In the last semester I learned homological algebra and higher category theory/homotopy theory.


But I am kind of confused when I try to really understand the link between the two subjects (this is really not my comfort zone ...)


Therefore I try to write (a kind of self-exercise) a text on homological algebra and homotopy theory but really introduce from $0$ the two subjects.


I would like to introduce the following concepts in homological algebra:


1. chain complex


1$\frac{1}{2}$. Grothendieck group


2. homotopy of a 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 these concepts 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 don't have any idea ...


This gives rise to my questions:


1. How to motivate infinity categories, or more generally 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 resolves the problem. Okay but why? And, do we need quasi categories, or would model categories be sufficient for doing that?


2. 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.



On the link between homology and homotopy
I'd encourage the OP to read the writings of others on this topic, before trying to write something from scratch. I attended lectures at OSU where Aaron Mazel-Gee motivated $\infty$-categories very much as the OP suggests in Question 1. It appears some of the ideas from those lectures have now appeared [here](https://etale.site/writing/zen-of-infty-cats.pdf).


As for Question 2, Weibel's book [Introduction to Homological Algebra](https://rads.stackoverflow.com/amzn/click/com/0521559871) does a great job with the first collection of topics, then [Hovey's book](https://web.math.rochester.edu/people/faculty/doug/otherpapers/hovey-model-cats.pdf) (or [Dwyer-Spalinski](http://folk.uio.no/paularne/SUPh05/DS.pdf)) gives the first three items in the second collection, and Lurie's books give you everything you could want about quasicategories and their connection to model categories and homological algebra (seriously, the introductions he writes for each chapter are phenomenal). As for the Dold-Kan correspondence, while I'm sure it appears somewhere in Lurie's writings, the [clearest exposition I've read](https://math.uchicago.edu/%7Eamathew/doldkan.pdf) anywhere is by Akhil Mathew.


I agree with Arthur that, if you were more categorically minded, you could reverse the order (e.g., starting with Lurie, if you knew about simplicial sets already). For myself, I'd rather start with something concrete and then build the abstraction on top of that, little by little, as this ordering suggests. Weibel's book really is written in such a way to make it easy to step from there into triangulated categories, model categories, and quasi-categories. But it starts at a place very accessible to algebraists.


