Categories with some but not all exponentials

Question

The introductory examples typically given of exponential objects in categories in fact involve categories which have all exponentials.

Are there not-too-esoteric examples of categories of independent interest for which it is significant that they have some exponentials (and more than just trivia like $A^1$) but they don't have all exponentials?

(For example, are there interesting cases where in effect we can't iterate exponentiation?)

---

Ooops. I really should not have asked this question as is, since the topological example given by Najib Idrissi is already there (in headline form) in this Wikipedia entry which I indeed recall reading not so very long ago. A senior moment, I fear! But given the helpful pointers to extra detail in Najib Idrissi's answer, I'll certainly leave this question in place.

But are there also nice non-topological examples, I wonder?

Answer 1

Here's a silly example: the category of countable (but possibly finite) sets has finite products, but Cantor's theorem implies that only the finite sets are exponentiable.

A not-so-silly variation on the above is the category of classes (in NBG, say). Again, this has finite products, and it is not hard to check that every set is exponentiable. What is less clear is whether every exponentiable object is a set, but it is certainly true that there is some proper class that is not exponentiable: see here.

It should also be pointed out that $\mathbf{CRing}^\mathrm{op}$ has some non-trivial exponentiable objects. For instance, $\mathbb{Z} [x] / (x^2)$ is exponentiable in $\mathbf{CRing}^\mathrm{op}$. Indeed, $$\mathbf{CRing} (A, B \otimes \mathbb{Z} [x] / (x^2) ) \cong \mathbf{CRing} (\mathrm{Sym}_A (\Omega^1_A), B)$$ where $\mathrm{Sym}_A (\Omega^1_A)$ denotes the commutative $A$-algebra generated by the Kähler differentials of $A$ over $\mathbb{Z}$.

The common thread running through all three examples above is that "small" objects are exponentiable: in the first example, finite sets are small in the context of countable sets; in the second example, sets are small in the context of classes; and in the third example, $\mathbb{Z} [x] / (x^2)$ is in some sense an infinitesimally fattened version of $\mathbb{Z}$, which is small in the sense of being a terminal object in $\mathbf{CRing}^\mathrm{op}$.

Answer 2

The category $\mathsf{Top}$ of topological spaces is a famous example. A topological space $X$ is exponentiable (meaning that $Y^X$ exists for all $Y$) iff it is core-compact, for example if it is compactly generated. Not all spaces are core-compact, which lead to the search for a nice category of spaces, a subcategory of $\mathsf{Top}$ that satisfies nice properties like "cartesian closed" (see this question for more information).

It's why many texts about algebraic topology begin with "we will assume all spaces involved are compactly generated" of something similar. It's also why some people prefer working with e.g. simplicial sets instead of topological spaces: they are "just as good as" topological spaces from the point of view of homotopy theory, but their category is much better behaved.

Answer 3

For a very small example you can take the well known five-element nondistributive lattice $ \{0,a,b,c,1\} $ with $ 0<b<a<1 $ and $ 0<c<1 $, regarded as a category.

Then e.g. $ b^c=a $ but $ b^a $ does not exist.