Timeline for Projectives in functor category
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Nov 22, 2017 at 10:38 | comment | added | მამუკა ჯიბლაძე | In fact I now looked at the "Related" column to the right and this is almost duplicate of mathoverflow.net/q/162795/41291 | |
| Nov 22, 2017 at 10:30 | comment | added | მამუკა ჯიბლაძე | @PeterLeFanuLumsdaine To be honest I am not sure this is research level, most of this material is standard and can be found in several books on abelian categories | |
| Nov 22, 2017 at 8:54 | comment | added | Peter LeFanu Lumsdaine | @მამუკაჯიბლაძე: Why not make those comments an answer? | |
| Nov 21, 2017 at 11:44 | comment | added | მამუკა ჯიბლაძე | Yes, and if $\mathscr A$ has enough projectives then so does $\mathscr A^C$ - this is fairly standard. It follows from considering the functors $P[h^c]$ for $P$ a projective in $\mathscr A$ and $c$ an object of $C$, with $P[h^c](c')$ the direct sum of $\hom(c,c')$ many copies of $P$. Then$$\hom(P[h^c],F)\cong\hom(P,F(c))$$for any $F\in\mathscr A^C$, which implies that each $P[h^c]$ is projective. Moreover it is more or less clear from the above that each $F$ can be covered by a direct sum of $P[h^c]$'s. | |
| Nov 21, 2017 at 9:32 | comment | added | მამუკა ჯიბლაძე | To prove the last claim: if $C$ is a nonempty small category and $\mathscr A^C$ has enough projectives then so does $\mathscr A$. Indeed, first note that $\varinjlim:\mathscr A^C\to\mathscr A$ preserves projectives (because its right adjoint $\operatorname{const}$ preserves epis, having itself a right adjoint $\varprojlim$). Given that, for an object $A$, cover $\operatorname{const}(A)$ by a projective $P$, apply $\varinjlim$ and take the composite $ \varinjlim P\twoheadrightarrow \varinjlim\operatorname{const}(A)\twoheadrightarrow A $. | |
| Nov 21, 2017 at 6:40 | history | edited | Martin Sleziak | CC BY-SA 3.0 |
typo
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| Nov 21, 2017 at 6:37 | review | Close votes | |||
| Nov 21, 2017 at 22:10 | |||||
| Nov 21, 2017 at 6:17 | comment | added | მამუკა ჯიბლაძე | It is a Grothendieck category again, so has enough injectives. It has enough projectives if and only if the target does. I think. | |
| Nov 21, 2017 at 5:39 | review | First posts | |||
| Nov 21, 2017 at 8:13 | |||||
| Nov 21, 2017 at 5:35 | history | asked | iis24 | CC BY-SA 3.0 |