Figure

In Lawvere, the term figure is used to describe particular maps used for detecting structure of an object in a category.

Figure

A map from an object A in a small class of objects in a category to an object we wish to study, call it X, is called a "figure of shape A in X," or more suggestively, "a singular figure of shape A in X." The singular creeps in because the structure of A may be lost or collapsed by the map from it to the object being studied.

Examples

• A terminal object can be used as a probe to study more complex objects in a category.
  • In the category of sets, a map from the singular set to some set X is a point of X. But this is also a figure of shape {1} in X.
  • In the category of endomaps, a figure of shape three-cycle would be a map from the three-cycle to an object X. This map is singular if, for example, the three cycle is mapped to a fixed-point in X. This probe finds cycles of length 3.