RAlg.tex

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\chapter{$\mathbb{R}$-ALG AND $\mathbb{L}$-COALG} \label{Ch:RAlg}

For this chapter, we will continue to let $\mathbb{D}=\Sq(\twocat{D})$ be the double category of squares in a 2-category $\twocat{D}$ with arrow objects.

A weak factorization system on a category $C$ is defined by two classes of morphisms, $\mathcal{L}$ and $\mathcal{R}$.
In an algebraic weak factorization system, these classes of morphisms are replaced by categories $\LCoalg$ and $\RAlg$ equipped with functors to $C^2$. In this chapter, we will discuss the universal property satisfied by these categories, allowing us to define analagous objects in other 2-categories, and record several technical lemmas which we will need in~\cref{Ch:Composition}. We will focus on comonads, but there are dual results for monads which we leave to the reader.

Recall from~\cite{street:ftm} the following proposition.

\begin{proposition}\label{Prop:EMObject}
	Let $C$ be a category, and $\mathbb{L}=(L,\epsilon,\delta)$ be a comonad on $C$. The category of coalgebras $\LCoalg$ has a universal property as follows:
	\begin{itemize}
	 	\item There is a forgetful functor $U\colon \LCoalg \to C$ and a natural transformtion $\alpha\colon U \Rightarrow LU$, satisfying $\epsilon U\circ\alpha = \id_U$ and $\delta U \circ \alpha = L\alpha\circ\alpha$.
	 	\item $(U,\alpha)$ is universal among such pairs satisfying such equations. Given another such pair $(F,\beta)$, where $F\colon X\to C$, there exists a unique functor $\hat{F}\colon X\to \LCoalg$ such that $U\hat{F}=F$ and $\alpha\hat{F}=\beta$.
	 \end{itemize}
	 Any colax morphism of comonads $(F,\phi)\colon(C,L_1,\epsilon_1,\delta_1)\to(D,L_2,\epsilon_2,\delta_2)$ induces a functor $\tilde{F}\colon\LCoalg[1]\to\LCoalg[2]$ such that $U_2\tilde{F}=FU_1$ and $\alpha_2\tilde{F}=\phi U_1\circ F\alpha_1$.

	 A natural transformation
	 \[
	 \begin{tikzcd}[bend angle=30]
	 	X \rar[bend left][domA]{\hat{F}_1} \rar[bend right][codA,swap]{\hat{F}_2} & \LCoalg
	 	\twocellA{\hat{\theta}}
	 \end{tikzcd}
	 \]
	 is uniquely determined by the functors $F_1=U\hat{F}_1$ and $F_2=U\hat{F}_2$ and natural transformations $\beta_1=\alpha\hat{F}_1$ and $\beta_2=\alpha\hat{F}_2$, and the natural transformation $\theta=U\hat{\theta}\colon F_1\Rightarrow F_2$, satisfying $L\theta\circ\beta_1=\beta_2\circ\theta$.
\end{proposition}

For the rest of this chapter, assume that $\twocat{D}$ \emph{has EM-objects for comonads}, i.e. for every comonad $\mathbb{L}$ in $\twocat{D}$ there is an object $\LCoalg$ satisfying the universal property above.

It is not too hard to use this universal property to construct the free/forgetful adjunction:

\begin{proposition}
	For any comonad $\mathbb{L}$ on an object $C$ in $\twocat{D}$, the 1-cell $U\colon\LCoalg\to C$ has a right adjoint $\hat{L}$ with $U\hat{L}=L$ and $\alpha\hat{L}=\delta$. The counit of this adjunction is simply the counit of $\mathbb{L}$, $\epsilon\colon U\hat{L}\Rightarrow\id_C$, while the unit is a 2-cell $\hat{\alpha}\colon\id_{\LCoalg}\Rightarrow\hat{L}U$ satisfying $U\hat{\alpha}=\alpha$.
\end{proposition}
\begin{proof}
	By \cref{Prop:EMObject}, to prove the existence of the 1-cell $\hat{L}$, it suffices to verify the equations $\epsilon L\circ\delta=\id_L$ and $\delta L\circ\delta=L\delta\circ\delta$, which are simply two of the comonad axioms.

	Using the 2-dimensional part of \cref{Prop:EMObject}, the existence of the 2-cell $\hat{\alpha}$ follows from the equation $L\alpha\circ\alpha=\delta U\circ\alpha$, which is the remaining comonad axiom.

	We leave the verification of the triangle identities for the adjunction to the reader.
\end{proof}

As our interest is in (co)monads in $\FFD$, which induce (co)monads on arrow objects, it will be useful to record the universal property that results from the interaction of the EM-object and arrow object universal properties.

Consider a comonad in $\FFD$ on an object $C$, i.e. a functorial factorization with half of the awfs structure. We can combine the universal properties of EM-objects and arrow objects into a universal property for $\LCoalg$, where now $\mathbb{L}$ is the comonad in $\twocat{D}$ arising from the comonad in $\FFD$.

\begin{lemma}\label{Lem:FFLCoalgUniversalProperty}
	Let $(E,\eta,\epsilon,\delta)$ be a comonad in $\FFD$ on an object $C$. There is a 2-cell
	\[
	\begin{tikzcd}[row sep=0ex, column sep=4ex, bend angle=15]
		{} & |[alias=domA]| C^2 \drar[bend left][]{\cod} & \\
		\LCoalg \urar[bend left][]{U} \drar[bend right][swap]{U} && C \\
		& |[alias=codA]| C^2 \urar[bend right][swap]{E}
		\twocellA{\alpha}
	\end{tikzcd}
	\]
	satisfying equations
	\begin{gather}
	\begin{tikzcd}[ampersand replacement=\&, row sep=tiny, baseline=(B.base)]
		|[alias=B]| \LCoalg \rar{U} \drar[bend right=20][swap]{U}
			\& |[alias=domA]| C^2 \rar[bend left=60][domB]{\dom} \rar[][codB,swap]{\cod}
			\& C \\
		\& |[alias=codA]| C^2 \urar[bend right=20][swap]{E} \&
		\twocellA{\alpha}
		\twocellB{\kappa}
	\end{tikzcd}
	=
	\begin{tikzcd}[ampersand replacement=\&, bend angle=30, baseline=(B.base)]
		|[alias=B]| \LCoalg \rar{U}
			\& C^2 \rar[bend left][domA]{\dom} \rar[bend right][codA,swap]{E}
			\& C
		\twocellA{\eta}
	\end{tikzcd} \label{Eq:LCoalg1}
	\\
	\begin{tikzcd}[ampersand replacement=\&, row sep=tiny, baseline=(B.base)]
		{} \& |[alias=domA]| C^2 \drar[bend left=20][]{\cod} \& \\
		|[alias=B]| \LCoalg \urar[bend left=20][]{U} \rar[swap]{U}
			\& |[alias=codA]| C^2 \rar[][domB]{E} \rar[bend right=60][codB,swap]{\cod}
			\& C
		\twocellA{\alpha}
		\twocellB{\epsilon}
	\end{tikzcd}
	=
	\begin{tikzcd}[ampersand replacement=\&]
		X \rar{U} \& C^2 \rar{\cod} \& C
	\end{tikzcd} \label{Eq:LCoalg2}
	\\
	\begin{tikzcd}[ampersand replacement=\&, row sep=tiny, baseline=(B.base)]
		{} \& |[alias=domA]| C^2 \ar[bend left=20]{drr}{\cod} \&[-2em]\&[-2em] \\
		|[alias=B]| \LCoalg \urar[bend left=20][]{U} \rar[swap]{U}
			\& |[alias=codA]| C^2 \ar{rr}[domB]{E} \drar[bend right=30][swap]{L}
			\&\& C \\
		\&\& |[alias=codB]| C^2 \urar[bend right=30][swap]{E} \&
		\twocellA{\alpha}
		\twocellB{\delta}
	\end{tikzcd}
	=
	\begin{tikzcd}[ampersand replacement=\&, row sep=tiny, baseline=(B.base)]
		{} \&[-2em]\&[-2em] |[alias=domA]| C^2 \drar[bend left=20]{\cod} \& \\
		|[alias=B]| \LCoalg \ar[bend left=20]{urr}{U} \ar{rr}[domB]{U}  \drar[bend right=30][swap]{U}
			\&\& |[alias=codA]| C^2 \rar[swap]{E}
			\& C. \\
		\& |[alias=codB]| C^2 \urar[bend right=30][swap]{L} \&\&
		\twocellA{\alpha}
		\twocellB{\vec{\alpha}}
	\end{tikzcd} \label{Eq:LCoalg3}
	\end{gather}
	where $\vec{\alpha}$ is the unique 2-cell such that $\dom\vec{\alpha}=\id_{\dom U}$ and $\cod\vec{\alpha}=\alpha$, the existence of which is implied by equation~\eqref{Eq:LCoalg1}.

	Given any object $X$, together with a morphism $F\colon X\to C^2$ and a 2-cell $\beta\colon \cod F\Rightarrow EF$ satisfying equations
	\begin{compactenum}
		\item $\beta\circ\kappa F=\eta F$
		\item $\epsilon F\circ\beta = \id_{\cod F}$
		\item $\delta F\circ\beta = E\vec{\beta}\circ\beta$
	\end{compactenum}
	where $\vec{\beta}\colon F\Rightarrow LF$ is the unique 2-cell such that $\dom\vec{\beta}=\id_{\dom F}$ and $\cod\vec{\beta}=\beta$; there is a unique morphism $\hat{F}\colon X\to \LCoalg$ such that $U\hat{F}=F$ and $\alpha\hat{F}=\beta$.

	Given any pair of morphisms $\hat{F}_1,\hat{F}_2\colon X\to\LCoalg$ and a 2-cell $\vec{\theta}\colon F_2\Rightarrow F_2$ such that
	\[
		E\vec{\theta}\circ\beta_1 = \beta_2\circ\cod\vec{\theta}
	\]
	(where $F_i=U\hat{F}_i$ and $\beta_i=\cod\alpha\hat{F}_i$ as in the previous paragraph), there is a unique 2-cell $\hat{\theta}\colon\hat{F}_1\Rightarrow\hat{F}_2$ such that $U\hat{\theta}=\vec{\theta}$.
\end{lemma}
\begin{proof}
	$U$ is simply the $U$ from \cref{Prop:EMObject}, while the 2-cell $\alpha$ there is the 2-cell $\vec{\alpha}$ here. The equation $\vec{\epsilon}U\circ\vec{\alpha}=\id_{F}$ implies that $\dom \vec{\alpha}=\id_{\dom U}$. With that observation, the rest of the equations follow immediately from the universal property of $C^2$ and the equations $\epsilon U\circ\alpha = \id_U$ and $\delta U \circ \alpha = L\alpha\circ\alpha$ from \cref{Prop:EMObject}.
\end{proof}