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Pausing to speak metaphorically for a bit, I have always thought of partial orders as being something like a simplified version of topological spaces. (In the finite case at least, complete partial orders and all topological spaces are identical)partial orders). This leads to thinking about continuous maps as a parallel concept to the idea of a poset homomorphism. But is there a corresponding parallel notion for a homotopy of two continuous maps?

Here is a sketch of how I imagine this working:

Since all preorders are categories, and posets are preorders, we can think of any partial order $(\leq, C)$ as a category whose morphisms are just given by the relation $\leq$. Now a homomorphism of posets is just a functor of these categories. Taking this one step further, we can define a natural transformation of any two homomorphisms of posets, which I am thinking of as a candidate definition for a poset homotopy.

Now it is also true that CW complexes are technically posets under the incidence relation. If we replace the posets in the above sketch with CW complexes and the functors with cellular maps, does the proposed definition for poset homotopy match the known definition for a CW complex homotopy'? (EDIT: Or is this true for some variation of a CW complex; for example a finite or simplicial complex?)

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Pausing to speak metaphorically for a bit, I have always thought of partial orders as being something like a simplified version of topological spaces. (In the finite case at least, complete partial orders and topological spaces are identical). This leads to thinking about continuous maps as a parallel concept to the idea of a poset homomorphism. But is there a corresponding parallel notion for a homotopy of two continuous maps?

Here is a sketch of how I imagine this working:

Since all preorders are categories, and posets are preorders, we can think of any partial order $(\leq, C)$ as a category whose morphisms are just given by the relation $\leq$. Now a homomorphism of posets is just a functor of these categories. Taking this one step further, we can define a natural transformation of any two homomorphisms of posets, which I am thinking of as a candidate definition for a poset homotopy.

Now it is also true that CW complexes are technically posets under the incidence relation. If we replace the posets in the above sketch with CW complexes and the functors with cellular maps, does the proposed definition for poset homotopy match the known definition for a CW complex homotopy'? (EDIT: Or is this true for some variation of a CW complex; for example a finite or simplicial complex?)

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# Is it reasonable to define poset homotopy' as a natural transformation of posets'?

Pausing to speak metaphorically for a bit, I have always thought of partial orders as being something like a simplified version of topological spaces. (In the finite case at least, complete partial orders and topological spaces are identical). This leads to thinking about continuous maps as a parallel concept to the idea of a poset homomorphism. But is there a corresponding parallel notion for a homotopy of two continuous maps?

Here is a sketch of how I imagine this working:

Since all preorders are categories, and posets are preorders, we can think of any partial order $(\leq, C)$ as a category whose morphisms are just given by the relation $\leq$. Now a homomorphism of posets is just a functor of these categories. Taking this one step further, we can define a natural transformation of any two homomorphisms of posets, which I am thinking of as a candidate definition for a poset homotopy.

Now it is also true that CW complexes are technically posets under the incidence relation. If we replace the posets in the above sketch with CW complexes and the functors with cellular maps, does the proposed definition for poset homotopy match the known definition for a `CW complex homotopy'?