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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{exponential law for spaces} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{topology}{}\paragraph*{{Topology}}\label{topology} [[!include topology - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{exponentiable_spaces}{Exponentiable spaces}\dotfill \pageref*{exponentiable_spaces} \linebreak \noindent\hyperlink{corecompactness}{Core-compactness}\dotfill \pageref*{corecompactness} \linebreak \noindent\hyperlink{ViaConvergence}{In terms of convergence}\dotfill \pageref*{ViaConvergence} \linebreak \noindent\hyperlink{general_exponential_laws}{General exponential laws}\dotfill \pageref*{general_exponential_laws} \linebreak \noindent\hyperlink{based_exponential_laws}{Based exponential laws}\dotfill \pageref*{based_exponential_laws} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} When in a [[convenient category of topological spaces]], e.g. [[compactly generated space]]s, the category is [[cartesian closed category|cartesian closed]], so that there is an [[adjunction]] between the mapping space and the cartesian product in that category. For general [[topological space]]s there is no globally defined adjunction, but we can instead characterize exactly which spaces are exponentiable. \hypertarget{exponentiable_spaces}{}\subsection*{{Exponentiable spaces}}\label{exponentiable_spaces} For $C$ a [[finitely complete category|category with finite products]], recall that an object $c$ is \textbf{[[exponentiable object|exponentiable]]} if the functor $c \times -: C \to C$ has a [[right adjoint]], usually denoted $(-)^c: C \to C$. \begin{utheorem} Let [[Top]] be the category of all topological spaces. An object $X$ of $Top$ is exponentiable if and only if $X \times -: Top \to Top$ preserves [[coequalizer]]s, or equivalently [[quotient space]]s. \end{utheorem} This functor always preserves [[coproducts]], so this condition is equivalent to saying that $X \times -$ preserves all small [[colimits]]. This is then equivalent to exponentiability by the [[adjoint functor theorem]]. This condition, however, is not really any more explicit. More interesting is to characterize the exponentiable spaces in terms of a point-set-topological condition. \hypertarget{corecompactness}{}\subsubsection*{{Core-compactness}}\label{corecompactness} For open subsets $U$ and $V$ of a topological space $X$, we write $V\ll U$ to mean that any open cover of $U$ admits a finite subcover of $V$; this is read as \textbf{$V$ is relatively compact under $U$} or \textbf{$V$ is way below $U$}. We say that $X$ is \textbf{core-compact} if for every open neighborhood $U$ of a point $x$, there exists an open neighborhood $V$ of $x$ with $V\ll U$. In other words, $X$ is core-compact iff for all open subsets $V$, we have $V = \bigcup \{ U | U\ll V \}$. This says essentially the same thing as saying that the open-set lattice of $X$ is a [[continuous lattice]], which yields the corresponding definition for [[locales]]. \begin{utheorem} An object $X$ of $Top$ is exponentiable if and only if it is core-compact. \end{utheorem} If $X$ is [[Hausdorff space|Hausdorff]], then core-compactness is equivalent to [[locally compact space|local compactness]]; thus in particular all locally compact Hausdorff spaces are exponentiable. In fact, local compactness implies exponentiability even without the Hausdorff condition, if [[locally compact space|local compactness]] is defined appropriately (for every point the compact neighborhoods form a neighborhood basis). For this reason, that core-compactness is also called \textbf{quasi local compactness}. When $X$ is core-compact, we can explicitly describe the exponential topology on $Y^X$ (whose points are continuous maps $f: X \to Y$). It is generated by [[topological basis|subbasis]] elements $O_{U,V}$, for $U$ an open subset of $X$ and $V$ an open subset of $Y$, where a continuous map $f\colon X \to Y$ belongs to $O_{U,V}$ iff $U\ll f^{-1}(V)$: \begin{displaymath} O_{U, V} = \{f \in Y^X: U \ll f^{-1}(V)\} . \end{displaymath} If $X$ and $Y$ are Hausdorff, then this topology on $Y^X$ coincides with the [[compact-open topology]]. \hypertarget{ViaConvergence}{}\subsubsection*{{In terms of convergence}}\label{ViaConvergence} Exponentiable (i.e. core-compact) spaces can also be characterized in terms of [[ultrafilter]] convergence. Recall that a topological space can equivalently be defined as a [[relational beta-module|lax algebra for the ultrafilter monad]] $U$ on the [[(1,2)-category]] [[Rel]] of sets and relations. In other words, it consists of a set $X$ and a relation $R\colon U X \to X$ called ``convergence'', such that $id_X \subseteq R \circ \eta$ and $R\circ U R \subseteq R\circ \mu$, where $\eta$ and $\mu$ are the unit and multiplication of the ultrafilter monad, regarded as relations. In the paper \begin{itemize}% \item [[Claudio Pisani]], \emph{Convergence in exponentiable spaces}, \href{http://www.tac.mta.ca/tac/volumes/1999/n6/5-06abs.html}{TAC} \end{itemize} it is shown that a space is exponentiable (i.e. core-compact) if and only if we have equality in the multiplication law $R\circ U R = R\circ \mu$. Some intuition for this characterization can be obtained as follows. Consider the standard non-locally-compact space, the rationals $\mathbb{Q}$ as a subspace of the reals $\mathbb{R}$. Suppose that $x$ is a rational number and that $y_n$ is a sequence of irrationals converging to $x$. Then for each $n$ we can find a sequence $z^n_m$ of rationals which converges to $y_n$; hence the $z^n_m$ form a ``sequence of sequences'' which ``globally converges'' to $x$ in $\mathbb{Q}$, i.e. which are related to $x$ by the composite relation $R\circ \mu$, but for which does not converge elementwise to an intermediate sequence which in turn converges to $x$, i.e. it is not related to $x$ by the relation $R \circ U R$. It turns out that when generalized to ultrafilter convergence, this sort of behavior exactly characterizes what it means to fail to be (quasi) locally compact. \hypertarget{general_exponential_laws}{}\subsection*{{General exponential laws}}\label{general_exponential_laws} If $X$ is exponentiable, then the exponential law gives us an isomorphism of \emph{sets} $Map(Y,B^X) \cong Map(X\times Y,B)$ for any other spaces $B$ and $Y$. If $Y$ is also exponentiable, then the [[Yoneda lemma]] yields from this a homeomorphism $B^{X\times Y} \cong (B^X)^Y$. However, we can also say some things in general without all spaces involved being exponentiable. We now agree to denote by $Map(X,Y)=X^Y$ the space of continuous maps $X\to Y$ in the [[compact-open topology]]. \begin{utheorem} Let $X,Y,B$ be [[topological spaces]]. For any $f\in B^{X\times Y}$, the formula \begin{displaymath} [(\theta f)(y)](x) = f(x,y) \end{displaymath} defines a [[continuous map]] $\theta f:Y\to B^X$ which we call the map adjoint to $f$, or the [[adjunct]] of $f$. The adjunction map \begin{displaymath} \theta : Map(X\times Y,B)\to Map(Y,B^X), \,\,\,\,\,\,\,\theta:f\mapsto \theta f \end{displaymath} is a [[one-to-one function]], and if $X$ is [[locally compact space|locally compact and Hausdorff]] then $\theta$ is a [[bijection]]. Independently from that assumption on $X$, if $Y$ is [[Hausdorff space|Hausdorff]], then $\theta$ is continuous in the [[compact-open topology]] \begin{displaymath} \theta : B^{X\times Y}\to (B^X)^Y. \end{displaymath} If both assumptions (on $X$ and $Y$) are satisfied, then $\theta$ is not only a continuous bijection, but also open, hence a [[homeomorphism]]. \end{utheorem} \hypertarget{based_exponential_laws}{}\subsection*{{Based exponential laws}}\label{based_exponential_laws} There is also a version for based (= [[pointed space|pointed]]) topological spaces. The [[cartesian product]] then needs to be replace by the [[smash product]] of the based spaces. Regarding that the maps preserve the base point, the adjunction map $\theta$ induces the adjunction map \begin{displaymath} \theta_*:Map_*(X\wedge Y,B)\to Map_*(Y,B^X) \end{displaymath} where the mapping space $Map_*$ for based spaces is the subspace of the usual mapping space, in the compact-open topology, which consists of the mappings preserving the base point. It appears that $\theta_*$ is again one-to-one and continuous, and it is bijective if $X$ is locally compact Hausdorff. If $Y$ is also Hausdorff then $\theta_*$ is a homeomorphism. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item M. Escardo and R. Heckmann, \href{http://www.cs.bham.ac.uk/~mhe/papers/newyork.pdf}{Topologies of spaces of continuous functions}, 2001. \item Eva Lowen-Colebunders and G\"u{}nther Richter, \emph{An elementary approach to exponential spaces}, \href{http://www.ams.org/mathscinet-getitem?mr=1836256}{MR}. \item Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, section 1.3 of \emph{Algebraic topology from a homotopical viewpoint}, Springer (2002) (\href{http://tocs.ulb.tu-darmstadt.de/106999419.pdf}{toc pdf}) \item [[Peter Johnstone]], \emph{[[Stone Spaces]]} \end{itemize} [[!redirects exponential law for topological spaces]] [[!redirects exponential law for based spaces]] [[!redirects core-compact space]] [[!redirects core-compact spaces]] [[!redirects core-compact topological space]] [[!redirects core-compact topological spaces]] [[!redirects quasi locally compact space]] [[!redirects quasi locally compact spaces]] [[!redirects quasi-locally-compact space]] [[!redirects quasi-locally-compact spaces]] [[!redirects exponentiable space]] [[!redirects exponentiable spaces]] [[!redirects exponentiable topological space]] [[!redirects exponentiable topological spaces]] \end{document}