in the last semester I learn homological algebra and higher category theory\homotopy theory but I am kind of confused when I try to realy understate the link between the two subject ( this is really not my comfort zone ...) Therefore I try to write(a kind of self exersice) a text on homological algebra and homotopy theory but starting at a low level (accesible for students which do basis courses in algebaic topology , commutative algebra and category theory ) I would like to introduce the following concepts in homological algebra : 1. chain complex 1$\frac{1}{2}$.grothendieck group 2. homotopy of complex 3. derived category 4. t-structures And also I would like to introduce the following concepts in homotopy theory : 1. Model categories 2. Homotopy category of a model category 3. Derivation in the setting of model categories 4. Quasi categories 4.5. simplicial object in a category and homotopy in this context 5. Dold kan equivalence Now the "hard" part start : how to organize this concept in a good way? for 1-3 (either in homology\homotopy) I think that I know how to do that but for 3-5 especially in homotopy I dont have any idea ... This gives rise to my questions : 1. how to give motivation infinite category or with more generallity homotopy theory\higher category theory but from a homological point of view I read somewhere a maybe good idea : For an abelian category $\mathcal{A}$ the derived category $\mathcal{D(A)}$ is not defined by a universal property I read somewhere that in some sense higher category theory resolve the problem,Okay but why ? and do we need quasi categories or model categories will be sufficient for doing that? 2. and if someone have some idea to organize this text I open to any suggestion I will be grateful if someone could give me some clues for doing this self exercise .