= Classifying topos of partial orders
{title2=$[\mathbf{FinPos},\mathbf{Set}]$}
Let $\mathbf{FinPos}$ contain finite <partially ordered sets> and all <order-preserving functions>, including the empty object. The covariant functor category $[\mathbf{FinPos},\mathbf{Set}]$ classifies weak <partial orders>. Finite positive presentations reduce to finite preorders with mutually comparable cycles identified. An internal model $M$ gives a flat functor $P\mapsto\operatorname{Hom}(P,M)$ on $\mathbf{FinPos}^{\rm op}$; conversely the corresponding filtered presentation by finite models recovers its carrier and order relation.
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