category theory

Contents

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 $I$.

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

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). See also at reflective subuniverse.

References

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)