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Do the exponential object and the product coincide for $Set$? If they do, then why are they different for general categories?

If they do not coincide for $Set$, can the reason why be extended to general categories?

This is a follow-up to my previous question here: Difference Between Product and Function Spaces

As I understand it, at least for the category $Set$, the exponential object $X^J$ and the Cartesian product $\prod\limits_{j \in J} X_j$ coincide, namely both can represent the set of all functions $J \to X$ (where $J$ is our index set).

The typical example one often learns at a very young age, albeit not in this language, is: $$\mathbb{R}^2 \simeq \mathbb{R} \times \mathbb{R}$$ which can be written as (to make my point even more clear): $$\mathbb{R}^{\{0,1\}} \simeq \prod\limits_{j \in \{0,1\}} \mathbb{R}$$

The conclusion I reached from my previous question was that $X^J \simeq \prod\limits_{j \in J} X_j \subset \mathcal{P} (X \times J)$ (since $\mathcal{P}(X \times J)$ can represent the set of all binary relations between $J$ and $X$, and obviously all functions $J \to X$ are a subset of these relations).

However, at least for topological spaces, the exponential object and the product clearly do not coincide. For one, the exponential object does not even always exist, which is expressed by saying that $Top$ is not Cartesian closed, whereas (to the best of my knowledge) the product topology on the space defined by the Cartesian product (defined by using the forgetful functor into $Set$) always exists.

Moreover, the compact-open topology (on the exponential object topological space) is not the same as the product topology on the product topological space (I think).

This is very confusing for me, because defining a category to be Cartesian closed iff it is closed under exponential objects seems to imply that it would be closed under the analog of Cartesian products, which I thought were just general categorical products. But either I am deeply misunderstanding something or this is clearly not the case.

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  • $\begingroup$ Just as a sort of follow-up, the reason why the two are related (in a dumbed-down version of Malice Vidrine's answer below) is because of the adjunction between the product and exponential functors. See: youtube.com/watch?v=8fZmdhLLgs4 $\endgroup$ – Chill2Macht Jul 21 '16 at 20:37
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There is a sense in which they always coincide if $\prod_{j\in J}$ is understood to be an internal product--that is, a right adjoint to the functor $J^*:\mathbf{C}/1\to\mathbf{C}/J$ given by pullback along the unique morphism $J\to 1$. The intuition for thinking of such a right adjoint as a product is that such a pullback is like taking an object in $\mathbf{C}/1\simeq\mathbf{C}$ to the "constant familiy" indexed by $J$--i.e. a diagonal functor--so a right adjoint should be like a product of that family. It's easy to tell from right adjointness and the universal property of pullbacks that $\Pi_J J^*(X)\simeq X^J$.

However, there's no reason to think that this internal notion of product and the external notion (the one you usually see in category theory texts) are going to be nicely related. Any Cartesian closed category has internal products, but there may not be a way in such a category to translate an internal indexing family into a set-indexed one (the category of directed multigraphs, for instance, resists this kind of translation), nor might there be a limit for every set-indexed family (as in, say, $\mathbf{FinSet}$). The reason they coincide so nicely in $\mathbf{Set}$ is because set-indexed products are, obviously, just internal products in $\mathbf{Set}$.

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  • $\begingroup$ I like your answer, although it does me too much credit by assuming that I am more intelligent than I actually am. So if $J$ is a discrete category (or we can consider it as an index set) then the product (power/limit over diagram) and exponential coincide? Ie if we assume $J$ has "minimal structure" and thus can serve merely indexing purposes. $\endgroup$ – Chill2Macht Jul 18 '16 at 15:22
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    $\begingroup$ $J$ has to be an internal discrete category to ensure they coincide; that is, it has to be an object in $\mathbf{C}$, which may not be describable as a set in any sensible way. Categorists like to think of morphisms $A\to J$ in $\mathbf{C}$ as the "native" version of a $J$-indexed family in $\mathbf{C}$. If, no matter what category we're taking products in, we're thinking about products indexed by a set $J$, there's no sensible way to talk about whether they coincide or not. $\endgroup$ – Malice Vidrine Jul 18 '16 at 15:51
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    $\begingroup$ There are several good resources for learning about internal categories and limits, including Categories for the Working Mathematician, the beginning of Ch. XII; Johnstone's Topos Theory, Ch. 2; the last chapter of Borceux's Handbook of Categorical Algebra, vol 1; and Ch. V.7 of Sheaves in Geometry and Logic. $\endgroup$ – Malice Vidrine Jul 18 '16 at 15:55
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What you're saying seems confused.

First, you say from your previous question you have concluded $\mathrm{Hom}_{\mathbf{Set}}(J,X)=X^J\cong\prod_{j\in J}X$ – do you not see that this is exactly the product (really power) and exponential coinciding in $\mathbf{Set}$?

Second, you say that in the category of topological spaces, the product (really power) and the exponential object clearly do not coincide. Yet when $J$ is a discrete set and $X$ is a topological space, we have isomorphisms of sets $\mathrm{Hom}_{\mathbf C}(J,X)\cong\mathrm{Hom}_{\mathbf{Set}}(J,X)\cong\prod_{j\in J}X$ (the second isomorphism since power and exponential coincide in $\mathbf{Set}$). Since $J$ is discrete, on the one hand it is locally compact Hausdorff and the compact-open topology on $\mathrm{Hom}_{\mathbf C}(J,X)$ makes $\mathrm{Hom}_{\mathbf C}(J,X)$ into an exponential object $J^X$ in the category of topological space, and on the other hand the compact-open topology is exactly the product topology on $\prod_{j\in J}X$, which you can check by hand.

Third, it barely makes sense to ask for an arbitrary (but locally small) category whether powers and exponentials coincide since we take the power of an object $A\in\mathbf C$ by a set $X\in\mathbf{Set}$, and exponential of an object $A\in\mathbf C$ by another object $B\in\mathbf C$. An arbitrary category need not have a way of considering its objects as sets. Even if you had some kind of forgetful functor $\mathbf{Set}\xleftarrow{U}\mathbf C$, it could happen (as it does in $\mathbf{Top}$) that there are multiple objects $B$ with the same underlying set $UB$, then obviously only one of the various exponentials $A^B$ could be the product object $\prod_{b\in UB}A$, if it even exists. I think that for the product and exponential to coincide, $B$ would have to be some kind of "discrete object of $\mathbf C$ with underlying set $UB$".

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  • $\begingroup$ I like your answer, although it does me too much credit by assuming that I am more intelligent than I actually am. So if $J$ is a discrete category (or we can consider it as an index set) then the product (power/limit over diagram) and exponential coincide? Ie if we assume $J$ has "minimal structure" and thus can serve merely indexing purposes. $\endgroup$ – Chill2Macht Jul 18 '16 at 15:21

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