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Let $\bf{C}$ be a category with finite products.

(1) An object $X$ of $\bf{C}$ is called cartesian if the functor $(-)\times X$ has a right adjoint.

(2) A morphism $s:X\rightarrow B$ in category $\bf{C}$ is called cartesian if $(X,s)$ is cartesian in the comma category $\bf{C}$$/B$.

(3) A category $\bf{C}$ is called cartesian closed if every object is cartesian.

My question is that whether the comma category $\bf{C}$$/B$ ($B$ is a $\bf{C}$-object) is cartesian closed when $\bf{C}$ is cartesian closed?

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    $\begingroup$ Those for which this is true are called locally cartesian closed. See on the nLab at ncatlab.org/nlab/show/locally+cartesian+closed+category . $\endgroup$ – Urs Schreiber Feb 2 '14 at 14:38
  • $\begingroup$ $Cat$, the category of categories, and $Pos$, the category of posets, are each cartesian closed, but their slices are typically not cartesian closed. $\endgroup$ – Todd Trimble Sep 11 '14 at 19:45
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The answer to the main question is no. There are plenty of cartesian closed categories with slices (comma categories) that are not cartesian closed.

Also, the usual name for an object $X$ for which functor $(-)\times X$ has a right adjoint is exponentiable.

The word cartesian is grossly over-used and there should be a moratorium on any further use.

I would also note, with regard to my general opinion that MathOverflow questions should not be closed quite so readily, that a textbook question like this in numer theory or functional analysis would have been closed within minutes of being asked.

"Lots of cartesian closed categories" have been provided by the discipline called Domain Theory in Theoretical Computer Science. This was begun by Dana Scott with a view to finding topological models of the untyped $\lambda$-calculus and is also rooted in questions of exponentiblity in General Topology, of which I gave some of the history here. Scott and others went on to use Domain Theory to provided mathematical semantics of features of programming languages.

The simplest of these categories to describe is perhaps $\mathbf{Dcpo}$, the category of directed-complete partial orders and so-called Scott-continuous functions, ie those that preserve directed joins. A directed join is a "purely infinitary" one, ie of a diagram that already contains a bound for any finite sub-diagram. $\mathbf{Dcpo}$ is readily seen to be cartesian closed, where the exponential $Y^X$ is given by the dcpo of Scott-continuous functions $X\to Y$.

A necessary condition for a functor to have a right adjoint is that it preserve colimits, in particular (the property of being) epis. Therefore if an object $f:X\to B$ of a slice category is to be exponentiable in the slice, product in the slice, ie pullback along $f$, must preserve epis.

Let $\varpi$ be the dcpo consisting of the natural numbers with their order and a great element $\top$ and let $\mathbb{N}$ be discrete natural number object. Then the obvious map $e:\mathbb{N}\to\varpi$ is epi. However the pullback $f^*e$ along the element $f=\top:\mathbf{1}\to\varpi$ is $\mathbf{0}\to\mathbf{1}$.

Note also that there is an ambiguity between the usage of Theoretical Computer Science and older Category Theory as to whether a "cartesian closed" category has to have all finite limits, ie equalisers and pullbacks as well as binary products and a terminal object. In fact, $\mathbf{Dcpo}$ does have the all, but most other CCCs that have been used in Domain Theory do not.

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