As a student, I'm always looking for organizing principles in mathematics to help me keep track of all of the mathematics I learn. It's easy to get lost in a deluge of definitions unless I organize them in some way. For a long time category theory was my main organizing principle (e.g. the idea of adjoint functors alone is already a very helpful way to organize constructions in mathematics), but at some point ordinary category theory became inadequate and I needed the language of higher categories (all that nLab stuff).

Here is the sort of thing higher category theory helps me keep organized in my head:

Where do long exact sequences come from? Why was that a thing it should have occurred to us to invent? Where can we expect them to show up in mathematics?

A standard answer is that long exact sequences come from short exact sequences of chain complexes, but this is inadequate for describing at least one very important long exact sequence in mathematics, namely the long exact sequence of a fibration. This is perhaps the first long exact sequence one learns about which involves nonabelian groups, and so cannot come from homological algebra in the usual sense at all. So where does it come from?

From the perspective of higher category theory, long exact sequences are shadows of two dual and more fundamental constructions, namely fiber sequences and cofiber sequences. These in turn come from repeatedly taking homotopy pullbacks resp. homotopy pushouts, which one can think of as the "nonabelian derived functors" of ordinary pullbacks resp. pushouts (which are not homotopically well-behaved and must be corrected). The long exact sequence of a fibration in particular comes from a fiber sequence of the form

$$\cdots \to \Omega F \to \Omega E \to \Omega B \to F \to E \to B$$

where $F \to E \to B$ is a fibration, whereas the long exact sequences in ordinary homology and cohomology come from a cofiber sequence of the form

$$A \to X \to X/A \to \Sigma A \to \Sigma X \to \Sigma X/A \to \cdots$$

where $A \to X$ is a cofibration. The higher categorical point of view tells you at least one interesting thing right off the bat, which is that these two constructions are categorically dual to each other: running the first one in the opposite category gets you the second one! (And if there's one thing any mathematician should respect, it's a duality.)

More generally, there's no reason to restrict our attention to spaces: the higher categorical machinery runs in any higher category with the right structure, and in particular running it in chain complexes also gets us the long exact sequence associated to a short exact sequence of chain complexes, while also explaining that the homological mapping cone is not only analogous to but is precisely the same construction as the topological mapping cone: they're both homotopy pushouts.

Let me also make some other comments.

What's the deal with model categories and simplicial sets?

You say that you've come to love the language of cohomology and sheaves. Great: if you're happy with the idea of using chain complexes as resolutions of objects to compute things like cohomology, then the main thing to know about the model category story is that

- model categories are a setting for understanding and computing with "nonabelian resolutions" (in particular to make sense of "nonabelian derived functors"), and
- simplicial objects can be used to build these resolutions; in particular, they can be thought of as "nonabelian chain complexes."

That second claim can be made precise using the Dold-Kan theorem, which tells you that the category of simplicial objects in an abelian category is equivalent to the category of chain complexes concentrated in nonnegative degree.

Here's a relatively concrete example. The Cech nerve of a nice cover $U \to X$ of a space is a simplicial object which resolves the space $X$ in a particular sense; in particular, if the cover has the property that every finite intersection of opens in the cover is contractible, the resulting resolution can be thought of as a nonabelian analogue of a free resolution of a module. The abelian version of this story, where you're mapping $X$ into abelian objects like Eilenberg-MacLane spaces, gives you Cech cohomology, but the nonabelian version of this story gives you, for example, the Cech cocycle description of a principal $G$-bundle.

The Cech cocycle description of a principal $G$-bundle is a great place to start seeing higher category theory at work. First, let me recall the following: if $U_{\alpha}$ is an open cover of a space $X$, then to specify a continuous function $f : X \to Y$ is precisely the same data as specifying

- continuous functions $f_{\alpha} : U_{\alpha} \to Y$ for all $\alpha$
- having the property that the restrictions of $f_{\alpha}$ and $f_{\beta}$ to their intersection $U_{\alpha \beta} = U_{\alpha} \cap U_{\beta}$ agree.

This is a Cech $0$-cycle description of $\text{Hom}(X, Y)$; we are just using the fact that $X$ is the coequalizer of a certain diagram built out of the $U_{\alpha}$s.

Why doesn't this suffice to describe principal bundles? It's because the functor $\text{Hom}(-, Y)$ preserves colimits, but the functor $[-, BG]$ doesn't, because we're taking homotopy classes. Another way to say this is that there is really a *groupoid* of principal $G$-bundles on a space $X$ (the fundamental groupoid of the mapping space $\text{Maps}(X, BG)$, in fact) and we want to know this groupoid, or at least its set of isomorphism classes.

The functor $[-, BG]$ doesn't preserve colimits, morally because taking colimits isn't guaranteed to play nicely with taking homotopy classes. However, it does play nicely with homotopy colimits, and the precise sense in which a Cech nerve of a space $X$ is a resolution of a space is that, under nice hypotheses, $X$ is the homotopy colimit of that Cech nerve. You can think of this as a fancier version of the coequalizer we talked about above, which is why to specify a principal $G$-bundle you need to talk about triple intersections instead of double intersections: you need

- continuous functions $g_{\alpha \beta} : U_{\alpha \beta} \to G$ for all $\alpha, \beta$
- having the property that the restrictions of $g_{\alpha \beta}, g_{\beta \gamma}, g_{\alpha \gamma}$ to their common intersection $U_{\alpha \beta \gamma} = U_{\alpha} \cap U_{\beta} \cap U_{\gamma}$ satisfy the cocycle relation $g_{\alpha \beta} g_{\beta \gamma} = g_{\alpha \gamma}$.

This is precisely a morphism between truncations of the simplicial objects, or nonabelian resolutions, given on the one hand by the Cech nerve of the cover $U \to X$ and on the other hand by the bar resolution of $BG$!

Thinking about algebraic topology this way has made it more topological for me: the above story can be adapted to explain ordinary Cech cohomology in a way that doesn't involve passing to chain complexes at any step, for example (the fact that you can in fact use chain complexes comes from Dold-Kan), and more generally doing algebraic topology this way lets you replace homological constructions with constructions that are genuinely about spaces.

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