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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

Last revised on September 23, 2016 at 13:02:26. See the history of this page for a list of all contributions to it.