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My naive opinion is that this question factors into two questions:

1. How do facts from algebraic topology shed light on posets?

2. How do facts about posets shed light on number theory?

The second question is, I think, a little easier to answer. The poset most obviously relevant to number theory is $\mathbb{N}$ under divisibility. In some vague sense the study of this poset is equivalent to the study of Dirichlet series (see for example this MO question). This perspective is, I think, due to Rota.

The first question is not about number theory. Once you realize that you can associate an abstract simplicial complex to a poset in a canonical way then I don't think this idea sounds so strange; it is commonly used by combinatorialists (see for example these notes). There are also some motivational theorems in this area, e.g. every finite CW-complex is weak homotopy equivalent to a finite topological space, and the category of finite $T_0$ spaces is isomorphic to the category of finite posets, but I'm sure an expert could say more here.

My naive opinion is that this question factors into two questions:

1. How do facts from algebraic topology shed light on posets?

2. How do facts about posets shed light on number theory?

The second question is, I think, a little easier to answer. The poset most obviously relevant to number theory is $\mathbb{N}$ under divisibility. In some vague sense the study of this poset is equivalent to the study of Dirichlet series (see for example this MO question). This perspective is, I think, due to Rota.

The first question is not about number theory. Once you realize that you can associate an abstract simplicial complex to a poset in a canonical way then I don't think this idea sounds so strange; it is commonly used by combinatorialists (see for example these notes). There are also some motivational theorems in this area, e.g. every finite CW-complex is weak homotopy equivalent to a finite topological space, and the category of finite $T_0$ spaces is isomorphic to the category of finite posets, but I'm sure an expert could say more here.

My naive opinion is that this question factors into two questions:

1. How do facts from algebraic topology shed light on posets?

2. How do facts about posets shed light on number theory?

The second question is, I think, a little easier to answer. The poset most obviously relevant to number theory is $\mathbb{N}$ under divisibility. In some vague sense the study of this poset is equivalent to the study of Dirichlet series (see for example this MO question). This perspective is, I think, due to Rota.

The first question is not about number theory. Once you realize that you can associate an abstract simplicial complex to a poset in a canonical way (I think it is a special case of the nerve functor) then I don't think this idea sounds so strange; it is commonly used by combinatorialists. There are also some motivational theorems in this area, e.g. every finite CW-complex is weak homotopy equivalent to a finite topological space, and the category of finite $T_0$ spaces is isomorphic to the category of finite posets, but I'm sure an expert could say more here.

My naive opinion is that this question factors into two questions:

1. How do facts from algebraic topology shed light on posets?

2. How do facts about posets shed light on number theory?

The second question is, I think, a little easier to answer. The poset most obviously relevant to number theory is $\mathbb{N}$ under divisibility. In some vague sense the study of this poset is equivalent to the study of Dirichlet series (see for example this MO question). This perspective is, I think, due to Rota.

The first question is not about number theory. Once you realize that you can associate an abstract simplicial complex to a poset in a canonical way then I don't think this idea sounds so strange; it is commonly used by combinatorialists (see for example these notes). There are also some motivational theorems in this area, e.g. every finite CW-complex is weak homotopy equivalent to a finite topological space, and the category of finite $T_0$ spaces is isomorphic to the category of finite posets, but I'm sure an expert could say more here.

My naive opinion is that this question factors into two questions:

1. How do facts from algebraic topology shed light on posets?

2. How do facts about posets shed light on number theory?

The second question is, I think, a little easier to answer. The poset most obviously relevant to number theory is $\mathbb{N}$ under divisibility. In some vague sense the study of this poset is equivalent to the study of Dirichlet series (see for example this MO question). This perspective is, I think, due to Rota.

The first question is not about number theory. Once you realize that you can associate an abstract simplicial complex to a poset in a canonical way (I think it is a special case of the nerve functor) then I don't think this idea sounds so strange. There are also some strong motivational theorems in this area, e.g. every finite CW-complex is weak homotopy equivalent to a finite topological space, and the category of finite topological spaces is equivalent to the category of finite posets, but I'm sure an expert could say more here.

My naive opinion is that this question factors into two questions:

1. How do facts from algebraic topology shed light on posets?

2. How do facts about posets shed light on number theory?

The second question is, I think, a little easier to answer. The poset most obviously relevant to number theory is $\mathbb{N}$ under divisibility. In some vague sense the study of this poset is equivalent to the study of Dirichlet series (see for example this MO question). This perspective is, I think, due to Rota.

The first question is not about number theory. Once you realize that you can associate an abstract simplicial complex to a poset in a canonical way (I think it is a special case of the nerve functor) then I don't think this idea sounds so strange; it is commonly used by combinatorialists. There are also some motivational theorems in this area, e.g. every finite CW-complex is weak homotopy equivalent to a finite topological space, and the category of finite $T_0$ spaces is isomorphic to the category of finite posets, but I'm sure an expert could say more here.