Topology and localization on 3-Type algebraic models

This master thesis, titled Topology and Localization on Homotopy Algebraic 3-types, consists of 4 chapters. In the first chapter, we will give the notion and examples of categorical algebras and crossed modules which corresponds to homotopy 3-types. Then we will give the notion and some properties of adjointness and exactness which we will use in chapter3 and chaphter 4 for algebraic homotpy 3-types In the second chapter we will show that 2-crossed modules which corresponds to homotopy algebraic 3-type has pullbacks. Since algebraic models must have a pullback to construct topology on the means of Barr-Wells. Also we Show that the category of 2-crossed modules is an exact category. In the third chapter firstly we will give the notion and examples of crossed squares. Then we will show that there is a fibration from crossed squares to crossed corners. Since fibration requires pullbacks it is possible to construct a topology by means of Barr-Wells on crossed squares. In the fourth chapter we will construct topology and localization on categorical albegras which corresponds to homotopy 3-type. For this we will define a system what we cal categorical idempotent filter. And construct topology on 2-crossed modules which corresponds to homotopy 3-type