Completed proof of reflection
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\subsubsection{Proof of H1}
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\subsubsection{Proof of H1}
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\todo{Relire + réeexpliquer pourquoi ça prouve}
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\todo{Relire + réeexpliquer pourquoi ça prouve}
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\label{sec:coproductConstr}
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We will define the sums of the form $X \oplus_i L_0^i Y$ in $\BB_i$.
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We will define the sums of the form $X \oplus_i L_0^i Y$ in $\BB_i$.
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\[
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\[
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X \oplus_i L_0^i Y := \left(R_{i-1}^i \oplus_{i-1} L_0^{i-1} Y, (R_0^{i-1} \inj_1^{i-1})_\UU \circ \Cstr_i^X \circ (\inj_1^{i-1} \circ \dash)^{-1}\right)
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X \oplus_i L_0^i Y := \left(R_{i-1}^i X \oplus_{i-1} L_0^{i-1} Y, (R_0^{i-1} \inj_1^{i-1})_\UU \circ \Cstr_i^X \circ (H_iF_{i-1}\inj_1^{i-1})^{-1}\right)
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\]
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\]
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Here, $(\inj_1^{i-1} \circ \dash)^{-1}$ is the inverse of the isomorphism of hypothesis H3', and
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Here, $(\inj_1^{i-1} \circ \dash)^{-1}$ is the inverse of the isomorphism of hypothesis H3', and
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The second injector is defined as follows:
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The second injector is defined as follows:
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\[
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\[
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inj_2^i := (\varepsilon_i \oplus_i \id_{L_0^i Y}) \circ L_{i-1}^i \inj_2^{i-1}
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\inj_2^i := (\varepsilon_i \oplus_i \id_{L_0^i Y}) \circ L_{i-1}^i \inj_2^{i-1}
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\]
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\]
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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 $X$.
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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 $X$.
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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 $X \oplus_i L_0^i Y$.
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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 $X \oplus_i L_0^i Y$.
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We will now show that this definition is actually a definition of the coproduct in $\BB_i$.
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We will now show that this definition is actually a definition of the coproduct in $\BB_i$.
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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 $\varphi_2 : L_0^i Y \to Z$.
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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 $\varphi_2 : L_0^i Y \to Z$.
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\subsubsection{Proof of H3}
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\subsubsection{Proof of H3}
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We need to prove that, for any objects $(X,\Cstr)$ in $\BB_i$ and $Y$ in $\TSet$, that the morphism
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$F_i(\inj_1^i) : F_i(X,\Cstr) \to F_i((X,\Cstr) \oplus L_0^i Y)$ is an isomorphism.
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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.
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We also know from the last induction step that $F_{i-1}\inj_1^{i-1}$ is an isomorphism.
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\[
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(X,\Cstr) \oplus L_0^iY \simeq (X \oplus L_0^{i-1}Y, (R_0^{i-1}\inj_1^{i-1})_\UU \circ \Cstr \circ (H_iF_{i-1}\inj_1^{i-1})^{-1})
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\]
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\section{Misc}
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\section{Misc}
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\section{$F_i \vdash G_i$ reflection}
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\section{$F_i \vdash G_i$ reflection}
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\label{apx:FG-refl}
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\label{apx:FG-refl}
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\todo{La preuve :/}
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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 $H_i(X)$
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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 $H_i(X)$
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\[\begin{array}{rcl}
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\[\begin{array}{rcl}
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F_iG_i(X,\Rtsc)
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F_iG_i(X,(B,g))
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&=& F_iW_i(G_{i-1}X,\Rtsc)\\
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&=& F_iW_i(G_{i-1}X,(B,g))\\
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&=& F_i(G_{i-1}X \oplus L_0^{i-1}H_\bullet(G_{i-1}X,\Rtsc),\widetilde{\inj_2})\\
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&=& F_i(G_{i-1}X \oplus L_0^{i-1}H_\bullet(G_{i-1}X,(B,g)),\widetilde{\inj_2})\\
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&=&(F_{i-1} \times \id)\left(G_{i-1}X \oplus L_0^{i-1}\Hbar_\bullet(G_{i-1}X,\Rtsc),(A,h)\right)\\
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&=&(F_{i-1} \times \id)\left(G_{i-1}X \oplus L_0^{i-1}\Hbar_\bullet(G_{i-1}X,(B,g)),(A,h)\right)\\
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&=&\left(F_{i-1}(G_{i-1}X \oplus L_0^{i-1}\Hbar_\bullet(G_{i-1}X,\Rtsc)),(A,h)\right)
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&=&\left(F_{i-1}(G_{i-1}X \oplus L_0^{i-1}\Hbar_\bullet(G_{i-1}X,(B,g))),(A,h)\right)
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\end{array}\]
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\end{array}\]
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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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% 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 using the two isomorphisms given by the induction hypothesis and hypothesis H3.
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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. $B$.
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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 E2.json
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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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\begin{center}
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% YADE DIAGRAM E3.json
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% GENERATED LATEX
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\input{graphs/E3.tex}
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% END OF GENERATED LATEX
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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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\begin{center}
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% YADE DIAGRAM E4.json
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% GENERATED LATEX
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\input{graphs/E4.tex}
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% END OF GENERATED LATEX
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\end{center}
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\end{document}
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\end{document}
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