Timeline for Motivation for working with augmented objects in homological or higher algebra
Current License: CC BY-SA 4.0
10 events
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| Jun 16, 2023 at 13:52 | comment | added | user267839 | @BenjaminSteinberg: for sake of completeness here another example I found where Allen Hatcher gave in his AT book (page 121 in the online available source) a concrete situation where it might be "convenient" to make use of this passing to augmented complex procedure to make a proof more "handy" - or say amenable | |
| Jun 3, 2023 at 14:17 | comment | added | Benjamin Steinberg | 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 | |
| Jun 3, 2023 at 10:51 | comment | added | user267839 | @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? | |
| Jun 3, 2023 at 0:41 | comment | added | Mariano Suárez-Álvarez | 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. | |
| Jun 2, 2023 at 21:21 | comment | added | Denis T | Do you have any particular motivation to not use augmented complexes? | |
| Jun 2, 2023 at 19:55 | comment | added | Benjamin Steinberg | In topology it comes about naturally by allowing empty simplices and making the boundary of a 0-simplex empty. | |
| Jun 2, 2023 at 13:58 | history | edited | user267839 | CC BY-SA 4.0 |
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| Jun 2, 2023 at 13:45 | history | edited | user267839 | CC BY-SA 4.0 |
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| Jun 2, 2023 at 13:34 | history | edited | user267839 | CC BY-SA 4.0 |
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| Jun 2, 2023 at 13:28 | history | asked | user267839 | CC BY-SA 4.0 |