= Classifying topos of strict linear orders
{title2=$[\mathbf{FinSLin},\mathbf{Set}]$}
Finite <strict total orders> and increasing injections, including the empty order, give a presheaf classifier for strict total orders. Trichotomy locally merges two finite sorted tuples, identifying equal entries. A strict map from a chain is injective, so parallel maps admitting the same realization are already equal. Reflexification gives a geometric morphism to the <classifying topos of linear orders>; pulling back weak totality gives exactly strict trichotomy.
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