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Stream: learning: questions

Topic: Intuition for Brown representability

Srikanth Pai B (Jan 07 2022 at 08:44):

I have to present the proof of Grothendieck duality in triangulated categories (from a paper by Neeman). The proof is apparently motivated by the proof of brown representability in topological category. For the proof of brown representability, we have two steps: One shows that it sufficient to check representability on spheres and the second is an actual construction of the representing object.
The construction of the object that represents the functor seems very unintuitive. What is really going on here?

Jens Hemelaer (Jan 07 2022 at 10:25):

In the topological case, the idea is that by Whitehead's theorem a CW-complex $X$ is in some sense completely determined by its homotopy groups $\pi_n(X) = [S^n, X]$ (homotopy classes of maps from the $n$-sphere to $X$). Intuitively, this is because CW-complexes are themselves constructed by gluing spheres of different dimensions together (I don't know whether this intuition can be made precise).

If a contravariant functor $F$ is represented by a CW-complex $X$, then we know what the homotopy groups of $X$ should be: they are given by $\pi_n(X) = F(S^n)$. So now it remains to construct a CW-complex with the right homotopy groups. This is done inductively. You can make the homotopy group $\pi_n(X)$ bigger by attaching $n$-spheres, and you can quotient out relations in $\pi_n(X)$ by attaching $(n+1)$-dimensional disks between these $n$-spheres. This should be done in a careful way in order to make sure that the $\pi_k(X)$'s for $k<n$ stay the same.

In the topological case, I believe it helps to look at the case where $F = H^n(-,\mathbb{Z})$ is singular cohomology. In this case, $F$ is represented by the Eilenberg--Mac Lane space $K(\mathbb{Z},n)$. So you can look at the construction of $K(\mathbb{Z},n)$ as a special case. For example, $K(\mathbb{Z},1) = S^1$.

Zhen Lin Low (Jan 07 2022 at 10:52):

I believe it's made precise in (∞, 1)-categorical language by saying that the ∞Grpd is freely generated under colimits by the point. I think the Brown representation theorem is morally the truncation of this to the homotopy category.

Mike Shulman (Jan 07 2022 at 16:23):

In other words, it's essentially a form of [[adjoint functor theorem]].