support of a set in nLab

This article is about support of a set. For other notions of support, see support.

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Definition

The support [X][X] of any set XX is the image of the unique function into any singleton X→1X \to 1. By definition of image the support is thus a subsingleton, and a singleton if XX is pointed. Note that this is different from the support of the unique function X→1X \to 1, which is always the empty set ∅\emptyset.

Generalizing to other categories

The above definition could be interpreted not just in Set but in any category with a terminal object. This leads to the notions of a support object.

The support object of an object AA of a category is the image of its map to the terminal object. In the internal logic of a category, this corresponds to the propositional truncation.

• split support
• support object
• propositional truncation