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FabrizioRomanoGenovese committed Dec 6, 2023
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17 changes: 7 additions & 10 deletions _posts/2023-11-28-mechanics-intent-search-ii.md
Original file line number Diff line number Diff line change
Expand Up @@ -100,15 +100,15 @@ If we have two open graphs, and if their interfaces match:
\rotatebox{0}{
\scalebox{1}{
\begin{tikzpicture}
\begin{scope}[xshift=-2cm]
\begin{scope}[xshift=-1cm]
\node[circle, fill, minimum size=5pt, inner sep=0pt, label=left:{$1$}] (al1) at (-2,0) {};
\node[circle, fill, minimum size=5pt, inner sep=0pt, label=right:{$1$}] (ar1) at (0,0) {};
\node[circle, fill, minimum size=5pt, inner sep=0pt, label=right:{$2$}] (ar2) at (0,-1) {};
\node[circle, fill, minimum size=5pt, inner sep=0pt, label=right:{$3$}] (ar3) at (0,-2) {};
\draw[thick] (al1) to (ar1);
\draw[thick, out=180, in=180, looseness=2] (ar2) to (ar3);
\end{scope}
\begin{scope}[xshift=2cm]
\begin{scope}[xshift=1cm]
\node[circle, fill, minimum size=5pt, inner sep=0pt, label=right:{$1$}] (br3) at (2,-2) {};
\node[circle, fill, minimum size=5pt, inner sep=0pt, label=left:{$1$}] (bl1) at (0,0) {};
\node[circle, fill, minimum size=5pt, inner sep=0pt, label=left:{$2$}] (bl2) at (0,-1) {};
Expand Down Expand Up @@ -523,14 +523,11 @@ As $\pi_1$ is defined in terms of $\pi_0$ on the space of parallel paths, in the
>$$
>\require{extpfeil}
>\begin{align*}
>(0,0) \xmapsto{h_0} 0
>&\qquad
>(0,0) \xmapsto{h_0} 0\\
>(0,1) \xmapsto{h_0} 0\\
>(1,1) \xmapsto{h_0} 1\\
>(0,0) \xmapsto{h_1} 0\\
>(0,1) \xmapsto{h_0} 0
>&\qquad
>(0,1) \xmapsto{h_1} 1\\
>(1,1) \xmapsto{h_0} 1
>&\qquad
>(1,1) \xmapsto{h_1} 1
>\end{align*}
>$$
Expand All @@ -552,8 +549,8 @@ $\\{(0,1)\\} \leq [(id_{\\{0,1\\}},id_{\\{0,1\\}})]$ if we can find some other c
All in all, the poset looks like this:

{% quiver %}
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<iframe class="quiver-embed" src="https://q.uiver.app/#q=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&embed" width="1797" height="288" style="border-radius: 8px; border: none;"></iframe>
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<iframe class="quiver-embed" src="https://q.uiver.app/#q=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&embed" width="600" height="400" style="border-radius: 8px; border: none;"></iframe>
{% endquiver %}

As for $\pi_0$, any $g:X \to \\{0,1\\}$ that is an obstruction of the first kind to $f$ will exhibit a number of non equal paths $(h_0,h_1)$, each one to be equal to one of the elements in the poset above.
Expand Down
11 changes: 11 additions & 0 deletions assets/style.scss
Original file line number Diff line number Diff line change
Expand Up @@ -494,6 +494,17 @@ nav {
flex-direction: wrap;
justify-content: center;
}
.quiver {
position: relative;
padding-bottom: 56.25%;
}
iframe{
position: absolute;
top: 0;
left: 0;
width: 100%;
height: 100%;
}

.quiverCaption, .tikzCaption {
flex-direction: wrap;
Expand Down

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