THE DOLD-KAN CORRESPONDENCE 1. Simplicial sets

We have just seen that the category ∆ is equivalent to the subcategory consisting of the [n]. As a result, a simplicial set X• is given by specifying sets Xn for each n ∈ Z≥0, together with maps Xn → Xm for each map [m] → [n] in ∆. These maps are required to satisfy compatibility conditions (i.e., form a functor). The set Xn is called the set of n-simplices of X•. Example 1.3. Let X be a topological space. Then we define its singular simplicial set SingX• as follows. We let (SingX)n = homTop(∆n, X). Using the functoriality of ∆n discussed above, it is clear that there are maps (SingX)n → (SingX)m for each [m]→ [n]. Example 1.4. Given n ∈ Z≥0, we define the standard n-simplex ∆[n]• via ∆[n]m = hom∆([m], [n]). Given a category C, we know that there is a way of generating presheaves on C. For each X ∈ C, we consider the presheaf hX defined as Y 7→ homC(Y,X); the presheaves obtained are the representable presheaves. The standard simplices are a special case of that.