Timeline for Projectives and Injectives in Functor Categories
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Apr 9, 2014 at 7:34 | vote | accept | Bugs Bunny | ||
| Apr 8, 2014 at 13:48 | comment | added | Andreas Blass | I believe this sort of thing was done by Charles Watts in "A homology theory for small categories" [Proc. Conf. Categorical Algebra, La Jolla; Springer (1966) 331-335]. I don't remember the details, nor do I have the paper handy, but the MathSciNet review sounds as though this paper may be relevant. | |
| Apr 8, 2014 at 13:34 | comment | added | Fernando Muro | @BugsBunny: Yes, it's also Grothendieck. Projectives are easy, they are just 'tensor products' of representables and projectives in $A$. It's actually the easiest part and doesn't need the strength of Grothendieck categories. As for references, if the target is Ab then probably Freyd's book on abelian categories. Otherwise Ulmer maybe? | |
| Apr 8, 2014 at 13:26 | answer | added | Zhen Lin | timeline score: 14 | |
| Apr 8, 2014 at 13:16 | comment | added | Bugs Bunny | @ Fernando: what is the reference for this? I am kind of see why this helps with injectives (probably, $A^C$ will be Grothendieck too, will it not?) but I am not sure about projectives... | |
| Apr 8, 2014 at 13:12 | history | edited | Bugs Bunny | CC BY-SA 3.0 |
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| Apr 8, 2014 at 13:11 | comment | added | Bugs Bunny | Good point and a counter-example too: take your $C$ and $A$ the category of finite dimensional vector spaces... I am editing the question to address this. | |
| Apr 8, 2014 at 13:10 | comment | added | Fernando Muro | If $A$ were a Grothendieck category the answer would be yes, even under smaller assumptions. I wonder what you have in mind when you require $A$ to have just 'very small' (co)products. | |
| Apr 8, 2014 at 13:07 | comment | added | Johannes Hahn | @ZhenLin Moreover: If one chooses $\mathcal{A}$ to be the category of finite dimensional $k$-vector spaces, then $\mathcal{A}$ has enough projective for the cardinality of $Ob(\mathcal{C})=\{\ast\}$ but there are no finite dimensional $k[x]$-modules. | |
| Apr 8, 2014 at 13:02 | comment | added | Zhen Lin | Even if you could choose a functorial projective cover in $\mathcal{A}$, there is still the fact that diagrams that are componentwise projective need not be projective. For instance, $\mathcal{C}$ could be the category freely generated by one endomorphism and $\mathcal{A}$ could be the category of $k$-vector spaces; then a diagram $\mathcal{C} \to \mathcal{A}$ is the same thing as a $k [x]$-module. | |
| Apr 8, 2014 at 12:28 | history | edited | Bugs Bunny | CC BY-SA 3.0 |
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| Apr 8, 2014 at 12:17 | history | asked | Bugs Bunny | CC BY-SA 3.0 |