Timeline for What is homology anyway?
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7 events
| when toggle format | what | by | license | comment | |
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| Nov 24, 2021 at 10:57 | comment | added | Z. M | Neither do I know about cosheaves, but my impression is that, sheaves look like "functions" on the space while cosheaves look like "objects themselves". I think that, for example, for a map $f\colon E\to X$, you could consider the functor for every open $U\subset X$ to the homotopy type (I don't know what version exactly, but maybe we could take the shape) of $f^{-1}(U)$ — this looks like a cosheaf and if $f$ is a "good" fibration, then this should be a local system that you mentioned. | |
| Nov 24, 2021 at 4:55 | comment | added | Mike Shulman | @Z.M I suppose one could perhaps consider homology with coefficients in cosheaves. I don't know anything about cosheaves, so I don't know whether that would work. I do know that sheaves arise very naturally and I doubt cosheaves are as natural. A topos, for instance, is essentially defined to be a category of sheaves. And classical the classical "local coefficient systems" used for both homology and cohomology are (locally constant) sheaves. | |
| Nov 23, 2021 at 22:25 | comment | added | Z. M | I wonder why you would consider the homology with coefficients in some sheaf instead of cosheaf? The later seems to be more natural coefficient system for homology? | |
| Sep 11, 2017 at 18:13 | comment | added | Mike Shulman | @DavidCorfield I've added some more explanation. | |
| Sep 11, 2017 at 18:13 | history | edited | Mike Shulman | CC BY-SA 3.0 |
add paragraph about locally-connected / proper duality
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| Sep 11, 2017 at 13:26 | comment | added | David Corfield | Could you say a little more about your final sentence, Mike? Looking at the pages -- ncatlab.org/nlab/show/locally+connected+geometric+morphism and ncatlab.org/nlab/show/proper+geometric+morphism -- a fundamental duality isn't terribly obvious to me. | |
| Sep 11, 2017 at 11:51 | history | answered | Mike Shulman | CC BY-SA 3.0 |