Categorical homotopy theory

Abstract

Homotopik cebirin bazı temel yapılarını inceledik.Aynı zamanda 'Category theory' nin 'Homotopy theori'ye nasıl bir katkıda bulunduğunu sunduk.Homotopy theory nin temeli topolojik uzaylara dayanıyor.'Chain Complexes' i kullanarak Homotopi kavramını diğer kategorilere genişlettik.Daha sonra,Kategorilerin ortak özelliklerini içeren 'Closed Model Category'ler için aksiyomları sunduk.İkinci kısımda,her 'Closed Model Category' e uygun 'Homotopy Category' modeli ile bağlantı kurduk.Bu bağlamın ardından,'Homotopy Category' sadece bir yerelleştirme.Uçüncü kısımda ,'Closed Model Category' de homotopi yi 'Homotopy Category' nin bazı özelliklerini kullanarak yaptık.Daha sonra,homotopi de yer alan 'limits' ve 'colimits' i tanımladık.Uygulamada 'Topolojik Uzayları' ve 'Chain Complexes' i kullandık.

In this thesis we present and discuss some basic aspects of homotopic algebra. We present how elements from category theory can be helpful in studying homotopy theory. Homotopy theory arises in studying topological spaces. Here we present a meaningful way to extend the notion of homotopy to other categories such as chain complexes. Then we present the axioms for closed model categories, which capture common features of categories in which we can talk about homotopy. After describing how we can do some homotopy theory starting with a few axioms, we construct the main object of study which is the homotopy category associated to a closed model category. After construction we will prove that the homotopy category is just a localization. Finally we present some features of the homotopy category in order to emphasize the advantages of doing homotopy in closed model categories and give brief mention of homotopy limits and colimits in the final section. All of the above are done keeping in mind the main examples which are topological spaces and chain complexes.