From dd14277e0bfc6270a488eb3b9ec231484065b9d8 Mon Sep 17 00:00:00 2001
From: spiceghello <stefano.piceghello@uib.no>
Date: Mon, 26 Mar 2018 09:11:55 +0200
Subject: [PATCH] additions on pentagons with interchange
---
Notes/smash.tex | 167 +++++++++++++++++++++++++++++++++---------------
1 file changed, 116 insertions(+), 51 deletions(-)
diff --git a/Notes/smash.tex b/Notes/smash.tex
index d0dc55a..2855ede 100644
--- a/Notes/smash.tex
+++ b/Notes/smash.tex
@@ -357,49 +357,59 @@ We define the pointed equivalences:
Suppose that we have maps $A_1\lpmap{f_1}A_2\lpmap{f_2}A_3$ and $B_1\lpmap{g_1}B_2\lpmap{g_2}B_3$
and suppose that either $f_1$ or $f_2$ is constant. Then there are two homotopies
$(f_2 \o f_1)\smsh (g_2 \o g_1)\sim 0$, one which uses the interchange law and one which does not. These two homotopies are equal. Specifically, the following two diagrams commute:
- \begin{center}
- \begin{tikzcd}
+ \begin{equation}\label{eq:pent-statement}
+ \begin{tikzcd}[column sep=2em]
(f_2 \o 0)\smsh (g_2 \o g_1)
\arrow[r, equals, "i"]
\arrow[dd, swap, equals, "\zeroh' \smsh (g_2 \o g_1)"]
&(f_2 \smsh g_2)\o (0 \smsh g_1)
\arrow[d, equals, "(f_2 \smsh g_2) \o t_{g_1}"]
- \\
- & (f_2 \smsh g_2)\o 0
- \arrow[d,equals, "\zeroh'"]
- \\
- 0\smsh (g_2 \o g_1)
- \arrow[r,equals, swap, "t_{g_2 \o g_1}"]
- & 0
- \end{tikzcd}
- \qquad
- \begin{tikzcd}
- (0 \o f_1)\smsh (g_2 \o g_1)
+ & (0 \o f_1)\smsh (g_2 \o g_1)
\arrow[r, equals, "i"]
\arrow[dd, swap, equals, "\zeroh \smsh (g_2 \o g_1)"]
& (0 \smsh g_2)\o (f_1 \smsh g_1)
\arrow[d,equals, "t_{g_2} \o (f_1 \smsh g_1)"]
\\
- & 0\o (f_1 \smsh g_1)
+ & (f_2 \smsh g_2)\o 0
+ \arrow[d,equals, "\zeroh'"]
+ && 0\o (f_1 \smsh g_1)
\arrow[d,equals, "\zeroh"]
\\
0\smsh (g_2 \o g_1)
+ \arrow[r,equals, swap, "t_{g_2 \o g_1}"]
+ & 0
+ & 0\smsh (g_2 \o g_1)
\arrow[r,swap, equals, "t_{g_2 \o g_1}"]
& 0
\end{tikzcd}
- \end{center}
+% \qquad
+% \begin{tikzcd}
+% (0 \o f_1)\smsh (g_2 \o g_1)
+% \arrow[r, equals, "i"]
+% \arrow[dd, swap, equals, "\zeroh \smsh (g_2 \o g_1)"]
+% & (0 \smsh g_2)\o (f_1 \smsh g_1)
+% \arrow[d,equals, "t_{g_2} \o (f_1 \smsh g_1)"]
+% \\
+% & 0\o (f_1 \smsh g_1)
+% \arrow[d,equals, "\zeroh"]
+% \\
+% 0\smsh (g_2 \o g_1)
+% \arrow[r,swap, equals, "t_{g_2 \o g_1}"]
+% & 0
+% \end{tikzcd}
+ \end{equation}
\end{lem}
\begin{proof}
% We will only do the case where $f_1\jdeq 0$, i.e. fill the diagram on the left. The other case is similar (and slightly easier).
- \textbf{Case $f_1\judgeq 0$ (diagram on the left)}. First apply induction on the paths that $f_2$, $g_1$ and $g_2$
+ \textbf{Case $f_1\judgeq 0$}. In order to fill the diagram on the left in (\ref{eq:pent-statement}), we first apply induction on the paths that $f_2$, $g_1$ and $g_2$
respect the basepoint. In this case $f_2\o0$ is definitionally equal to $0$, and the canonical
proof that $f_2\o 0\sim0$ is (definitionally) equal to reflexivity. This means that the homotopy
$(f_2 \o 0)\smsh (g_2 \o g_1)\sim0\smsh (g_2 \o g_1)$ is also equal to reflexivity, and also the
path that $f_2 \smsh g_2$ respects the basepoint is reflexivity, hence the homotopy
$(f_2 \smsh g_2)\o 0\sim0$ is also reflexivity. This means we need to fill the following square:
- \begin{center}
+ \begin{equation}\label{eq:pent-left-all}
\begin{tikzcd}
(f_2 \o 0)\smsh (g_2 \o g_1)
\arrow[r, equals,"i"]
@@ -411,7 +421,7 @@ We define the pointed equivalences:
\arrow[r, swap, equals,"t_{g_1 \o g_2}"]
& 0
\end{tikzcd}
- \end{center}
+ \end{equation}
For the underlying homotopy, take $x : A_1\smsh B_1$ and apply induction on $x$. Suppose
$x\equiv(a,b)$ for $a:A_1$ and $b:B_1$. With the notational convention for basepoints as in \autoref{lem:interchange}, we have to fill the square (we use that the paths that the maps respect the basepoints are reflexivity):
\begin{equation}\label{eq:pent-left-ab}
@@ -477,7 +487,35 @@ We define the pointed equivalences:
\arrow[from=uu, equals, crossing over, very near start, "\mapfunc{f_2 \smsh g_2}(\gluer\tr\gluer\sy)"]
\end{tikzcd}
\end{equation}
- Similarly, if $x$ varies over $\gluer_b$, we need to fill the cube below: the front and the back are the squares in (\ref{eq:pent-left-ab}) for $(a_1,b)$ and (\ref{eq:pent-left-auxr}) respectively; the left square is again degenerate; the other three sides come from the fact that $i$ and $t$ respect $\gluer_b$ (given in (\ref{eq:i-gluer}) and (\ref{eq:t-gluer}) respectively). Again, we omit the arguments of $\gluer$ in $\gluer\tr\gluer\sy$ (in this case, not a priori judgementally equal).
+ Such a cube can be generalized to the cube:
+ \begin{center}
+ \begin{tikzcd}[column sep=3em]
+ & h(y)
+ \arrow[rr, equals,"1"]
+ \arrow[dd, swap, equals, near end,"1"]
+ && h(y)
+ \arrow[dd,equals,"\mapfunc{h}(p_l\sy)"]
+ \\
+ h(x)
+ \arrow[rr, equals, near end, crossing over, "1"]
+ \arrow[dd, swap, equals, "1"]
+ \arrow[ur, equals, "q_l"]
+ && h(x)
+ \arrow[ur, equals, near start, "\mapfunc{h}(p_l)"]
+ \\
+ & h(y)
+ \arrow[rr, swap, equals, near start, "q_l\sy"]
+ && h(x)
+ \\
+ h(x)
+ \arrow[rr, swap, equals,"q_r\tr q_r\sy"]
+ \arrow[ur, equals, "q_l"]
+ && h(x)
+ \arrow[from=uu, equals, crossing over, near start, "\mapfunc{h}(p_r\tr p_r\sy)"]
+ \arrow[ur, swap, equals, "1"]
+ \end{tikzcd}
+ \end{center}
+ for $X$ and $X'$ pointed types; a map $h : X \to X'$; terms $x$, $y$ $z : X$; paths $p_l : x = y$, $p_r : x = z$, $q_l : h(x) = h(y)$, $q_r : h(x) = h(z)$; and 2-paths $s_l : \mapfunc{h}(p_l) = q_l$ (for the back and the top) and $s_r : \mapfunc{h}(p_r) = q_r$ (for the right side). This cube is filled by path induction on $s_l$, $s_r$, $p_l$ and $p_r$. Finally, if $x$ varies over $\gluer_b$, we need to fill the cube below: the front and the back are the squares in (\ref{eq:pent-left-ab}) for $(a_1,b)$ and (\ref{eq:pent-left-auxr}) respectively; the left square is again degenerate; the other three sides come from the fact that $i$ and $t$ respect $\gluer_b$ (given in (\ref{eq:i-gluer}) and (\ref{eq:t-gluer}) respectively). Again, we omit the arguments of $\gluer$ in $\gluer\tr\gluer\sy$ (in this case, not a priori judgementally equal).
\begin{equation}\label{eq:pent-left-gluer}
\begin{tikzcd}[column sep=4em]
& \auxr
@@ -506,35 +544,7 @@ We define the pointed equivalences:
\end{tikzcd}
\end{equation}
%After canceling applications of $\mapfunc{h\smsh k}(\gluer_z)=\gluer_{k(z)}$ on various sides of the squares (TODO).
- In order to fill the cubes in (\ref{eq:pent-left-gluel}) and (\ref{eq:pent-left-gluer}), we generalize the paths and fill the cubes by path induction. The cube in (\ref{eq:pent-left-gluel}) can be generalized to a cube:
- \begin{center}
- \begin{tikzcd}[column sep=3em]
- & h(y)
- \arrow[rr, equals,"1"]
- \arrow[dd, swap, equals, near end,"1"]
- && h(y)
- \arrow[dd,equals,"\mapfunc{h}(p_l\sy)"]
- \\
- h(x)
- \arrow[rr, equals, near end, crossing over, "1"]
- \arrow[dd, swap, equals, "1"]
- \arrow[ur, equals, "q_l"]
- && h(x)
- \arrow[ur, equals, near start, "\mapfunc{h}(p_l)"]
- \\
- & h(y)
- \arrow[rr, swap, equals, near start, "q_l\sy"]
- && h(x)
- \\
- h(x)
- \arrow[rr, swap, equals,"q_r\tr q_r\sy"]
- \arrow[ur, equals, "q_l"]
- && h(x)
- \arrow[from=uu, equals, crossing over, near start, "\mapfunc{h}(p_r\tr p_r\sy)"]
- \arrow[ur, swap, equals, "1"]
- \end{tikzcd}
- \end{center}
- for $X$ and $X'$ pointed types; a map $h : X \to X'$; terms $x$, $y$ $z : X$; paths $p_l : x = y$, $p_r : x = z$, $q_l : h(x) = h(y)$, $q_r : h(x) = h(z)$; and 2-paths $s_l : \mapfunc{h}(p_l) = q_l$ (for the back and the top) and $s_r : \mapfunc{h}(p_r) = q_r$ (for the right side). This cube is filled by path induction on $s_l$, $s_r$, $p_l$ and $p_r$. The cube in (\ref{eq:pent-left-gluer}) can be generalized to a similar cube:
+ This cube can be generalized to the cube:
\begin{center}
\begin{tikzcd}[column sep=3em]
& h(y)
@@ -562,14 +572,69 @@ We define the pointed equivalences:
\arrow[ur, swap, equals, "1"]
\end{tikzcd}
\end{center}
- for paths $p_l : x = y$, $p_b : y = z$, $q_l : h(x) = h(y)$, $q_b : h(y) = h(z)$ and for 2-paths $s_l : \mapfunc{h}(p_l) = q_l$ (for the top) and $s_b : \mapfunc{h}(p_b) = q_b$ (for the back).
+ for paths $p_l : x = y$, $p_b : y = z$, $q_l : h(x) = h(y)$, $q_b : h(y) = h(z)$ and for 2-paths $s_l : \mapfunc{h}(p_l) = q_l$ (for the top) and $s_b : \mapfunc{h}(p_b) = q_b$ (for the back). Again, a filler is provided by path induction on the squares and paths defining the cube.
- \textbf{Case $f_2\judgeq 0$ (diagram on the right)}. (TODO)
+ (TODO: this homotopy is pointed)
- To show that this homotopy is pointed, (TODO)
+ \textbf{Case $f_2\judgeq 0$}. The case for the diagram on the right in (\ref{eq:pent-statement}) is easier and can be resolved analogously. Again, we first apply induction on the paths that $f_1$, $g_1$ and $g_2$ respect the basepoint, reducing the diagram to the following square:
+ \begin{equation}\label{eq:pent-right-all}
+ \begin{tikzcd}
+ (0 \o f_1)\smsh (g_2 \o g_1)
+ \arrow[r, equals,"i"]
+ \arrow[d, swap, equals,"1"]
+ & (0 \smsh g_2)\o (f_1 \smsh g_1)
+ \arrow[d,equals,"t_{g_1}\o (f_2\smsh g_2)"]
+ \\
+ 0 \smsh (g_2 \o g_1)
+ \arrow[r, swap, equals,"t_{g_1 \o g_2}"]
+ & 0
+ \end{tikzcd}
+ \end{equation}
+ For the underlying homotopy. let $x : A_1 \smsh B_1$ and apply induction on $x$. For the point constructors: if $x \judgeq (a,b)$, $x \judgeq \auxl$ or $x \judgeq \auxr$, we have to fill (respectively) the squares:
+ \begin{equation}\label{eq:pent-right-ab}
+ \begin{tikzcd}[column sep=5em]
+ (a_3,g_2(g_1(b)))
+ \arrow[r, equals,"1"]
+ \arrow[d,swap,equals,"1"]
+ %\arrow[d,equals,"\gluer_{g_2(g_1(b))}\tr\gluer_{g_2(g_1(b_1))}\sy"]
+ & (a_3,g_2(g_1(b)))
+ \arrow[d,equals,"\gluer_{g_2(g_1(b))}\tr\gluer_{b_3}\sy"]
+ \\
+ (a_3,g_2(g_1(b)))
+ \arrow[r,swap,equals,"\gluer_{g_2(g_1(b))}\tr\gluer_{b_3}\sy"]
+ & (a_3,b_3)
+ \end{tikzcd}
+ \end{equation}
+ \begin{equation}\label{eq:pent-right-aux}
+ \begin{tikzcd}[column sep=5em]
+ \auxl
+ \arrow[r, equals,"1"]
+ \arrow[d,swap,equals,"1"]
+ & \auxl
+ \arrow[d,equals,"\gluel_{a_3}\sy"]
+ & \auxr
+ \arrow[r, equals, "1"]
+ \arrow[d, swap, equals, "1"]
+ & \auxr
+ \arrow[d, equals, "\gluer_{b_3}\sy"]
+ \\
+ \auxl
+ \arrow[r,swap, equals,"\gluel_{a_3}\sy"]
+ & (a_3, b_3)
+ & \auxr
+ \arrow[r, swap, equals, "\gluer_{b_3}\sy"]
+ & (a_3, b_3)
+ \end{tikzcd}
+ \end{equation}
+ which can be done by reflexivity. The cubes to be filled for $x$ varying over $\gluel_a$ or $ gluel_b$ are similar to the ones in (\ref{eq:pent-left-gluel}) and (\ref{eq:pent-left-gluel}) and can be generalized accordingly. (ADD REMARK cubically, ap id p reduces to refl, so the cases are exactly the same).
+ (TODO: this homotopy is pointed)
\end{proof}
+\begin{lem}
+ (TODO: the fillers in \autoref{lem:smash-coh} are the same)
+\end{lem}
+
\begin{thm}\label{thm:smash-functor-right}
Given pointed types $A$, $B$ and $C$, the functorial action of the smash product induces a map
$$({-})\smsh C:(A\pmap B)\pmap(A\smsh C\pmap B\smsh C)$$