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:

- chain complex

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

homotopy of a complex

derived category

t-structures

And also I would like to introduce the following concepts in homotopy theory:

Model categories

Homotopy category of a model category

Derivation in the setting of model categories

Quasi-categories

4.5. simplicial object in a category and homotopy in this context

- 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:

- 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?

- 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.

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