# Any comparison between the category of cubes and its opposite?

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 $$\square^{op}$$?