F a C U L T Y O F S C I E N C E Master Project in Mathematics Quasi-categories -basic Constructions

0.1 Abstract ii 0.1 Abstract The project consists of two chapters and an appendix. In the rst chapter dene quasi-categories, and perform basic constructions with these, some of which are motivated by ordinary category theory. Among these, we shall build under-categories and colimits. Also we briey review some dierent notions of brations needed for the next chapter, and introduce a pair of adjoint functors called straightening and unstraightening. The second chapter is devoted to a certain application of the tools developed in chapter 1. However we will have to apply auxillary results since developing all we need is outside the scope of this project. There will be some loose discussion of the results we apply and those we obtain. We will try to relate the work of the chapter to a theorem in topology, which states that in certain cases (not too restrictive), there is a weak equivalence G ad / /G LBG for a simplicial group G. A classical proof of this is given in the appendix.