Browsing by Author "Kurki, Joonas"
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Kurki, Joonas (2021)The goal of the thesis is to prove the DoldKan Correspondence, which is a theorem stating that the category of simplicial abelian groups sAb and the category of positively graded chain complexes Ch+ are equivalent. The thesis also goes through these concepts mentioned in the theorem, starting with categories and functors in the first section. In this section, the aim is to give enough information about category theory, so that the equivalence of categories can be understood. The second section uses these category theoretical concepts to define the simplex category, where the objects are ordered sets n = { 0 > 1 > ... > n }, where n is a natural number, and the morphisms are order preserving maps between these sets. The idea is to define simplicial objects, which are contravariant functors from the simplex category to some other category. Here is also given the definition of coface and codegeneracy maps, which are special kind of morphisms in the simplex category. With these, the cosimplicial (and later simplicial) identities are defined. These identities are central in the calculations done later in the thesis. In fact, one can think of them as the basic tools for working with simplicial objects. In the third section, the thesis introduces chain complexes and chain maps, which together form the category of chain complexes. This lays the foundation for the fourth section, where the goal is to form three different chain complexes out of any given simplicial abelian group A. These chain complexes are the Moore complex A*, the chain complex generated by degeneracies DA* and the normalized chain complex NA*. The latter two of these are both subcomplexes of the Moore complex. In fact, it is later on shown that there exists an isomorphism An = NAn +DAn between the abelian groups forming these chain complexes. This connection between these chain complexes is an important one, and it is proved and used later on in the seventh section. At this point in the thesis, all the knowledge for understanding the DoldKan Correspondence has been presented. Thus begins the forming of the functors needed for the equivalence, which the theorem claims to exist. The functor from sAb to Ch+ maps a simplicial abelian group A to its normalized chain complex NA*, the definition of which was given earlier. This direction does not require that much additional work, since most of it was done in the sections dealing with chain complexes. However, defining the functor in the opposite direction does require some more thought. The idea is to map a chain complex K* to a simplicial abelian group, which is formed using direct sums and factorization. Forming it also requires the definition of another functor from a subcategory of the simplex category, where the objects are those of the simplex category but the morphisms are only the injections, to the category of abelian groups Ab. After these functors have been defined, the rest of the thesis is about showing that they truly do form an equivalence between the categories sAb and Ch+.
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