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Tuesday, November 1, 2022

The category of elements and section preorders

The category of elements of a copresheaf F denoted el(F) provides an algebraic context to the section preordering. In particular, we have that the section preordering on F is just the object preordering of el(F).

Proposition. let F : C \to Sets be a copresheaf, then the object preordering of el(F) is the section preordering of F.

This is useful because now we know that the section preordering of F can be constructed from the object preorder of its category of elements. The subobject lattice of F is then basically the Heyting algebra of the Alexandrov topology of open subsets of the section preorder.

See also:
Subobject lattices of presheaves

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