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\section*{nice category of spaces}

Within the nLab, ``nice category of spaces'' is a general but inexact term referring to nice or convenient properties one would like a category of spaces to have for some purpose (``space'' here connoting something along topological lines), but typically not satisfied by the category of [[topological spaces]] and [[continuous maps]] itself, thus necessitating a move to some nicer category to suit one's purposes.

The category [[Top]] of [[topological space|topological spaces]] lacks many good categorical properties. It is [[complete category|complete]] and [[cocomplete category|cocomplete]], and is [[extensive category|extensive]], but:

\begin{itemize}%
\item not [[cartesian closed category|cartesian closed]] or [[locally cartesian closed category|locally cartesian closed]],
\item not [[locally presentable category|locally presentable]], and
\item not a [[topos]] or even a [[quasitopos]],
\item nor is it even a [[pretopos]] or an [[exact category|exact]] or even a [[regular category]].

\end{itemize}
The lack of cartesian closure and, to a lesser extent, local presentability, is especially problematic for [[homotopy theory]]. Many different solutions for repairing lack of cartesian closure have been proposed, generally involving either restricting to a subcategory of [[Top]] (usually [[reflective subcategory|reflective]] or coreflective, so that it inherits completeness and cocompleteness), enlarging it to a supercategory, or some combination thereof. Most involve restricting the topologies to those that can be specified on ``small'' (and in particular, [[compact space|compact]]) subsets.

In particular, a [[convenient category of topological spaces]] is, in the technical sense of the \href{http://ncatlab.org/nlab/show/HomePage}{nLab}, a cartesian-closed category of spaces together with some other useful properties (q.v.).

\hypertarget{examples}{}\section*{{Examples}}\label{examples}

\begin{itemize}%
\item The frequent choice among algebraic topologists today is to use the subcategory of [[compactly generated space|compactly generated spaces]], which is cartesian closed, but not locally cartesian closed. It is a coreflective subcategory of a reflective subcategory of $Top$.


\item Homotopy theorists often find the category of [[simplicial sets]] to be an especially nice environment. It is for example a [[Grothendieck topos]], thus a [[locally cartesian closed category]] and satisfying all [[exactness properties]] one expects of toposes, and is a [[locally presentable category]]. The fact that every topological space has a simplicial set as its [[singularization]] then becomes an application of the homotopy theory of simplicial sets to the study of topological spaces, rather than a way to use simplicial sets to study the homotopy theory of topological spaces.

For more on this see \emph{[[Top]]}, \emph{[[homotopy theory]]} and [[infinity-groupoid]].


\item The subcategory of [[Delta-generated space]]s, recently proposed by J. H. Smith, is also both cartesian closed and locally presentable.


\item An approach of mainly historical interest is to use [[quasitopological space|quasitopological spaces]], an enlargement of $Top$ which is cartesian closed.


\item The category $PsTop$ of [[pseudotopological space|pseudotopological spaces]] (also called Choquet spaces) is a quasitopos containing $Top$ as a full reflective subcategory. In particular, $PsTop$ is [[locally cartesian closed category|locally cartesian closed]] (but not locally presentable).


\item In his paper \emph{On a topological topos}, Peter Johnstone described a [[Grothendieck topos]] $E$ which contains the category of [[sequential space|sequential]] topological spaces as a full reflective subcategory which is closed under many colimits (including all those used to define [[CW complex|CW complexes]]). Again, since $E$ is a Grothendieck topos, it is locally presentable and locally cartesian closed. Moreover, the [[geometric realization]] and [[singular complex]] functors form a [[geometric morphism]] between $E$ and the category of [[simplicial set|simplicial sets]]. The ``underlying set'' functor $E\to Set$ is not [[faithful functor|faithful]], but it is faithful on the full subcategory of [[subsequential space|subsequential spaces]], which contain the sequential spaces and form a [[quasitopos]]. See [[topological topos]].


\item The category of [[compact Hausdorff spaces]] is perfectly nice for some purposes. While neither cartesian closed nor locally presentable, it is however a complete and cocomplete [[pretopos]]. In this way it is both a category of [[nice topological spaces]] and a nice category of topological spaces, thus an exception proving the ``rule'' described at [[dichotomy between nice objects and nice categories]].


\item The category of [[locales]] and the full subcategory of [[sober spaces]] can be considered nice for certain purposes. The category of locales is extensive and is opposite to the category of [[frames]] (which is monadic over $Set$ and thus [[exact category|exact]]). The category of locales is however neither cartesian closed nor locally presentable, although there is a nice description of [[exponential object|exponentiable locales]]. Johnstone's Stone Spaces gives an account of topology via locale theory.


\item John Milnor proposed the category of spaces having the homotopy type of a CW complex as a nice category of spaces. If $X$ and $Y$ are objects and $X$ is compact, then there is an exponential object $Y^X$ in this category. This by the way is also a category of [[nice topological spaces]].


\item The category of [[algebraic lattices]], considered as a full subcategory of $T_0$-[[separation axioms|spaces]], is a nice cartesian closed category of spaces in which to do [[domain theory]]. Related to this is the category of [[equilogical spaces]], which is locally cartesian closed (and thus also regular) and arises as the [[exact completion|reg/ex completion]] of the category of $T_0$ spaces.



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Peter May]], \emph{[[A Concise Course in Algebraic Topology]]} (Chapter 5, for compactly generated spaces)


\item O. Wyler, \emph{Convenient categories for topology}


\item L. Fajstrup and J. Rosicky, \emph{A convenient category for directed homotopy} (for Delta-generated spaces)


\item E. Spanier, \emph{Quasi-topologies} (for quasi-topological spaces)


\item O. Wyler, \emph{Lecture notes on topoi and quasitopoi} (for pseudotopological spaces)


\item [[Peter Johnstone]], \emph{On a [[topological topos]]}


\item [[Peter Johnstone]], [[Stone Spaces]]


\item J. Milnor, On spaces having the homotopy type of a CW-complex, Trans. Amer. Math. Soc. 90, no. 2 (1959), 272-280.



\end{itemize}
[[!redirects nice categories of spaces]]



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