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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 17th 2018
   • (edited Jun 17th 2018)
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   Author: Dmitri Pavlov
   Format: MarkdownItexSome categories of topological spaces, such as [[compactly generated topological spaces]], are [[cartesian closed]]. Is there a convenient/nice category of topological spaces that is [[locally cartesian closed]]? For instance, are [[Δ-generated topological spaces]] locally cartesian closed?

   Some categories of topological spaces, such as compactly generated topological spaces, are cartesian closed.

   Is there a convenient/nice category of topological spaces that is locally cartesian closed?

   For instance, are Δ-generated topological spaces locally cartesian closed?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJun 18th 2018
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   Author: Mike Shulman
   Format: MarkdownItexThe closest thing to an answer that I know of is that if $\mathcal{K}$ is the category of not-necessarily-weak-Hausdorff [[k-spaces]], and $A$ and $B$ are k-spaces that *are* weak Hausdorff, then the pullback functor $\mathcal{K}/B \to \mathcal{K}/A$ has a right adjoint. This is what May and Sigurdsson used in their book *Parametrized homotopy theory*. If you give up on the desire that your category be a subcategory of [[Top]], then there are closely related categories that are [[quasitopoi]] and hence lccc, such as [[subsequential spaces]], [[convergence spaces]], and [[pseudotopological spaces]].

   The closest thing to an answer that I know of is that if $\mathcal{K}$ is the category of not-necessarily-weak-Hausdorff k-spaces, and $A$ and $B$ are k-spaces that are weak Hausdorff, then the pullback functor $\mathcal{K}/B \to \mathcal{K}/A$ has a right adjoint. This is what May and Sigurdsson used in their book Parametrized homotopy theory.

   If you give up on the desire that your category be a subcategory of Top, then there are closely related categories that are quasitopoi and hence lccc, such as subsequential spaces, convergence spaces, and pseudotopological spaces.

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 18th 2018
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   Author: Dmitri Pavlov
   Format: MarkdownItexThanks! I transplanted your description into the article [[compactly generated topological space]].

   Thanks! I transplanted your description into the article compactly generated topological space.