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Upon reading Hatcher's Algebraic Topology, I found mention of the subject of homological algebra while he was introducing chain complexes. I've had a brief look at the subject, and while I understand the basic premise of the subject at a technical level and some of its applications, I'm at a complete loss as to what it's intuitively "about". Every other field of mathematics I've encountered has been "about" something that can, regardless of technical background, be immediately seen to be interesting in and of itself. Chain complexes are not certainly not that. On the other hand, while some of these may only be partial characterisations of the subjects, I think that they are broadly acceptable as partial characterisations:

  • Group theory is "about" symmetry.
  • Measure theory is "about" size.
  • Topology is "about" space.
  • Category theory is "about" structure.
  • Logic is "about" truth.
  • Number theory is "about" repetition.
  • Algebraic topology is "about" holes or maybe movement.
  • Differential topology is "about" smoothness.

I can't think of a nice informal bird's eye view description for homological algebra though. Do researchers in the subject have something like that in their heads? If not, do they just study it for the applications? Or is it because the subject is orderly enough to have results, but disorderly enough for them to be non-trivial, and intuitive meaning is irrelevant? If the last is the case, are there other advanced areas that are studied for that reason?

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  • $\begingroup$ I think you can also post this in Mathoverflow , if something related not been posted yet , for some highly valuable ideas and insight $\endgroup$
    – A S D
    Commented Jul 16, 2021 at 14:46

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I would say that homological algebra is about obstruction.

Specifically, the homology of a chain complex measures the obstruction of a cycle to be a boundary. In chain complexes arising from geometry, this can usually be interpreted as a geometric obstruction. So for instance for the cellular chain complex of a CW complex the homology measures the obstruction of a chain of cells to be the boundary of the cell of one higher dimension. The Tor functors measure the obstruction of a module to be flat, etc.

PS: The name obstruction theory is already taken (https://en.wikipedia.org/wiki/Obstruction_theory).

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    $\begingroup$ This answer is absolutely right in what the meaning of an exact sequence is and the obstruction that homology actually measures. Zooming out, it is also very interesting to wonder in what type of category one can actually talk about exact sequences. This leads for example to the notion of an exact category. Looking at all chain complexes over an exact category $\mathcal{E}$ gives rise to the triangulated category $D(\mathcal{E})$, the derived category of $\mathcal{E}$. This is a beautiful invariant attached to $\mathcal{E}$ that sees a lot of 'homological information' of $\mathcal{E}$. $\endgroup$ Commented Jul 18, 2021 at 9:24

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