\documentclass[draft]{article} \input homework \input theorem \usepackage{color} \title{A Comonadic Generalization of {\bf Top}} \author{Jason Reed} \def\ppf{p} \def\T{\mathcal{T}} \def\C{\mathbf{C}} \def\Cat{\mathbf{Cat}} \def\Top{\mathbf{Top}} \def\Sets{\mathbf{Sets}} \def\Pspa{P\mathbf{Spa}} \def\CAT{\mathbf{CAT}} \begin{document} \maketitle Consider an object $P : \C \to \Cat$ of the coslice category $\CAT/\Cat$. A {\em $P$-space} is defined as pair $(C, U, \epsilon, \delta)$ where $C$ is an object of $\C$, and $U$ is a comonad (with counit $\epsilon$ and comultiplication $\delta$) in the category $PC$. We usually just refer to $C$ when the naming of the remaining pieces is evident. A {\em $P$-continuous map} $C_1 \to C_2$ between $P$-spaces is given by a pair $(f , \gamma)$ where $f : C_1 \to C_2$ and $\gamma$ is a natural transformation $Pf \circ U_1 \to U_2 \circ Pf$ such that (abbreviating $Pf = \ppf$) $$\begin{diagram} \ppf U_1 & \rTo^{\gamma} & U_2 \ppf \\ \dTo<{\ppf \delta_1}\\ \ppf U_1^2 &&\dTo>{\delta_2}\\ \dTo<{\gamma_{U_1}} \\ U_2 \ppf U_1 & \rTo_{U_2 \gamma} & U_2^2 \ppf \\ \end{diagram} \qquad \begin{diagram} \ppf U_1 & \rTo^{\gamma} & U_2\ppf \\ \dTo<{\ppf \epsilon_1}& &\dTo>{\epsilon_2}\\ \ppf &\rEq& \ppf \\ \end{diagram}$$ In other words, $\gamma$ is a coalgebra morphism $\gamma_{U_1} \circ (p\delta_1) \to \delta_2$, and also $p\epsilon_1 \to \epsilon_2$, acting on coalgebras for the functor $U_2$, and the constantly-$p$ functor, respectively. Composition and identities are defined by $$(f',\gamma') \circ (f,\gamma) = (f' \circ f, (\gamma' * Pf) \circ (Pf' * \gamma))$$ $$id_{C,U} = (id_C,id_U)$$ % It's easy to check that they satisfy unit and associativity properties. Thus we get a category $\Pspa$ of $P$-spaces and $P$-continuous maps. An {\em open object} of a $P$-space $C$ is a $U$-coalgebra, an arrow $a : X \to UX$ in $PC$ satisfying `comonoid action' axioms with respect to the comonad: $$\begin{diagram} X&\rTo^a& UX\\ \dTo{\delta}\\ UX&\rTo_{Ua}& U^2X\\ \end{diagram} \qquad \begin{diagram} X&\rTo^a&UX\\ &\rdEq&\dTo>\epsilon\\ &&X\\ \end{diagram}$$ \begin{lemma} There is a functor $Op : \Pspa \to \Cat$, which takes a $P$-space and yields the category its of open objects. \end{lemma} \begin{proof} Arrows in $Op(C)$ are the standard notion of coalgebra morphism. The effect of $Op$ on an arrow in $\Pspa$ is as follows. We take in $(f,\gamma)$ a $P$-continuous map $C_1 \to C_2$, and must output a functor $Op(C_1) \to Op(C_2)$. First we define the object part of this functor: if $a : X \to UX$ is an open object in $C_1$, then we claim $\gamma_X \circ Pf(a)$ is an open object in $C_2$, with underlying object $Pf(X)$. We must check that the comonad algebra axioms hold. Abbreviate again $Pf = \ppf$. Cells marked $\mathsf{A}$ follow by hitting assumptions with $p$, $\mathsf{N}$ follows by naturality of $\gamma$, and $\star$ are from the definition of $P$-continuous. $$\begin{diagram} \ppf X&\rTo^{\ppf a}& \ppf U_1X & \rTo^{\gamma_X} & U_2 \ppf X\\ \dTo<{\ppf a}& \mathsf{A} &\dTo>{\ppf \delta_1}\\ \ppf U_1X&\rTo_{\ppf U_1a}& \ppf U_1^2X &\star&\dTo>{\delta_2}\\ \dTo<{\gamma_X} & \mathsf{N} & \dTo>{\gamma_{U_1X}} \\ U_2 \ppf X& \rTo_{U_2 \ppf a} & U_2 \ppf U_1 X & \rTo_{U_2 \gamma_X} & U_2^2 \ppf X\\ \end{diagram}$$ $$\begin{diagram} \ppf X&\rTo^{\ppf a}&\ppf U_1X & \rTo^{\gamma_X} & U_2\ppf X\\ &\rdEq>{\mathsf{A}}&\dTo>{\ppf \epsilon_1}&\star &\dTo>{\epsilon_2}\\ &&\ppf X&\rEq& \ppf X\\ \end{diagram}$$ For the arrow part of the functor $Op(C_1) \to Op(C_2)$ we must consider composition of coalgebra morphisms, but these are preserved by $a \mapsto \gamma_X \circ p a$: $$\begin{diagram} \ppf X&\rTo^{\ppf a}& \ppf U_1X & \rTo^{\gamma_X} & U_2 \ppf X\\ \dTo<{\ppf f}& \mathsf{A} &\dTo~{\ppf U_1 f}&\mathsf{N}&\dTo>{U_2 \ppf f}\\ \ppf Y&\rTo_{\ppf b}& \ppf U_1Y &\rTo_{\gamma_Y}&U_2 \ppf Y\\ \dTo<{\ppf g}& \mathsf{A} &\dTo~{\ppf U_1 g}&\mathsf{N}&\dTo>{U_2 \ppf g}\\ \ppf Z&\rTo_{\ppf c}& \ppf U_1Z &\rTo_{\gamma_Z}&U_2 \ppf Z\\ \end{diagram}$$ as are identities: $$\begin{diagram} \ppf X&\rTo^{\ppf a}& \ppf U_1X & \rTo^{\gamma_X} & U_2 \ppf X\\ \dTo<{\ppf (id)}& \mathsf{A} &\dTo~{\ppf U_1 (id)}&\mathsf{N}&\dTo>{U_2 \ppf (id)}\\ \ppf X&\rTo_{\ppf a}& \ppf U_1X &\rTo_{\gamma_X}&U_2 \ppf X\\ \end{diagram}$$ \cqed \end{proof} \begin{theorem} Let $P$ be the functor $\Sets^\op \to \Cat$ that takes $X$ to the evident poset category arising from its powerset $\ps X$ ordered by inclusion, and takes a function $f : X \to Y$ to its inverse image map $f^< : \ps Y \to \ps X$. Then the category $\Top$ is isomorphic to $\Pspa^\op$. \end{theorem} \begin{proof} For a $P$-space $C$, take $C$ as the underlying set of the topology, and the open objects of $C$ as the open sets of the topology. Given a topological space $(X, \T)$, let $U$ be the interior operation. The comonad data $\epsilon$ and $\delta$ simply record the decreasing and idempotent properties of the interior operation in a topological space. Then $(X, U, \epsilon, \delta)$ is a $P$-space. Check that the two definitions of continuity match up. \cqed \end{proof} \begin{conjecture} There is a nice class of maps from the open objects of $C_1$ to the open objects of $C_2$ such that every map that belongs to this class arises from a $P$-continuous map. \end{conjecture} %% Is $\gamma_{U_1} \circ p \delta_1$ a $U_2$-algebra on $pU_1$? %% $$\begin{diagram} %% \ppf U_1 & \rTo^{p\delta_1} & pU_1^2&\rTo^{\gamma_{U_1}}& U_2 \ppf U_1\\ %% \dTo<{\ppf \delta_1}&\mathsf{Ass}&\dTo~{p\delta_{1(U_1)}}\\ %% \ppf U_1^2 &\rTo~{p U_1 \delta_1 }&pU_1^3&&\dTo>{\delta_2}\\ %% \dTo<{\gamma_{U_1}} &\mathsf{N}&\dTo~{\gamma_{U_1^2}}\\ %% U_2 \ppf U_1 & \rTo_{U_2 p\delta_1} & U_2 p U_1^2 & \rTo_{U_2 \gamma_{U_1}} & U_2^2 \ppf U_1 \\ %% \end{diagram}$$ %% uh, it is only if it is already an $U_2$-algebra morphism. This saves me no definitional effort! \end{document}