Timeline for Are there examples where one proves something about the functor represented by an object using the functor it corepresents?
Current License: CC BY-SA 2.5
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| Apr 20, 2010 at 3:38 | comment | added | JBorger | OK, here's an example: a group is initial if and only if it is terminal. It would be nice to have a meatier example. For instance, it might be reasonable to say that Lefschetz fixed-point theorem gives an example. If you think of the cohomology of $X$ as being $\mathrm{Hom}(X,E)$, where $E$ is some generalized space, then for $X$ over a finite field, you can prove there exist rational points, i.e. certain maps into $X$, by analyzing the cohomology of $X$. It would be nice to have lots of other examples so that maybe we could draw some general lessons. | |
| Apr 20, 2010 at 3:30 | comment | added | JBorger | I just meant that the Yoneda functor is fully faithful, so passing from a category to its image under the Yoneda functor involves no loss of information. In other words, you can recover an object, up to unique isomorphism, from the functor it represents. I don't know of any nice examples. That's why I'm asking the question! There must be lots, but I don't usually think of things in these terms. | |
| Apr 19, 2010 at 22:59 | history | answered | Martin Brandenburg | CC BY-SA 2.5 |