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I would like to understand if there is deeper reason/motivation behind augmentations in homological algebra. Recall classically in homology if there is a complex of free $R$-modules ($R commutative unitary ring)

$$ ... \xrightarrow{\partial_{n+1}} C_n \xrightarrow{\partial_{n}} C_{n-1} \xrightarrow{\partial_{n-1}} ... \xrightarrow{\partial_{2}} C_1 \xrightarrow{\partial_{1}} C_0 \to 0 $$

then it can be turned to augmented chain complex by extending the right part to

$$ ... \xrightarrow{\partial_{2}} C_1 \xrightarrow{\partial_{1}} C_0 \xrightarrow{\epsilon} R \to 0 $$

where $\epsilon(\sum_i a_i \sigma_i)):= \sum_i a_i \in R$

such that the $0$-th cohomology $H^0$ of the original complex and of the augmented complex $H^0_a$ are related by $H^0= H^0_a \oplus R$.

Fine, but the question is if there is any deeper philosophical reason behind using augmentations or is it just a historical leftover (I'm not sure who introduced this concept first. Eilenberg?) Moreover was it to pass to augmented objects just a "matter of taste" or was there some deeper mathematical reason behind?

(Maybe it's in certain sence more flexible for certain generalization directions. For example to work in some situations more "uniformly", I don't know, that's just a guess) Obviously there is no information gain/loss when passing to augmented objects. Are there nevertheless any advantages to pass to them? For example does this concept maybe provide better structural /functorial compatibilities or say "more natural framework" when working in more general setting (eg simplicial objects in higher categoy theory)?

Another guess: does augmentation play some interesting interpretation or give some interesting viewpoint in context of classical Dold-Kan corresponence?

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  • $\begingroup$ In topology it comes about naturally by allowing empty simplices and making the boundary of a 0-simplex empty. $\endgroup$
    – Benjamin Steinberg
    Commented Jun 2, 2023 at 19:55
  • $\begingroup$ Do you have any particular motivation to not use augmented complexes? $\endgroup$
    – Denis T
    Commented Jun 2, 2023 at 21:21
  • $\begingroup$ Some years ago someone asked in this site why people like reduced (co)homologies. The answers there probable can illuminate the reason for using augmented complexes. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Jun 3, 2023 at 0:41
  • $\begingroup$ @BenjaminSteinberg: That looks like a reasonable motivation i was looking for, thanks! So as far as I understand your point correctly, consequently the advantage is then that this provides seemingly the kind of ' uniformity' in the sense that arguments using induction on degree of the chain starting at $ 0$ circumvent treating more sometimes annoying exceptional cases when we add artificially this 'auxilary' -1-simplex aka the 'empty' simplex as generator of -1 piece? Is that the crucial point in the story? $\endgroup$
    – user267839
    Commented Jun 3, 2023 at 10:51
  • $\begingroup$ Yes. For instance notice if you try to define the relative homology of a pair since the empty simplex belongs to the subspace you get the usual relative homology. And this explains why you have the long exact sequence of relative homology groups $\endgroup$
    – Benjamin Steinberg
    Commented Jun 3, 2023 at 14:17

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