Renamed Graphs
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@ -104,9 +104,9 @@
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The category $\BB$ is equipped with a forgetful functor $R$ to $\BB_0$, the category of models of the two-sort sort specification $(\mathcal{O},\El)$.
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The category $\BB$ is equipped with a forgetful functor $R$ to $\BB_0$, the category of models of the two-sort sort specification $(\mathcal{O},\El)$.
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\begin{center}
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\begin{center}
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% YADE DIAGRAM G1-0.json
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% YADE DIAGRAM GlobalConstructionSimple.json
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% GENERATED LATEX
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% GENERATED LATEX
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\input{graphs/G1-0.tex}
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\input{graphs/GlobalConstructionSimple.tex}
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% END OF GENERATED LATEX
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% END OF GENERATED LATEX
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\end{center}
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\end{center}
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@ -156,9 +156,9 @@
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Here is a figure that describes the recursive construction of some of the above objects
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Here is a figure that describes the recursive construction of some of the above objects
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\begin{center}
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\begin{center}
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% YADE DIAGRAM G1.json
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% YADE DIAGRAM GlobalRecursiveConstruction.json
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% GENERATED LATEX
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% GENERATED LATEX
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\input{graphs/G1.tex}
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\input{graphs/GlobalRecursiveConstruction.tex}
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% END OF GENERATED LATEX
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% END OF GENERATED LATEX
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\end{center}
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\end{center}
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@ -334,9 +334,9 @@
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A morphism $X \to X'$ of $\BB_i$ is a morphism $f : R_{i-1}^iX \to R_{i-1}^iX'$ in $\BB_{i-1}$ such that the following diagram commutes.
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A morphism $X \to X'$ of $\BB_i$ is a morphism $f : R_{i-1}^iX \to R_{i-1}^iX'$ in $\BB_{i-1}$ such that the following diagram commutes.
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\begin{center}
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\begin{center}
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% YADE DIAGRAM D1.json
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% YADE DIAGRAM BiMorphismDiagram.json
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% GENERATED LATEX
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% GENERATED LATEX
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\input{graphs/D1.tex}
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\input{graphs/BiMorphismDiagram.tex}
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% END OF GENERATED LATEX
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% END OF GENERATED LATEX
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\end{center}
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\end{center}
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@ -351,9 +351,9 @@
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The adjunction $F_i \vdash G_i$ is built using the two functors from the adjunction $F_{i-1} \vdash G_{i-1}$ defined in the previous induction step. We use them to define the first part of the adjunction, and we compose them with two adjoint functors $W$ and $E$ that we will define in this section. The overall construction for this induction step is as follows:
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The adjunction $F_i \vdash G_i$ is built using the two functors from the adjunction $F_{i-1} \vdash G_{i-1}$ defined in the previous induction step. We use them to define the first part of the adjunction, and we compose them with two adjoint functors $W$ and $E$ that we will define in this section. The overall construction for this induction step is as follows:
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\begin{center}
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\begin{center}
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% YADE DIAGRAM G2.json
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% YADE DIAGRAM InductionStepDiagram.json
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% GENERATED LATEX
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% GENERATED LATEX
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\input{graphs/G2.tex}
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\input{graphs/InductionStepDiagram.tex}
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% END OF GENERATED LATEX
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% END OF GENERATED LATEX
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\end{center}
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\end{center}
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@ -403,9 +403,9 @@
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It is indeed a morphism of $\BB_{i}$ as it makes the following diagram commute.
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It is indeed a morphism of $\BB_{i}$ as it makes the following diagram commute.
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\begin{center}
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\begin{center}
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% YADE DIAGRAM D2.json
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% YADE DIAGRAM WghMorphismOfBi.json
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% GENERATED LATEX
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% GENERATED LATEX
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\input{graphs/D2.tex}
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\input{graphs/WghMorphismOfBi.tex}
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% END OF GENERATED LATEX
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% END OF GENERATED LATEX
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\end{center}
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\end{center}
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@ -420,18 +420,18 @@
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Where $(A,h)$ is defined as the following pullback:
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Where $(A,h)$ is defined as the following pullback:
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\begin{center}
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\begin{center}
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% YADE DIAGRAM D3a.json
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% YADE DIAGRAM EDefinitionPullback.json
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% GENERATED LATEX
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% GENERATED LATEX
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\input{graphs/D3a.tex}
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\input{graphs/EDefinitionPullback.tex}
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% END OF GENERATED LATEX
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% END OF GENERATED LATEX
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\end{center}
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\end{center}
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The action on a morphism $f$ from $X$ to $X'$ is defined as $E(f) = (R_{i-1}^i f, !)$, with $!$ being the only morphism making the following diagram commute (thanks to the pullback property):
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The action on a morphism $f$ from $X$ to $X'$ is defined as $E(f) = (R_{i-1}^i f, !)$, with $!$ being the only morphism making the following diagram commute (thanks to the pullback property):
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\begin{center}
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\begin{center}
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% YADE DIAGRAM D3b.json
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% YADE DIAGRAM EDefinitionMorphism.json
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% GENERATED LATEX
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% GENERATED LATEX
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\input{graphs/D3b.tex}
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\input{graphs/EDefinitionMorphism.tex}
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% END OF GENERATED LATEX
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% END OF GENERATED LATEX
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\end{center}
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\end{center}
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@ -464,11 +464,12 @@
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The second component is defined through the universal property of the pullback defined by $E(Z)$ according to the following diagram:
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The second component is defined through the universal property of the pullback defined by $E(Z)$ according to the following diagram:
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\begin{center}
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\begin{center}
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% YADE DIAGRAM D6.json
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% YADE DIAGRAM PhiXYZSndComponentPullback.json
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% GENERATED LATEX
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% GENERATED LATEX
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\input{graphs/D6.tex}
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\input{graphs/PhiXYZSndComponentPullback.tex}
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% END OF GENERATED LATEX
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% END OF GENERATED LATEX
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\end{center}
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\end{center}
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\subsubsection{Constructing $\phi^{-1}_{XYZ}$}
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\subsubsection{Constructing $\phi^{-1}_{XYZ}$}
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@ -485,20 +486,22 @@
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Where $\square$ is a morphism $K_{H_iF_{i-1}} (X,Y) \to R_0^i Z$ in $\TSet$ defined by the following diagram:
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Where $\square$ is a morphism $K_{H_iF_{i-1}} (X,Y) \to R_0^i Z$ in $\TSet$ defined by the following diagram:
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\begin{center}
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\begin{center}
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% YADE DIAGRAM D7.json
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% YADE DIAGRAM PhiXYZ-1Square.json
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% GENERATED LATEX
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% GENERATED LATEX
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\input{graphs/D7.tex}
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\input{graphs/PhiXYZ-1Square.tex}
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% END OF GENERATED LATEX
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% END OF GENERATED LATEX
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\end{center}
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\end{center}
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Finally, we check that $\phi^{-1}_{XYZ}(g,h)$ is a morphism of $\BB_i$ from $W(X,Y)$ to $Z$, i.e. that it makes the following diagram commute:
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Finally, we check that $\phi^{-1}_{XYZ}(g,h)$ is a morphism of $\BB_i$ from $W(X,Y)$ to $Z$, i.e. that it makes the following diagram commute:
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\begin{center}
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\begin{center}
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% YADE DIAGRAM D8.json
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% YADE DIAGRAM PhiXYZ-1MorphismOfBi.json
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% GENERATED LATEX
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% GENERATED LATEX
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\input{graphs/D8.tex}
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\input{graphs/PhiXYZ-1MorphismOfBi.tex}
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% END OF GENERATED LATEX
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% END OF GENERATED LATEX
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\end{center}
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\end{center}
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In order to complete this proof, we need to show that
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In order to complete this proof, we need to show that
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\begin{itemize}
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\begin{itemize}
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@ -544,9 +547,9 @@
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It is a morphism of $\BB_i$ as it makes the following diagram commute:
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It is a morphism of $\BB_i$ as it makes the following diagram commute:
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\begin{center}
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\begin{center}
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% YADE DIAGRAM D4.json
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% YADE DIAGRAM TlInj1MorphismOfBi.json
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% GENERATED LATEX
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% GENERATED LATEX
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\input{graphs/D4.tex}
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\input{graphs/TlInj1MorphismOfBi.tex}
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% END OF GENERATED LATEX
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% END OF GENERATED LATEX
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\end{center}
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\end{center}
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@ -564,9 +567,9 @@
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We define $\{g ; h\}_i = \{R_{i-1}^i g ; h\}_{i-1}$. This is a morphism of $\BB_i$ as it makes the following diagram commute:
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We define $\{g ; h\}_i = \{R_{i-1}^i g ; h\}_{i-1}$. This is a morphism of $\BB_i$ as it makes the following diagram commute:
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\begin{center}
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\begin{center}
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% YADE DIAGRAM D5.json
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% YADE DIAGRAM TlUniversalMorphismIsOfBi.json
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% GENERATED LATEX
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% GENERATED LATEX
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\input{graphs/D5.tex}
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\input{graphs/TlUniversalMorphismIsOfBi.tex}
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% END OF GENERATED LATEX
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% END OF GENERATED LATEX
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\end{center}
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\end{center}
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@ -774,9 +777,9 @@
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A morphism from $(X,\Cstr)$ to $(X', \Cstr')$ is a morphism from $X$ to $X'$ in $\TSet$ (i.e. a natural transformation $X \implies X'$) which makes the following diagram commute, for all $a$ in $S_{i-1}$.
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A morphism from $(X,\Cstr)$ to $(X', \Cstr')$ is a morphism from $X$ to $X'$ in $\TSet$ (i.e. a natural transformation $X \implies X'$) which makes the following diagram commute, for all $a$ in $S_{i-1}$.
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\begin{center}
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\begin{center}
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% YADE DIAGRAM D1.json
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% YADE DIAGRAM BiMorphismDiagram.json
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% GENERATED LATEX
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% GENERATED LATEX
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\input{graphs/D1.tex}
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\input{graphs/BiMorphismDiagram.tex}
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% END OF GENERATED LATEX
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% END OF GENERATED LATEX
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\end{center}
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\end{center}
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@ -840,11 +843,12 @@
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The second component of $\phi_{XYZ} (\phi_{XYZ}^{-1}(g,h))$ follows the following diagram
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The second component of $\phi_{XYZ} (\phi_{XYZ}^{-1}(g,h))$ follows the following diagram
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\begin{center}
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\begin{center}
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% YADE DIAGRAM D9.json
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% YADE DIAGRAM CompositionSecondComponent.json
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% GENERATED LATEX
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% GENERATED LATEX
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\input{graphs/D9.tex}
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\input{graphs/CompositionSecondComponent.tex}
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% END OF GENERATED LATEX
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% END OF GENERATED LATEX
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\end{center}
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\end{center}
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\todo{Justifier que la partie du haut commute, i.e. que \[
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\todo{Justifier que la partie du haut commute, i.e. que \[
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@ -863,9 +867,9 @@
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where $\square$ follows the following diagram
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where $\square$ follows the following diagram
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\begin{center}
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\begin{center}
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% YADE DIAGRAM D10.json
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% YADE DIAGRAM CompositionSquareConstruction.json
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% GENERATED LATEX
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% GENERATED LATEX
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\input{graphs/D10.tex}
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\input{graphs/CompositionSquareConstruction.tex}
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% END OF GENERATED LATEX
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% END OF GENERATED LATEX
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\end{center}
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\end{center}
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We want to show that the following diagram commutes, for any objects $X$,$Y$,$Z$,$X'$,$Y'$,$Z'$ and morphisms $f$,$g$,$h$.
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We want to show that the following diagram commutes, for any objects $X$,$Y$,$Z$,$X'$,$Y'$,$Z'$ and morphisms $f$,$g$,$h$.
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\begin{center}
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\begin{center}
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% YADE DIAGRAM D10.0.json
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% YADE DIAGRAM NaturalityDiagram.json
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% GENERATED LATEX
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% GENERATED LATEX
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\input{graphs/D10.0.tex}
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\input{graphs/NaturalityDiagram.tex}
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% END OF GENERATED LATEX
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% END OF GENERATED LATEX
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\end{center}
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\end{center}
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@ -904,9 +908,9 @@
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The second components are defined as described by the following diagram
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The second components are defined as described by the following diagram
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\begin{center}
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\begin{center}
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% YADE DIAGRAM D11.json
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% YADE DIAGRAM NaturalityDoublePullbackDefinition.json
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% GENERATED LATEX
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% GENERATED LATEX
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\input{graphs/D11.tex}
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\input{graphs/NaturalityDoublePullbackDefinition.tex}
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% END OF GENERATED LATEX
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% END OF GENERATED LATEX
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\end{center}
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\end{center}
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@ -934,35 +938,35 @@
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Where $(A,h)$ is the pullback defined as
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Where $(A,h)$ is the pullback defined as
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\begin{center}
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\begin{center}
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% YADE DIAGRAM E1.json
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% YADE DIAGRAM ReflectionFGPullback.json
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% GENERATED LATEX
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% GENERATED LATEX
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\input{graphs/E1.tex}
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\input{graphs/ReflectionFGPullback.tex}
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% END OF GENERATED LATEX
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% END OF GENERATED LATEX
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\end{center}
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\end{center}
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We can extend this pullback with the two isomorphisms $\en_0^i$ and $H_iF_{i-1}(\inj_\tl^{i-1})$ in another pullback. This new pullback is over the injection morphism $\inj_2$ of the coproduct of $\TSet$, so the new pullback object is the second component of the $\bullet \oplus B$ i.e. $B$.
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We can extend this pullback with the two isomorphisms $\en_0^i$ and $H_iF_{i-1}(\inj_\tl^{i-1})$ in another pullback. This new pullback is over the injection morphism $\inj_2$ of the coproduct of $\TSet$, so the new pullback object is the second component of the $\bullet \oplus B$ i.e. $B$.
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\begin{center}
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\begin{center}
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% YADE DIAGRAM E2.json
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% YADE DIAGRAM ReflectionFGExtendedPullback.json
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% GENERATED LATEX
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% GENERATED LATEX
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\input{graphs/E2.tex}
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\input{graphs/ReflectionFGExtendedPullback.tex}
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% END OF GENERATED LATEX
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% END OF GENERATED LATEX
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\end{center}
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\end{center}
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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.
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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.
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\begin{center}
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\begin{center}
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% YADE DIAGRAM E3.json
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% YADE DIAGRAM ReflectionFGIsomorphismFirst.json
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% GENERATED LATEX
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% GENERATED LATEX
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\input{graphs/E3.tex}
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\input{graphs/ReflectionFGIsomorphismFirst.tex}
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% END OF GENERATED LATEX
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% END OF GENERATED LATEX
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\end{center}
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\end{center}
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And the second component is made using the isomorphism constructed by the pullback, that makes the diagram commute.
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And the second component is made using the isomorphism constructed by the pullback, that makes the diagram commute.
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\begin{center}
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\begin{center}
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% YADE DIAGRAM E4.json
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% YADE DIAGRAM ReflectionFGIsomorphismSecond.json
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% GENERATED LATEX
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% GENERATED LATEX
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\input{graphs/E4.tex}
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\input{graphs/ReflectionFGIsomorphismSecond.tex}
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% END OF GENERATED LATEX
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% END OF GENERATED LATEX
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\end{center}
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\end{center}
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@ -1 +0,0 @@
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{"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":1,"id":4,"label":{"kind":"normal","label":"\\Gamma","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":0},{"from":2,"id":5,"label":{"kind":"normal","label":"A","style":{"alignment":"left","bend":0,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":1},{"from":2,"id":6,"label":{"kind":"normal","label":"\\Delta","style":{"alignment":"right","bend":0.20000000000000004,"color":"black","dashed":false,"head":"default","kind":"normal","position":0.5,"tail":"none"},"zindex":0},"to":0}],"nodes":[{"id":0,"label":{"isMath":true,"label":"\\Con","pos":[100,100],"zindex":0}},{"id":1,"label":{"isMath":true,"label":"\\Ty","pos":[300,100],"zindex":0}},{"id":2,"label":{"isMath":true,"label":"\\Tm","pos":[500,100],"zindex":0}},{"id":3,"label":{"isMath":true,"label":"\\circlearrowleft","pos":[302,76.8125],"zindex":0}}],"sizeGrid":200,"title":"1"}]},"version":12}
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