nLab exponential ideal

Definition

A class $I$ of objects in a cartesian closed category $C$ is called an exponential ideal if whenever $Y\in I$ and $X\in C$, the exponential object $Y^X$ is in $I$.

Properties

Theorem

If $I \hookrightarrow C$ is a reflective subcategory, then it is an exponential ideal if and only if its reflector $C\to I$ preserves finite products.

This appears for instance as (Johnstone, A4.3.1); it can also been seen as a consequence of Day's reflection theorem. See also at reflective subuniverse. Note that in this case $I$ is itself a cartesian closed category, since being a reflective subcategory it is also closed under finite products.

References

The relation of exponential ideals to reflective subcategories is discussed in section A4.3.1 of

• Peter Johnstone, Sketches of an Elephant