nLab
exponential ideal

Contents

Definition

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

Properties

Theorem

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

This appears for instance as (Johnstone, A4.3.1). See also at reflective subuniverse. Note that in this case II 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

Revised on September 23, 2016 13:02:26 by Mike Shulman (192.195.154.58)