To model topological spaces combinatorially, one can use simplicial sets -- or cubical sets. Simplicial sets are defined as presheaves on the simplex category $\Delta$, the category of non-empty finite ordered sets and order-preserving maps. In general $\Delta^{op}$ and $\Delta$ are quite different, but there is the following observation. Let $\Delta_{alg}$ be the category of all ordered finite sets (i.e. including the empty one). Then $\Delta_{alg}^{op}$ can be seen as a non-full subcategory in $\Delta^{alg}$ -- of non-empty sets with maps that preserve endpoints.

Now consider $\square$, the category of cubes (in any of its version, with connections or without). Is there any similar comparison between $\square$ and $\square^{op}$?


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