adjoint functor and therefore it preserves colimits; then goes to $X \oplus_i L_0^i Y$.
|
||||
This goes from $L_0^i Y = L_{i-1}^i L_0^{i-1} Y$ to $L_{i-1}^i(R_{i-1}^i X \oplus_{i-1} L_0^{i-1} Y)$, which is equivalent to $L_{i-1}^i R_{i-1}^i X \oplus_i L_0^i Y$ as $L_{i-1}^i$ is a left-adjoint functor and therefore it preserves colimits; then goes to $X \oplus_i L_0^i Y$.
|
||||
|
||||
We will now show that this definition is actually a definition of the coproduct in $\BB_i$.
|
||||
To that extent, we take two objects $X$ and $Z$ in $\BB_i$, $Y$ in $\TSet$ and two morphisms of $\BB_i$ $\varphi_1 : X \to Z$ and $\varphi_2 : L_0^i Y \to Z$.
|
||||
@ -514,6 +514,16 @@
|
||||
|
||||
\subsubsection{Proof of H3}
|
||||
|
||||
We need to prove that, for any objects $(X,\Cstr)$ in $\BB_i$ and $Y$ in $\TSet$, that the morphism
|
||||
$F_i(\inj_1^i) : F_i(X,\Cstr) \to F_i((X,\Cstr) \oplus L_0^i Y)$ is an isomorphism.
|
||||
|
||||
We know from \autoref{sec:coproductConstr} that $\inj_1^i := \inj_1^{i-1}$ as a morphism of $\BB_{i}$ is a morphism $\BB_{i-1}$ that verifies some equalities.
|
||||
|
||||
We also know from the last induction step that $F_{i-1}\inj_1^{i-1}$ is an isomorphism.
|
||||
\[
|
||||
(X,\Cstr) \oplus L_0^iY \simeq (X \oplus L_0^{i-1}Y, (R_0^{i-1}\inj_1^{i-1})_\UU \circ \Cstr \circ (H_iF_{i-1}\inj_1^{i-1})^{-1})
|
||||
\]
|
||||
|
||||
|
||||
\section{Misc}
|
||||
|
||||
@ -779,15 +789,14 @@
|
||||
\section{$F_i \vdash G_i$ reflection}
|
||||
\label{apx:FG-refl}
|
||||
|
||||
\todo{La preuve :/}
|
||||
We want to find, for each object $(X,(B,g))$ of $\CC_i = (X : \CC_{i-1}) \times (\Set/H_i(X))$, an isomorphism $(X,(B,g)) \to F_iG_i(X,(B,g))$. $g$ is a morphism from $B$ to $H_i(X)$
|
||||
|
||||
\[\begin{array}{rcl}
|
||||
F_iG_i(X,\Rtsc)
|
||||
&=& F_iW_i(G_{i-1}X,\Rtsc)\\
|
||||
&=& F_i(G_{i-1}X \oplus L_0^{i-1}H_\bullet(G_{i-1}X,\Rtsc),\widetilde{\inj_2})\\
|
||||
&=&(F_{i-1} \times \id)\left(G_{i-1}X \oplus L_0^{i-1}\Hbar_\bullet(G_{i-1}X,\Rtsc),(A,h)\right)\\
|
||||
&=&\left(F_{i-1}(G_{i-1}X \oplus L_0^{i-1}\Hbar_\bullet(G_{i-1}X,\Rtsc)),(A,h)\right)
|
||||
F_iG_i(X,(B,g))
|
||||
&=& F_iW_i(G_{i-1}X,(B,g))\\
|
||||
&=& F_i(G_{i-1}X \oplus L_0^{i-1}H_\bullet(G_{i-1}X,(B,g)),\widetilde{\inj_2})\\
|
||||
&=&(F_{i-1} \times \id)\left(G_{i-1}X \oplus L_0^{i-1}\Hbar_\bullet(G_{i-1}X,(B,g)),(A,h)\right)\\
|
||||
&=&\left(F_{i-1}(G_{i-1}X \oplus L_0^{i-1}\Hbar_\bullet(G_{i-1}X,(B,g))),(A,h)\right)
|
||||
\end{array}\]
|
||||
|
||||
Where $(A,h)$ is the pullback defined as
|
||||
@ -798,7 +807,7 @@
|
||||
% END OF GENERATED LATEX
|
||||
\end{center}
|
||||
|
||||
We can extend this pullback using the two isomorphisms given by the induction hypothesis and hypothesis H3.
|
||||
We can extend this pullback using the two isomorphisms given by the induction hypothesis and hypothesis H3. This pullback is over the injection morphism $\inj_2$ of the coproduct, so the new pullback object is the second component of the $\bullet \oplus B$ i.e. $B$.
|
||||
|
||||
\begin{center}
|
||||
% YADE DIAGRAM E2.json
|
||||
@ -807,6 +816,24 @@
|
||||
% END OF GENERATED LATEX
|
||||
\end{center}
|
||||
|
||||
The first component of the isomorphism is the following isomorphism, where $\eta_{i-1}^{FG}$ if the counit of the adjunction $F_{i-1} \vdash G_{i-1}$, that we know to be an isomorphism from the induction hypothesis.
|
||||
|
||||
\begin{center}
|
||||
% YADE DIAGRAM E3.json
|
||||
% GENERATED LATEX
|
||||
\input{graphs/E3.tex}
|
||||
% END OF GENERATED LATEX
|
||||
\end{center}
|
||||
|
||||
And the second component is made using the isomorphism constructed by the pullback, that makes the diagram commute.
|
||||
\begin{center}
|
||||
% YADE DIAGRAM E4.json
|
||||
% GENERATED LATEX
|
||||
\input{graphs/E4.tex}
|
||||
% END OF GENERATED LATEX
|
||||
\end{center}
|
||||
|
||||
|
||||
\end{document}
|
||||
|
||||
|
||||
@ -838,6 +865,20 @@
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
@ -1 +1 @@
|
||||
{"graph":{"latexPreamble":"\\newcommand\\ensuremath[1]{#1}\n\\newcommand\\BB{{\\ensuremath{\\mathcal{B}}}}\n\\newcommand\\TT{{\\ensuremath{\\mathcal{T}}}}\n\\newcommand\\UU{{\\ensuremath{\\mathcal{U}}}}\n\\newcommand\\El{{\\ensuremath{\\operatorname{\\mathcal{E}l}}}}\n\\newcommand\\ii{{\\ensuremath{\\mathbf{i}}}}\n\\newcommand\\Cstr{{\\ensuremath{\\operatorname{\\mathcal{C}str}}}}\n\\newcommand\\Set{{\\ensuremath{\\operatorname{\\mathcal{S}et}}}}\n\\newcommand\\Hom{{\\ensuremath{\\operatorname{\\mathcal{H}om}}}}\n\\newcommand\\this{{\\ensuremath{\\operatorname{\\texttt{this}}}}}\n\\newcommand\\Hbar{{\\ensuremath{\\overline{H}}}}\n\\newcommand\\dash{{\\;\\textrm{---}\\;}}\n\n\\newcommand\\inj{\\operatorname{inj}}\n\\newcommand\\id{\\operatorname{id}}","tabs":[{"active":true,"edges":[{"from":0,"id":6,"label":{"kind":"normal","label":"\\Cstr^X","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":1},{"from":1,"id":7,"label":{"kind":"normal","label":"(R_0^{i-1} (\\inj_1^{i-1})_\\UU","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":2},{"from":0,"id":8,"label":{"kind":"normal","label":"H_iF_{i-1}\\inj_1^{i-1}","style":{"alignment":"right","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":3},{"from":4,"id":9,"label":{"kind":"normal","label":"R_0^{i-1}(\\inj_1^{i-1})_\\UU","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":2},{"from":3,"id":10,"label":{"kind":"normal","label":"(\\inj_1^{i-1} \\circ \\dash)^{-1}","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"none","kind":"normal","position":0.5,"tail":"none"},"zindex":-3},"to":5},{"from":5,"id":11,"label":{"kind":"normal","label":"\\Cstr^X","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":4},{"from":0,"id":12,"label":{"kind":"normal","label":"","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"none","kind":"double","position":0.5,"tail":"none"},"zindex":0},"to":5},{"from":4,"id":13,"label":{"kind":"normal","label":"","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"none","kind":"double","position":0.5,"tail":"none"},"zindex":0},"to":1},{"from":3,"id":14,"label":{"kind":"normal","label":"\\Cstr^{X \\oplus L_0^i Y}","style":{"alignment":"left","bend":0.1,"color":"green","dashed":true,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":2}],"nodes":[{"id":0,"label":{"isMath":true,"label":"H_iF_{i-1}R_{i-1}^i X)","pos":[236.5,73.5],"zindex":0}},{"id":1,"label":{"isMath":true,"label":"(R_0^i X)_\\UU","pos":[1013,73.8125],"zindex":0}},{"id":2,"label":{"isMath":true,"label":"(R_0^{i-1} (R_{i-1}^i X \\oplus L_0^{i-1} Y))_\\UU","pos":[1017.5,249.5],"zindex":0}},{"id":3,"label":{"isMath":true,"label":"H_iF_{i-1}(R_{i-1}^i X \\oplus_{i-1} L_0^{i-1} Y)","pos":[236.5,249.5],"zindex":-10000}},{"id":4,"label":{"isMath":true,"label":"(R_0^i X)_\\UU","pos":[706,235.8125],"zindex":-10000}},{"id":5,"label":{"isMath":true,"label":"H_iF_{i-1}R_{i-1}^i X","pos":[556,234.8125],"zindex":-10000}}],"sizeGrid":147,"title":"1"}]},"version":12}
|
||||
{"graph":{"latexPreamble":"\\newcommand\\ensuremath[1]{#1}\n\\newcommand\\BB{{\\ensuremath{\\mathcal{B}}}}\n\\newcommand\\TT{{\\ensuremath{\\mathcal{T}}}}\n\\newcommand\\UU{{\\ensuremath{\\mathcal{U}}}}\n\\newcommand\\El{{\\ensuremath{\\operatorname{\\mathcal{E}l}}}}\n\\newcommand\\ii{{\\ensuremath{\\mathbf{i}}}}\n\\newcommand\\Cstr{{\\ensuremath{\\operatorname{\\mathcal{C}str}}}}\n\\newcommand\\Set{{\\ensuremath{\\operatorname{\\mathcal{S}et}}}}\n\\newcommand\\Hom{{\\ensuremath{\\operatorname{\\mathcal{H}om}}}}\n\\newcommand\\this{{\\ensuremath{\\operatorname{\\texttt{this}}}}}\n\\newcommand\\Hbar{{\\ensuremath{\\overline{H}}}}\n\\newcommand\\dash{{\\;\\textrm{---}\\;}}\n\n\\newcommand\\inj{\\operatorname{inj}}\n\\newcommand\\id{\\operatorname{id}}","tabs":[{"active":true,"edges":[{"from":0,"id":6,"label":{"kind":"normal","label":"\\Cstr^X","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":1},{"from":1,"id":7,"label":{"kind":"normal","label":"(R_0^{i-1} (\\inj_1^{i-1})_\\UU","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":2},{"from":0,"id":8,"label":{"kind":"normal","label":"H_iF_{i-1}\\inj_1^{i-1}","style":{"alignment":"right","bend":0,"color":"black","dashed":false,"head":"default","kind":"double","position":0.5,"tail":"none"},"zindex":0},"to":3},{"from":4,"id":9,"label":{"kind":"normal","label":"R_0^{i-1}(\\inj_1^{i-1})_\\UU","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":2},{"from":3,"id":10,"label":{"kind":"normal","label":"(H_iF_{i-1}\\inj_1^{i-1})^{-1}","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"double","position":0.5,"tail":"none"},"zindex":-3},"to":5},{"from":5,"id":11,"label":{"kind":"normal","label":"\\Cstr^X","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":4},{"from":0,"id":12,"label":{"kind":"normal","label":"","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"none","kind":"double","position":0.5,"tail":"none"},"zindex":0},"to":5},{"from":4,"id":13,"label":{"kind":"normal","label":"","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"none","kind":"double","position":0.5,"tail":"none"},"zindex":0},"to":1},{"from":3,"id":14,"label":{"kind":"normal","label":"\\Cstr^{X \\oplus L_0^i Y}","style":{"alignment":"left","bend":0.1,"color":"green","dashed":true,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":2}],"nodes":[{"id":0,"label":{"isMath":true,"label":"H_iF_{i-1}R_{i-1}^i X","pos":[236.5,73.5],"zindex":0}},{"id":1,"label":{"isMath":true,"label":"(R_0^i X)_\\UU","pos":[1013,73.8125],"zindex":0}},{"id":2,"label":{"isMath":true,"label":"(R_0^{i-1} (R_{i-1}^i X \\oplus L_0^{i-1} Y))_\\UU","pos":[1017.5,249.5],"zindex":0}},{"id":3,"label":{"isMath":true,"label":"H_iF_{i-1}(R_{i-1}^i X \\oplus_{i-1} L_0^{i-1} Y)","pos":[236.5,249.5],"zindex":-10000}},{"id":4,"label":{"isMath":true,"label":"(R_0^i X)_\\UU","pos":[706,235.8125],"zindex":-10000}},{"id":5,"label":{"isMath":true,"label":"H_iF_{i-1}R_{i-1}^i X","pos":[556,234.8125],"zindex":-10000}}],"sizeGrid":147,"title":"1"}]},"version":12}
|
||||
@ -1 +1 @@
|
||||
{"graph":{"latexPreamble":"\\newcommand\\ensuremath[1]{#1}\n\\newcommand\\BB{{\\ensuremath{\\mathcal{B}}}}\n\\newcommand\\TT{{\\ensuremath{\\mathcal{T}}}}\n\\newcommand\\UU{{\\ensuremath{\\mathcal{U}}}}\n\\newcommand\\CC{{\\ensuremath{\\mathcal{C}}}}\n\\newcommand\\El{{\\ensuremath{\\operatorname{\\mathcal{E}l}}}}\n\\newcommand\\ii{{\\ensuremath{\\mathbf{i}}}}\n\\newcommand\\Cstr{{\\ensuremath{\\operatorname{\\mathcal{C}str}}}}\n\\newcommand\\Set{{\\ensuremath{\\operatorname{\\mathcal{S}et}}}}\n\\newcommand\\Hom{{\\ensuremath{\\operatorname{\\mathcal{H}om}}}}\n\\newcommand\\this{{\\ensuremath{\\operatorname{\\texttt{this}}}}}\n\\newcommand\\Hbar{{\\ensuremath{\\overline{H}}}}\n\\newcommand\\dash{{\\;\\textrm{---}\\;}}\n\n\\newcommand\\inj{\\operatorname{inj}}\n\\newcommand\\id{\\operatorname{id}}","tabs":[{"active":true,"edges":[{"from":0,"id":4,"label":{"kind":"normal","label":"","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":1},{"from":1,"id":5,"label":{"kind":"normal","label":"R_0^{i-1}(G_{i-1}X \\oplus L_0^{i-1} \\Hbar(G_{i-1}X,\\Rtsc))_\\El","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":2},{"from":0,"id":6,"label":{"kind":"normal","label":"h","style":{"alignment":"right","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":3},{"from":3,"id":7,"label":{"kind":"normal","label":"\\widetilde{\\inj_2}","style":{"alignment":"right","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":2},{"from":6,"id":8,"label":{"kind":"pullshout","label":"","style":{"alignment":"","bend":0,"color":"black","dashed":false,"head":"","kind":"normal","position":0,"tail":""},"zindex":0},"to":4}],"nodes":[{"id":0,"label":{"isMath":true,"label":"A","pos":[201,81],"zindex":0}},{"id":1,"label":{"isMath":true,"label":"R_0^{i-1}(G_{i-1}X \\oplus L_0^{i-1} \\Hbar(G_{i-1}X,\\Rtsc))_\\El","pos":[624,81],"zindex":0}},{"id":2,"label":{"isMath":true,"label":"R_0^{i-1}(G_{i-1}X \\oplus L_0^{i-1} \\Hbar(G_{i-1}X,\\Rtsc))_\\UU","pos":[624,180],"zindex":0}},{"id":3,"label":{"isMath":true,"label":"H_iF_{i-1}(G_{i-1}X\\oplus L_0^{i-1}\\Hbar_\\bullet(\\bullet))","pos":[201,180],"zindex":0}}],"sizeGrid":200,"title":"1"}]},"version":12}
|
||||
{"graph":{"latexPreamble":"\\newcommand\\ensuremath[1]{#1}\n\\newcommand\\BB{{\\ensuremath{\\mathcal{B}}}}\n\\newcommand\\TT{{\\ensuremath{\\mathcal{T}}}}\n\\newcommand\\UU{{\\ensuremath{\\mathcal{U}}}}\n\\newcommand\\CC{{\\ensuremath{\\mathcal{C}}}}\n\\newcommand\\El{{\\ensuremath{\\operatorname{\\mathcal{E}l}}}}\n\\newcommand\\ii{{\\ensuremath{\\mathbf{i}}}}\n\\newcommand\\Cstr{{\\ensuremath{\\operatorname{\\mathcal{C}str}}}}\n\\newcommand\\Set{{\\ensuremath{\\operatorname{\\mathcal{S}et}}}}\n\\newcommand\\Hom{{\\ensuremath{\\operatorname{\\mathcal{H}om}}}}\n\\newcommand\\this{{\\ensuremath{\\operatorname{\\texttt{this}}}}}\n\\newcommand\\Hbar{{\\ensuremath{\\overline{H}}}}\n\\newcommand\\dash{{\\;\\textrm{---}\\;}}\n\n\\newcommand\\inj{\\operatorname{inj}}\n\\newcommand\\id{\\operatorname{id}}","tabs":[{"active":true,"edges":[{"from":0,"id":4,"label":{"kind":"normal","label":"","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":1},{"from":1,"id":5,"label":{"kind":"normal","label":"R_0^{i-1}(G_{i-1}X \\oplus L_0^{i-1} \\Hbar(G_{i-1}X,(B,g)))_\\El","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":2},{"from":0,"id":6,"label":{"kind":"normal","label":"h","style":{"alignment":"right","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":3},{"from":3,"id":7,"label":{"kind":"normal","label":"\\widetilde{\\inj_2}","style":{"alignment":"right","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":2},{"from":6,"id":8,"label":{"kind":"pullshout","label":"","style":{"alignment":"","bend":0,"color":"black","dashed":false,"head":"","kind":"normal","position":0,"tail":""},"zindex":0},"to":4}],"nodes":[{"id":0,"label":{"isMath":true,"label":"A","pos":[201,81],"zindex":0}},{"id":1,"label":{"isMath":true,"label":"R_0^{i-1}(G_{i-1}X \\oplus L_0^{i-1} \\Hbar(G_{i-1}X,(B,g)))_\\El","pos":[624,81],"zindex":0}},{"id":2,"label":{"isMath":true,"label":"R_0^{i-1}(G_{i-1}X \\oplus L_0^{i-1} \\Hbar(G_{i-1}X,(B,g)))_\\UU","pos":[624,180],"zindex":0}},{"id":3,"label":{"isMath":true,"label":"H_iF_{i-1}(G_{i-1}X\\oplus L_0^{i-1}\\Hbar_\\bullet(X,(B,g)))","pos":[201,180],"zindex":0}}],"sizeGrid":200,"title":"1"}]},"version":12}
|
||||
File diff suppressed because one or more lines are too long
1
Report/graphs/E3.json
Normal file
1
Report/graphs/E3.json
Normal file
@ -0,0 +1 @@
|
||||
{"graph":{"latexPreamble":"\\newcommand\\ensuremath[1]{#1}\n\\newcommand\\BB{{\\ensuremath{\\mathcal{B}}}}\n\\newcommand\\TT{{\\ensuremath{\\mathcal{T}}}}\n\\newcommand\\UU{{\\ensuremath{\\mathcal{U}}}}\n\\newcommand\\CC{{\\ensuremath{\\mathcal{C}}}}\n\\newcommand\\El{{\\ensuremath{\\operatorname{\\mathcal{E}l}}}}\n\\newcommand\\ii{{\\ensuremath{\\mathbf{i}}}}\n\\newcommand\\Cstr{{\\ensuremath{\\operatorname{\\mathcal{C}str}}}}\n\\newcommand\\Set{{\\ensuremath{\\operatorname{\\mathcal{S}et}}}}\n\\newcommand\\Hom{{\\ensuremath{\\operatorname{\\mathcal{H}om}}}}\n\\newcommand\\this{{\\ensuremath{\\operatorname{\\texttt{this}}}}}\n\\newcommand\\Hbar{{\\ensuremath{\\overline{H}}}}\n\\newcommand\\dash{{\\;\\textrm{---}\\;}}\n\n\\newcommand\\inj{\\operatorname{inj}}\n\\newcommand\\id{\\operatorname{id}}","tabs":[{"active":true,"edges":[{"from":0,"id":3,"label":{"kind":"normal","label":"\\eta^{FG}_{i-1}","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"double","position":0.5,"tail":"none"},"zindex":0},"to":1},{"from":1,"id":4,"label":{"kind":"normal","label":"F_{i-1}(\\inj_1^{i-1})","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"double","position":0.5,"tail":"none"},"zindex":0},"to":2}],"nodes":[{"id":0,"label":{"isMath":true,"label":"X","pos":[100,100],"zindex":0}},{"id":1,"label":{"isMath":true,"label":"F_{i-1}G_{i-1}X","pos":[300,100],"zindex":0}},{"id":2,"label":{"isMath":true,"label":"F_{i-1}(G_{i-1}X \\oplus L_0^{i-1}\\Hbar_\\bullet(G_{i-1}X,(B,g)))","pos":[626,100],"zindex":0}}],"sizeGrid":200,"title":"1"}]},"version":12}
|
||||
1
Report/graphs/E4.json
Normal file
1
Report/graphs/E4.json
Normal file
@ -0,0 +1 @@
|
||||
{"graph":{"latexPreamble":"\\newcommand\\ensuremath[1]{#1}\n\\newcommand\\BB{{\\ensuremath{\\mathcal{B}}}}\n\\newcommand\\TT{{\\ensuremath{\\mathcal{T}}}}\n\\newcommand\\UU{{\\ensuremath{\\mathcal{U}}}}\n\\newcommand\\CC{{\\ensuremath{\\mathcal{C}}}}\n\\newcommand\\El{{\\ensuremath{\\operatorname{\\mathcal{E}l}}}}\n\\newcommand\\ii{{\\ensuremath{\\mathbf{i}}}}\n\\newcommand\\Cstr{{\\ensuremath{\\operatorname{\\mathcal{C}str}}}}\n\\newcommand\\Set{{\\ensuremath{\\operatorname{\\mathcal{S}et}}}}\n\\newcommand\\Hom{{\\ensuremath{\\operatorname{\\mathcal{H}om}}}}\n\\newcommand\\this{{\\ensuremath{\\operatorname{\\texttt{this}}}}}\n\\newcommand\\Hbar{{\\ensuremath{\\overline{H}}}}\n\\newcommand\\dash{{\\;\\textrm{---}\\;}}\n\n\\newcommand\\inj{\\operatorname{inj}}\n\\newcommand\\id{\\operatorname{id}}","tabs":[{"active":true,"edges":[{"from":0,"id":6,"label":{"kind":"normal","label":"","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"none","kind":"double","position":0.5,"tail":"none"},"zindex":0},"to":1},{"from":1,"id":7,"label":{"kind":"normal","label":"","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"double","position":0.5,"tail":"none"},"zindex":0},"to":2},{"from":2,"id":8,"label":{"kind":"normal","label":"","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":3},{"from":1,"id":9,"label":{"kind":"normal","label":"H_i\\eta^{FG}_{i-1} \\circ g","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":4},{"from":4,"id":10,"label":{"kind":"normal","label":"H_iF_{i-1}\\inj_0^{i-1}","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":3},{"from":0,"id":11,"label":{"kind":"normal","label":"g","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":5},{"from":5,"id":12,"label":{"kind":"normal","label":"H_i\\text{HR}","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":4}],"nodes":[{"id":0,"label":{"isMath":true,"label":"B","pos":[100,100],"zindex":0}},{"id":1,"label":{"isMath":true,"label":"B","pos":[300,100],"zindex":0}},{"id":2,"label":{"isMath":true,"label":"A","pos":[678,100],"zindex":0}},{"id":3,"label":{"isMath":true,"label":"H_iF_{i-1}(G_{i-1}X \\oplus L_0^{i-1}\\Hbar_\\bullet(G_{i-1}X,(B,g)))","pos":[678,211],"zindex":0}},{"id":4,"label":{"isMath":true,"label":"H_iF_{i-1}G_{i-1}X","pos":[300,211],"zindex":-10000}},{"id":5,"label":{"isMath":true,"label":"H_iX","pos":[100,211],"zindex":-10000}}],"sizeGrid":200,"title":"1"}]},"version":12}
|
||||
Loading…
x
Reference in New Issue
Block a user