nLab exponential ideal




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.



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); it can also been seen as a consequence of Day's reflection theorem. 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.


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

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