张量积之间的箭头

Question

张量积 $A\otimes B\to C\otimes D$ 之间的箭头由其分量 $A\to C$ 和 $B\to D$ 决定。因此，用一对箭头来表示这样的箭头似乎很自然。我的问题是：如何在 tikz-cd 中绘制这样的箭头？

以下是我的尝试。但是，它太复杂了，中间的箭头$\otimes$无法匹配右侧张量积的正确边界。

\[
\begin{tikzcd}[row sep=0pt]
A\ar[r,phantom,shift right=0.8em,"{f}"]&
Cup\\
\otimes\ar[r,"\otimes"description]&\otimes\\
B\ar[r,phantom,shift left=0.8em,"{g}"]&
Down
\end{tikzcd}
\]

Answer 1

我真的很喜欢这个想法，但在这种情况下，我实际上想争辩说，人们可以使用tikzmark来注释方程式。换句话说，我并不 100% 确信人们应该tikz-cd只因为它的箭头而开始使用。毕竟，LaTeX 本身非常适合排版方程式，人们只需使用 Ti 添加箭头即可钾Z 而不是设计完整的方程。

\documentclass[fleqn]{article}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{tikzmark}
\begin{document}
\[
\left.\begin{array}{c}\tikzmarknode{A}{A}\\ \otimes \\ 
\tikzmarknode{B}{B}\end{array}\!\right\}
\xrightarrow{f\otimes g}
\left\{\!\begin{array}{c}\tikzmarknode{Cup}{\text{Cup}}\\ \otimes \\ 
\tikzmarknode{Down}{\text{Down}}\end{array}\right.
\]
\begin{tikzpicture}[overlay,remember picture]
\draw[-latex,preaction={draw,white,very thick,-}] (A) to[bend left=15] node[midway,above]{$\scriptstyle f$} (Cup);
\draw[-latex,preaction={draw,white,very thick,-}] (B)  to[bend right=15] node[midway,below]{$\scriptstyle g$} (Down);
\end{tikzpicture}

\[
\left.\begin{array}{c}\tikzmarknode{A'}{A}\\ \otimes \\ 
\tikzmarknode{B'}{B}\end{array}\!\right\}
\xrightarrow{f\otimes g}
\left\{\!\begin{array}{c}\tikzmarknode{Cup'}{\text{Cup}}\\ \otimes \\ 
\tikzmarknode{Down'}{\text{Down}}\end{array}\right.
\]
\begin{tikzpicture}[overlay,remember picture]
\draw[->,preaction={draw,white,very thick,-}] (A') to
node[midway,above]{$\scriptstyle f$} (Cup');
\draw[->,preaction={draw,white,very thick,-}] (B')  to
node[midway,below]{$\scriptstyle g$} (Down');
\end{tikzpicture}

\end{document}

Answer 2

和tikz-cd：

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{tikz-cd}
\usetikzlibrary{decorations.pathreplacing, calligraphy}

\begin{document}

\[
\begin{tikzcd}[
    row sep=0pt,
    every matrix/.append style={name=mycd, nodes={text centered, text height=height{"C"},text depth=depth{"p"}, inner ysep=2pt}
        },    
    /tikz/column 2/.append style={nodes={text width=width("Down")}},
    execute at end picture={
        \draw [decorate,decoration={calligraphic brace, amplitude=5pt}] (mycd-1-1.north east) -- (mycd-3-1.south east);
        \draw [decorate,decoration={calligraphic brace, mirror, amplitude=5pt}] (mycd-1-2.north west) -- (mycd-3-2.south west);
        },
    ]
A\ar[r,"f", shorten=-.1cm]&[2em]\text{Cup}\\
\otimes\ar[r,"f\otimes g", shorten=.3cm]&\otimes\\
B\ar[r,"g"', shorten=-.1cm]&\text{Down}
\end{tikzcd}
\]


\[
\begin{tikzcd}[
    row sep=0pt,
    every matrix/.append style={name=mycd, nodes={text centered, text height=height{"C"},text depth=depth{"p"}, inner ysep=2pt}
        },    
    /tikz/column 2/.append style={nodes={text width=width("Down")}},
    execute at end picture={
        \draw [decorate,decoration={calligraphic brace, amplitude=5pt}] (mycd-1-1.north east) -- (mycd-3-1.south east);
        \draw [decorate,decoration={calligraphic brace, mirror, amplitude=5pt}] (mycd-1-2.north west) -- (mycd-3-2.south west);
        },
    ]
A\ar[r,"f", shorten=-.1cm, bend left]&[2em]\text{Cup}\\
\otimes\ar[r,"f\otimes g", shorten=.3cm]&\otimes\\
B\ar[r,"g"', , shorten=-.1cm, bend right]&\text{Down}
\end{tikzcd}
\]

\end{document}