Timeline for Are there categories whose internal hom is somewhat 'exotic'?
Current License: CC BY-SA 4.0
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| Aug 24, 2019 at 6:19 | comment | added | Qiaochu Yuan | @Jakob: yes, thanks for that comment! I was going to mention this but then deleted it for some reason. I like that it drives home the intuition that we are looking at "$c$-parameterized families of natural transformations from $F$ to $G$." | |
| Aug 23, 2019 at 5:36 | comment | added | Jakob Werner | Via the Yoneda lemma it is of course obvious that the internal Hom in presheaf categories is given by $[F,G](c) = \mathrm{Hom}_{\mathrm{PSh}(C)}(c \times F, G)$, but I always like to think of it as $[F,G](c) = \mathrm{Hom}_{\mathrm{PSh}(C/c)}(F|_c, G|_c)$, where $C/c$ is the slice category over $c$ and $F|_c$, resp. $G|_c$ is the composition of $F$ with the forgetful functor $C/c \to C$. To me this looks more symmetrical and perhaps it is also closer to what one is maybe used to from the case of sheaves on a topological space. | |
| Aug 23, 2019 at 4:56 | history | answered | Qiaochu Yuan | CC BY-SA 4.0 |