= Classifying topos of strict partial orders
{title2=$[\mathbf{FinSPos},\mathbf{Set}]$}
Let $\mathbf{FinSPos}$ contain finite <strict partial orders> and <strict order-preserving functions>. Its covariant set-valued functor category classifies the strict theory. A finite presentation with a strict cycle is contradictory; every other finite diagram gives a finite strict partial order. Equalizing finitely many element identifications either gives such an order or forces a forbidden cycle, so the internal model-to-flat-functor construction remains valid. The <reflexification of a strict partial order> induces a surjective geometric morphism to the weak-order classifier, but not a geometric embedding because strict maps exclude collapse of comparable elements.
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