Timeline for How (if at all) does category theory deal with situations where the usual notion of isomorphism isn't right?
Current License: CC BY-SA 3.0
9 events
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| Sep 29, 2015 at 18:10 | history | edited | Peter LeFanu Lumsdaine | CC BY-SA 3.0 |
changed paragraph breaks, for better flow.
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| Sep 29, 2015 at 17:40 | comment | added | Peter LeFanu Lumsdaine | @goblin: sorry I missed that bit. Edited answer to address it. | |
| Sep 29, 2015 at 17:39 | history | edited | Peter LeFanu Lumsdaine | CC BY-SA 3.0 |
answered a bit of the question I missed at first
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| Sep 29, 2015 at 16:14 | comment | added | Eric Wofsey | @goblin: I don't understand your first sentence is supposed to mean, or how it applies to the examples you gave in the question. | |
| Sep 29, 2015 at 14:47 | comment | added | goblin GONE | But remember, I want to understand isomorphisms as "corresponding to" natural isomorphisms between $\mathrm{Hom}(X,−)$ and $\mathrm{Hom}(Y,−)$. I want to not have to choose them. So suppose $\mathbf{C}$ is an $\mathcal{M}$-category. Then, from the perspective of hom-functors, how can we justify the idea that an "isomorphism" $X \rightarrow Y$ is the same as a "tight isomorphism" $X \rightarrow Y$? | |
| Sep 29, 2015 at 14:32 | comment | added | Peter LeFanu Lumsdaine | @goblin: you choose the enrichment in $\mathbf{SDS}$ so that the special isomorphisms (i.e. special morphisms with a special inverse) are the ones you want. Nothing can automatically ensure in general that these are the “correct” ones — e.g. there is always the trivial $\mathbf{SDS}$-enrichment, where all morphisms are special, and its isos will just be the isos of the original category. But if you want a general result that ensures giving the “correct” isomorphisms, then you’ll need a more precise axiomatisation of the situations you want to cover. | |
| Sep 29, 2015 at 14:23 | comment | added | goblin GONE | How does enriching in $\mathbf{SDS}$ ensure that the isomorphisms are the "correct" ones? | |
| Sep 29, 2015 at 12:19 | comment | added | Todd Trimble | Here's the topic in the nLab: ncatlab.org/nlab/show/M-category | |
| Sep 29, 2015 at 11:46 | history | answered | Peter LeFanu Lumsdaine | CC BY-SA 3.0 |