diff --git a/draft/draft.tex b/draft/draft.tex index 8d19dcb..b2c1b62 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -2281,7 +2281,7 @@ Now, we want to abstract the given proof for an extensive category that has term \item $h=Fg\comp k$: In this case we take $k'=k$. Now, $k'\appr k$ and $g+1\comp k'=h$.\qed \end{itemize} \end{proof} -To have an abstraction of~\autoref{lem:maybe-func-set} recalling that our category extensive, we use the fact that for an arbitrary morphism $f\c X\to Y+Z$, we can have morphisms $f_Y\c X_Y\to Y$ and $f_Z\c X_Z\to Z$, such that $X_Y+X_Z\iso X$, so we follow with $f_Y+f_Z$. In our case, we take $\brks{\alpha\comp p_1,\alpha\comp p_2}\c R\to (X+1)\times(X+1)$, where $(X+1)\times (X+1)\iso X^2+(2\times X)+1$, and we assume the following pullbacks exist: +To have an abstraction of~\autoref{lem:maybe-func-set} recalling that our category is extensive, we use the fact that for an arbitrary morphism $f\c X\to Y+Z$, we can have morphisms $f_Y\c X_Y\to Y$ and $f_Z\c X_Z\to Z$, such that $X_Y+X_Z\iso X$, so we follow with $f_Y+f_Z$. In our case, we take $\brks{\alpha\comp p_1,\alpha\comp p_2}\c R\to (X+1)\times(X+1)$, where $(X+1)\times (X+1)\iso X^2+(2\times X)+1$, and we assume the following pullbacks exist: \begin{equation*} \begin{tikzcd}[ampersand replacement=\&] {R_{X^2}} \& R \& {R_{2\times X}} \& R \\