naturaltransfdefcompositions.tex

%% A tentative, eye-catchy, one-page definition of category (I would like to replicate the page frome my handwritten notes.
\documentclass[preview]{standalone}

\usepackage{amsmath}
\usepackage{verbatim}
\usepackage[italian,english]{babel}
\usepackage[utf8]{inputenc}

\usepackage[basic,cat]{./Math-Symbols-List/toninus-math-symbols}
\usepackage{./Latex-Theorem/theoremtemplate}
\usepackage{./visualcat}


\begin{document} 
\begin{definition}[Vertical composition]
	\begin{tikzcd}
		\beta \circ \alpha: F \xrightarrow[]{\cdot} H & = &
		\cat[C] 
			\arrow[rrr, bend left=60, "F"] 
			\arrow[rrr, phantom, bend left=60, ""{name=U}] 
			\arrow[rrr, bend right=60, "H"']
			\arrow[rrr, phantom, bend right=60, ""'{name=D,below}]		
		 	\arrow[rrr, "G"]
			\arrow[rrr, ""{name=C}]		 	
		 	& & & \cat[D]
		\arrow[Rightarrow,blue,"\alpha"', from=U, to=C,shift right=2ex]
		\arrow[Rightarrow,blue,"\beta"', from=C, to=D,shift right=2ex]
		\arrow[Rightarrow,red,"\beta \circ \alpha"'{below right}, from=U, to=D,shift left=2ex]
	\end{tikzcd}
	
	\begin{tikzcd}
		x \ar[d,"s"] & & F(x) \ar[rr,bend left = 30,red,"(\beta \circ \alpha)_x =\beta_x \alpha_x"] \ar[r,"\alpha_x"] \ar[d,"F(s)"] \ar[dr,phantom,"\commute"]& G(x) \ar[r,"\beta_x"] \ar[d,"G(s)"] \ar[dr,phantom,"\commute"]& H(x) \ar[d,"H(s)"] \\
		y & & F(y) \ar[rr,bend right = 30,red,"(\beta \circ \alpha)_y = \beta_y \alpha_y"] \ar[r,"\alpha_y"] & G(y) \ar[r,"\beta_y"] & H(y) 
	\end{tikzcd}
\end{definition}

\begin{definition}[Horizontal composition]
	\begin{tikzcd}
		\beta \ast \alpha: F' F \xrightarrow[]{\cdot} G' G & = &	
		\cat[C] 
			\arrow[rr, bend left=30, "F"{name=U1}] 
			\arrow[rr, bend right=30, "G"'{name=D1}]
			\arrow[rrrr, bend left=60, "F' \circ F"{name=U}]
			\arrow[rrrr, bend right=60, "G' \circ G"{name=D}] 		 	
		 & &\cat[D]
			\arrow[rr, bend left=30, "F'"{name=U2}] 
			\arrow[rr, bend right=30, "G'"'{name=D2}]		 
		 & &\cat[E]
		\arrow[Rightarrow,"\alpha"', from=U1, to=D1]
		\arrow[Rightarrow,"\beta", from=U2, to=D2]
		\arrow[Rightarrow,red,"\beta \ast \alpha"'{above left}, from=U, to=D,shift left=3ex]
	\end{tikzcd}
	
	$\St$ $(\beta\ast \alpha)_x = G' \alpha_x \cdot \beta_{F(x)} = \beta_{G(x)} \cdot F' \alpha_x$

	\begin{tikzcd}[row sep=scriptsize, column sep=scriptsize]
		x \ar[dd,"s"] & & F(x) \ar[rr,"\alpha_x"] \ar[dd,"F(s)"] \ar[ddrr,phantom,"\commute"]& & G(x) \ar[dd,"G(s)"] & & F'F(x)\ar[dd," "] \ar[rr,"F'\alpha_x "] \ar[drrr,"(\beta \ast \alpha)_x",red] \ar[dr,"\beta_{F(x)} "] & & F'G(x)\ar[dd," "]\ar[dr,"\beta_{G(x)} "]\\
		& & & & & & & G'F(x) \ar[dd," "] \ar[rr,"G'\alpha_x "] & & G'G(x)\ar[dd," "] \\
		y & & F(y) \ar[rr,"\alpha_y"] & & G(y) & & F'F(y) \ar[rr,"F'\alpha_y "]\ar[dr,"\beta_{F(y)}"]\ar[drrr,"(\beta \ast \alpha)_y",red] & & F'G(y)\ar[dr,"\beta_{G(y)}"] \\
		& & & & & & & G'F(y) \ar[rr,"G'\beta_y "] & & G'G(y)  		
	\end{tikzcd}

\end{definition}

\end{document}