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.

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

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.

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)