Timeline for Are inclusions "canonical" injections?
Current License: CC BY-SA 4.0
10 events
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| Dec 12, 2018 at 8:41 | comment | added | Simon L Rydin Myerson | Every injection from a singleton has to factor through a good map. So for each element of each ordinal there must be an $i$ having that element as its image. So the $i$ are injections, but can't all be inclusions. | |
| Dec 12, 2018 at 0:09 | comment | added | user44191 | Why couldn't they all be inclusions? The $F(X)$ are all ordinals, so they include the singleton ordinal. The $i$ in my example are order-preserving maps between ordinals, if I've understood your answer correctly, and so are injections - meaning the identity automorphism works. | |
| Dec 11, 2018 at 22:53 | comment | added | Simon L Rydin Myerson | @user44191 actually there is no such automorphism even for your original example, since in that case all good maps from singletons have the same domain, so all maps $i$ from singletons have the same domain, so they can't all be inclusions. I will think about what I should have said. | |
| Dec 11, 2018 at 7:11 | comment | added | user44191 | Let $F: \text{Set} \rightarrow \text{Set}$ be as in my answer, except that there are $3$ $1$-element sets (and $F$ takes each of those to itself). This should be a fine endofunctor, and the $f_X$ are obvious. This should lead to no possibility for such an automorphism, right? | |
| Dec 11, 2018 at 0:05 | history | edited | Simon L Rydin Myerson | CC BY-SA 4.0 |
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| Dec 10, 2018 at 19:48 | history | edited | Simon L Rydin Myerson | CC BY-SA 4.0 |
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| Dec 10, 2018 at 16:34 | history | edited | Simon L Rydin Myerson | CC BY-SA 4.0 |
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| Dec 10, 2018 at 12:38 | history | edited | Simon L Rydin Myerson | CC BY-SA 4.0 |
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| Dec 10, 2018 at 12:22 | history | edited | Simon L Rydin Myerson | CC BY-SA 4.0 |
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| Dec 10, 2018 at 12:15 | history | answered | Simon L Rydin Myerson | CC BY-SA 4.0 |