On the link between homology and homotopy

Question

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.

Answer 1

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.

As for Question 2, Weibel's book Introduction to Homological Algebra does a great job with the first collection of topics, then Hovey's book (or Dwyer-Spalinski) 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 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.

Answer 2

I'll answer your second question first. To some degree, the ordering you choose will largely depend on whether you want to lead with examples or with full abstraction. As an example, you can introduce projective resolutions and the derived category using only facts about $\text{Ch}(\mathcal{A})$ and Ore's calculus of fractions (see Weibel's book for a treatment like this) or you can introduce model categories, prove their properties, prove that $\text{Ch}(\mathcal{A})$ admits a projective model structure using a small object argument (see this nLab page for an outline of the argument), and arrive thus at a description of the derived category as a homotopy category.

Personally I think the second account would be unnecessarily convoluted and it would make more sense to introduce some homological algebra first, not least because that way you can introduce the projective model structure as an example of a model structure, projective resolution as an example of a cofibrant resolution, derived category as an example of a homotopy category et cetera; these concepts can be difficult to gain an intuition for without several examples! But both orderings are available to you.

On the question of model categories and quasicategories: model categories can be viewed as "presentations" for quasicategories (see this nLab page for this perspective, and Appendices A.2 and A.3 of Lurie's Higher Topos Theory for a development of the theory of model categories with this explicit goal). Quasicategories have several advantages over model categories: for example, there is a quasicategory of functors from any quasicategory to another, whereas the analogous statement doesn't hold for model categories. Model structures are heavily involved in many of the foundational proofs regarding quasicategories, however, so there's no two ways of ordering these topics.

On your first question: personally I don't believe homological algebra is sufficient motivation for introducing either model categories or infinity-categories. As raised in the comments, the triangulated category $\mathcal{D}(\mathcal{A})$ doesn't allow functorial cones and this is annoying in some applications, but people mostly got on fine with applying homological algebra for decades before people started talking about dg- and quasicategories. A stronger order for your text, in my opinion, would be to introduce basic concepts from homological algebra, then use these as examples when you start talking about model categories and finally quasicategories.

On the question of a universal property for $\mathcal{D}(\mathcal{A})$ using infinity categories, you might find section 1.3.3 of Lurie's Higher Algebra helpful. Note, however, that $\mathcal{D}(\mathcal{A})$ certainly does have a universal property in ordinary 1-categorical language: it is the localization of $\text{Ch}(\mathcal{A})$ at the quasi-isomorphisms.