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Floris van Doorn 2018-09-24 14:05:50 +02:00
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Notes/smash.tex
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@ -216,7 +216,7 @@ We define the pointed equivalences:
\begin{lem}\label{lem:smash-general} \begin{lem}\label{lem:smash-general}
The smash product is functorial: if $f:A\pmap A'$ and $g:B\pmap B'$ then The smash product is functorial: if $f:A\pmap A'$ and $g:B\pmap B'$ then
$f\smsh g:A\smsh B\pmap A'\smsh B'$. We write $A\smsh g$ or $f\smsh B$ if one of the $f\smsh g:A\smsh B\pmap A'\smsh B'$. We write $A\smsh g$ or $f\smsh B$ if one of the
functions is the identity function. Moreover, if $p:f\sim f'$ and $q:g\sim g'$ then $p\smsh q:f\smsh g\sim f'\smsh g'$; this operation preserves reflexivities, symmetries and transitivities. We will write $p \smsh g$ or $f \smsh q$ if one of the homotopies is reflexivity. functions is the identity function. Moreover, if $p:f\sim f'$ and $q:g\sim g'$ then $p\smsh q:f\smsh g\sim f'\smsh g'$; this operation preserves reflexivities, symmetries and transitivies. We will write $p \smsh g$ or $f \smsh q$ if one of the homotopies is reflexivity.
% The smash product satisfies the following properties. % The smash product satisfies the following properties.
% \begin{itemize} % \begin{itemize}
% \item The smash product is functorial: if $f:A\pmap A'$ and $g:B\pmap B'$ then % \item The smash product is functorial: if $f:A\pmap A'$ and $g:B\pmap B'$ then
@ -357,59 +357,49 @@ We define the pointed equivalences:
Suppose that we have maps $A_1\lpmap{f_1}A_2\lpmap{f_2}A_3$ and $B_1\lpmap{g_1}B_2\lpmap{g_2}B_3$ 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 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: $(f_2 \o f_1)\smsh (g_2 \o g_1)\sim 0$, one which uses the interchange law and one which does not. These two homotopies are equal. Specifically, the following two diagrams commute:
\begin{equation}\label{eq:pent-statement} \begin{center}
\begin{tikzcd}[column sep=2em] \begin{tikzcd}
(f_2 \o 0)\smsh (g_2 \o g_1) (f_2 \o 0)\smsh (g_2 \o g_1)
\arrow[r, equals, "i"] \arrow[r, equals, "i"]
\arrow[dd, swap, equals, "\zeroh' \smsh (g_2 \o g_1)"] \arrow[dd, swap, equals, "\zeroh' \smsh (g_2 \o g_1)"]
&(f_2 \smsh g_2)\o (0 \smsh g_1) &(f_2 \smsh g_2)\o (0 \smsh g_1)
\arrow[d, equals, "(f_2 \smsh g_2) \o t_{g_1}"] \arrow[d, equals, "(f_2 \smsh g_2) \o t_{g_1}"]
& (0 \o f_1)\smsh (g_2 \o 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)
\arrow[r, equals, "i"] \arrow[r, equals, "i"]
\arrow[dd, swap, equals, "\zeroh \smsh (g_2 \o g_1)"] \arrow[dd, swap, equals, "\zeroh \smsh (g_2 \o g_1)"]
& (0 \smsh g_2)\o (f_1 \smsh g_1) & (0 \smsh g_2)\o (f_1 \smsh g_1)
\arrow[d,equals, "t_{g_2} \o (f_1 \smsh g_1)"] \arrow[d,equals, "t_{g_2} \o (f_1 \smsh g_1)"]
\\ \\
& (f_2 \smsh g_2)\o 0 & 0\o (f_1 \smsh g_1)
\arrow[d,equals, "\zeroh'"]
&& 0\o (f_1 \smsh g_1)
\arrow[d,equals, "\zeroh"] \arrow[d,equals, "\zeroh"]
\\ \\
0\smsh (g_2 \o g_1) 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}"] \arrow[r,swap, equals, "t_{g_2 \o g_1}"]
& 0 & 0
\end{tikzcd} \end{tikzcd}
% \qquad \end{center}
% \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} \end{lem}
\begin{proof} \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). % 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 $f_1\judgeq 0$}. 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 $g_2$ \textbf{Case $f_1\judgeq 0$ (diagram on the left)}. 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 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 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 $(f_2 \o 0)\smsh (g_2 \o g_1)\sim0\smsh (g_2 \o g_1)$ is also equal to reflexivity, and also the
path that $f_2 \smsh g_2$ respects the basepoint is reflexivity, hence the homotopy 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: $(f_2 \smsh g_2)\o 0\sim0$ is also reflexivity. This means we need to fill the following square:
\begin{equation}\label{eq:pent-left-all} \begin{center}
\begin{tikzcd} \begin{tikzcd}
(f_2 \o 0)\smsh (g_2 \o g_1) (f_2 \o 0)\smsh (g_2 \o g_1)
\arrow[r, equals,"i"] \arrow[r, equals,"i"]
@ -421,7 +411,7 @@ We define the pointed equivalences:
\arrow[r, swap, equals,"t_{g_1 \o g_2}"] \arrow[r, swap, equals,"t_{g_1 \o g_2}"]
& 0 & 0
\end{tikzcd} \end{tikzcd}
\end{equation} \end{center}
For the underlying homotopy, take $x : A_1\smsh B_1$ and apply induction on $x$. Suppose 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): $x\equiv(a,b)$ 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} \begin{equation}\label{eq:pent-left-ab}
@ -487,35 +477,7 @@ We define the pointed equivalences:
\arrow[from=uu, equals, crossing over, very near start, "\mapfunc{f_2 \smsh g_2}(\gluer\tr\gluer\sy)"] \arrow[from=uu, equals, crossing over, very near start, "\mapfunc{f_2 \smsh g_2}(\gluer\tr\gluer\sy)"]
\end{tikzcd} \end{tikzcd}
\end{equation} \end{equation}
Such a cube can be generalized to the cube: 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).
\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{equation}\label{eq:pent-left-gluer}
\begin{tikzcd}[column sep=4em] \begin{tikzcd}[column sep=4em]
& \auxr & \auxr
@ -544,7 +506,35 @@ We define the pointed equivalences:
\end{tikzcd} \end{tikzcd}
\end{equation} \end{equation}
%After canceling applications of $\mapfunc{h\smsh k}(\gluer_z)=\gluer_{k(z)}$ on various sides of the squares (TODO). %After canceling applications of $\mapfunc{h\smsh k}(\gluer_z)=\gluer_{k(z)}$ on various sides of the squares (TODO).
This cube can be generalized to the cube: 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:
\begin{center} \begin{center}
\begin{tikzcd}[column sep=3em] \begin{tikzcd}[column sep=3em]
& h(y) & h(y)
@ -572,69 +562,14 @@ We define the pointed equivalences:
\arrow[ur, swap, equals, "1"] \arrow[ur, swap, equals, "1"]
\end{tikzcd} \end{tikzcd}
\end{center} \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). Again, a filler is provided by path induction on the squares and paths defining the cube. 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).
(TODO: this homotopy is pointed) \textbf{Case $f_2\judgeq 0$ (diagram on the right)}. (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: To show that this homotopy is pointed, (TODO)
\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} \end{proof}
\begin{lem}
(TODO: the fillers in \autoref{lem:smash-coh} are the same)
\end{lem}
\begin{thm}\label{thm:smash-functor-right} \begin{thm}\label{thm:smash-functor-right}
Given pointed types $A$, $B$ and $C$, the functorial action of the smash product induces a map Given pointed types $A$, $B$ and $C$, the functorial action of the smash product induces a map
$$({-})\smsh C:(A\pmap B)\pmap(A\smsh C\pmap B\smsh C)$$ $$({-})\smsh C:(A\pmap B)\pmap(A\smsh C\pmap B\smsh C)$$
@ -729,7 +664,7 @@ are filled by (corollaries of) \autoref{lem:smash-general}.
$\epsilon_{B,C}\equiv\epsilon : (B\pmap C)\smsh B \pmap C$ dinatural in $B$ and pointed natural in $C$. $\epsilon_{B,C}\equiv\epsilon : (B\pmap C)\smsh B \pmap C$ dinatural in $B$ and pointed natural in $C$.
These maps satisfy the unit-counit laws: These maps satisfy the unit-counit laws:
$$(A\to\epsilon_{A,B})\o \eta_{A\to B,A}\sim \idfunc[A\to B]\qquad $$(A\to\epsilon_{A,B})\o \eta_{A\to B,A}\sim \idfunc[A\to B]\qquad
\epsilon_{B,A\smsh B}\o \eta_{A,B}\smsh B\sim\idfunc[A\smsh B].$$ \epsilon_{B,B\smsh C}\o \eta_{A,B}\smsh B\sim\idfunc[A\smsh B].$$
\end{lem} \end{lem}
Note: $\eta$ is also dinatural in $B$, but we don't need this. Note: $\eta$ is also dinatural in $B$, but we don't need this.
\begin{proof} \begin{proof}