Let K be a convex. Some subsets are the image of an affine map g of K into itself. What should such subsets be called? The affine map g may be considered to be idempotent.
I guess that there exist a concept from category theory that describe objects L with the property that there exists morphisms f:L->K and g:K->L such that gf is the identity on L.
This is an interesting question.
One suggestion would be to call subsets of an idempotent map the image of a selfmap. A particular form of selfmap would be an affine map of g of K onto itself. For more about selfmaps, see
Fixed point sets of equivariant fiber-preserving maps
https://arxiv.org/pdf/1704.01413.pdf
See, also:
The Borsuk-Ulam property for homotopy classes of selfmaps of surfaces of Euler characteristic zero
https://arxiv.org/pdf/1608.00397.pdf
Actually I used Borsuk-Ulam to prove an improved version of the Caratheodory theorem that is related to the kind of problems that I am studying.
For a compact convex set the image of an idempotent and a fix-point set for a selfmap is the same. Therefore some papers call such sets fixpoint sets. To call it a fix-point set only makes sense for certain topological spaces but the concept is of a more general nature since it can be defined in any category.
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