exponential ideal

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 $I$.

Of course, in particular this implies that $I$ is itself cartesian closed.

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). See also at *reflective subuniverse*.

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

Revised on October 28, 2014 21:52:01
by Urs Schreiber
(141.0.9.60)