Timeline for are quotients by equivalence relations "better" than surjections?
Current License: CC BY-SA 4.0
6 events
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| Dec 8, 2018 at 0:12 | comment | added | Qiaochu Yuan | ...conceptually distinct even though they happen to agree as universal constructions in $\text{Set}$. Said another way, I'm either talking about your state of knowledge before you learn that $\text{coim}(f) \to \text{im}(f)$ is an isomorphism in $\text{Set}$, or I'm reserving the right to replace $\text{Set}$ with another category in which that's false. If you really want to stick to $\text{Set}$ then Andrej's answer is probably more relevant than mine. | |
| Dec 8, 2018 at 0:10 | comment | added | Qiaochu Yuan | @Kevin: yes, I'm eliding some philosophical points here that I think Andrej's answer tackles better, at least in $\text{Set}$. In particular I am really referring to a particular construction of the image and coimage rather than to anything which satisfies their universal properties, which is a sin I don't normally indulge in. My excuse is that I never just isolate myself to $\text{Set}$ if I can help it; morally the coimage is "the image as seen from $X$" and the image is "the image as seen from $Y$," as I think the example of $\text{Top}$ makes vividly clear, and so I think of them as... | |
| Dec 8, 2018 at 0:06 | comment | added | Kevin Buzzard | But these things like im and coim are I believe defined by category-theoretic properties and are hence only defined up to unique isomorphism. Aren't I trying to do something better than isolating the object up to unique isomorphism? I'm wondering if in Set there is some sort of canonical representative (and am still not sure about if there is one). | |
| Dec 7, 2018 at 22:42 | history | edited | Qiaochu Yuan | CC BY-SA 4.0 |
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| Dec 7, 2018 at 22:36 | history | edited | Qiaochu Yuan | CC BY-SA 4.0 |
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| Dec 7, 2018 at 22:29 | history | answered | Qiaochu Yuan | CC BY-SA 4.0 |