Timeline for Proving that the functor induced by some inclusion functor has a left adjoint
Current License: CC BY-SA 4.0
8 events
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| Aug 29, 2023 at 15:02 | comment | added | Juan C. Cala | @Sampah just merely additive, right exactness of $G$ is not required. When $G$ is right exact, $G\cong (LJ)G$, and the converse is also true. | |
| Aug 29, 2023 at 14:50 | comment | added | Sampah | Ah so the functors are taken to be right exact (makes sense) and additive. No worries glad you have solved it. | |
| Aug 29, 2023 at 14:46 | comment | added | Juan C. Cala | @Sampah no, but you made me check and now I found that Auslander is assuming that all functors are additive and that suffices... my bad. | |
| Aug 29, 2023 at 14:34 | comment | added | Sampah | Opps, you are definitely correct on that. My bad. But while I haven't checked the details, I feel the construction I outlined does give the required inverse. Does not Auslander assume the functor $\mathcal B(\mathcal{A})\rightarrow\mathcal D$ to preserve cokernels (I don't have access to the book)? It just feels natural to assume so as $\mathcal D$ is an additive category which has cokernels. | |
| Aug 29, 2023 at 14:27 | comment | added | Juan C. Cala | @Sampah but why universality of cokernel applies? Because $G(X_1\to X_0\to X)=G(0)$ is not necessarily zero. | |
| Aug 29, 2023 at 8:55 | comment | added | Sampah | I think given an arbitrary natural transformation (nt) $\tau:F\Rightarrow JG$, we construct the corresponding nt $LF\Rightarrow G$ as follows: for any $X\in\mathcal{B}(\mathcal{A})$, apply $LF$ and $G$ to $\eta_X$ and the resulting diagrams are now connected by $\tau$ for $X_1$ and $X_0$ in $\eta_X$ and then use the universality of cokernel $LF(X)$ to construct the component $LF(X)\rightarrow G(X)$. | |
| S Aug 29, 2023 at 5:14 | review | First questions | |||
| Aug 29, 2023 at 6:40 | |||||
| S Aug 29, 2023 at 5:14 | history | asked | Juan C. Cala | CC BY-SA 4.0 |