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Partial orders are the free conservative cocompletion of total orders
Theory and applications of categories, Tome 44 (2025), pp. 1-14
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We show that the category of partially ordered sets Pos is the free conservative cocompletion of the category of finite non-empty totally ordered sets Δ, also known as the simplex category. This means that Pos is the initial cocomplete category that contains Δ as a full subcategory and preserves the existing colimits of Δ.
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Classification : 18A35
Keywords: cocompletion, partial orders, simplex category 
@article{TAC_2025_44_a0,
     author = {Calin Tataru},
     title = {Partial orders are the free conservative cocompletion of total orders},
     journal = {Theory and applications of categories},
     pages = {1--14},
     year = {2025},
     volume = {44},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2025_44_a0/}
}
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Calin Tataru. Partial orders are the free conservative cocompletion of total orders. Theory and applications of categories, Tome 44 (2025), pp. 1-14. http://geodesic.mathdoc.fr/item/TAC_2025_44_a0/