Section Thick subcategories and saturation

(For this material see Schubert 1970, pp. 105–107).

changed to

(For this material see Gabriel and Zisman, Calculus of fractions and homotopy theory I.2.5.d) and Schubert 1970, pp. 105–107).

as Gabriel and Zisman I.2.5.d) seem to be the original reference for this.

]]>Previous eom link http://eom.springer.de/l/l060290.htm does not work, replaced with new version https://encyclopediaofmath.org/wiki/Localization_in_categories. It is a shame on Springer that upon change to a new system they did not ensure redirects and now tons of old eom links are broken.

]]>Need to use f hat prime unlike in the usual line.

]]>$M'' \overset{f'}\longrightarrow (N''+N')/N' \overset{can}\longrightarrow N''/(N''\cap N')\overset{g'}\longrightarrow P/P''$Let $h$ be the composition of $f'$, of the canonical isomorphism of $(N''+ N')/N'$ on $N''/(N''\cap N')$ and $g'$.

By the way the position of prime in $f'$ in overset is strange…

]]>Added to the section on quotient category,

It remains to define bilinear composition laws:

$Hom_{A/T}(M, N)\times Hom_{A/T}(N, P)\longrightarrow Hom_{A/T}(M,P).$For this, let $\overline{f}$ be an element of $Hom_{A/T}(M, N)$ and let $\overline{g}$ be an element of $Hom_{A/T}(N, P)$. The element $\overline{f}$ is the image of a morphism $f: M'\to N/N'$ where $M/M'$ and $N'$ are objects of $T$. Similarly, $\overline{g}$ is the image of a morphism $g:N'\to P/P'$, where $N/N'$ and $P'$ are objects of $T$. If $M''$ designates the inverse image $f^{-1}((N''+N')/N')$, it is easy to see that $M/M''$ belongs to $T$; we denote by $f'$ the morphism from $M''$ to $(N''+N')/N'$ which is induced by $f$. Likewise, $g(N''\cap N')$ is an object of $T$; if $P''$ denotes the sum $P'+g(N''\cap N')$, it is easy to see that $P''$ belongs to $T$; we denote $g'$ the morphism of $N''/(N''\cap N')$ in $P/P''$ which is induced by $g$.

Let $h$ be the composition of $f'$, of the canonical isomorphism of $(N''+ N')/N'$ on $N''/(N''\cap N')$ and $g'$. The image $\overline{h}$ of $h$ in $Hom_{A/T}(M, P)$ depends only on $\overline{f}$ and $\overline{g}$ and not on $f$ and $g$. It is therefore justified to define the composition in $A/T$ by equality $\overline{g}\circ\overline{f} = \overline{h}$. These composition laws are bilinear; they make $A/T$ a category.

(adapted from Gabriel, Des Catégories Abéliennes, III.1)

Translation and adaptation is mine with some help from Google Translate.

]]>Recently added definition that thick subcategory in an abelian category is closed “under direct summands, kernels of epis, cokernels of monos and extensions” is in the abelian categories precisely the same as it was/is stated below/before: closed under subobjects, quotients and extensions (or packed differently: topologizing+closed under extensions). Direct summands is a superfluous condition as it is included both under subobjects and under quotients.

I have split the definition part into two sections: one for derived and one for abelian context, as it should be as the definitions quite differ.

]]>Thanks for the alert.

I have now tried to briefly fix the first two paragraphs – stating closure under direct summands for thick triangulated subcategories and stating a definition of thick subcatgeories of abelian categories that I see in the literature.

I see there is still something weird going on as it goes to the third paragraph. I haven’t touched that, but left a warning.

(I didn’t write this stuff, nor am I expert on it, and, most importantly, I am about to leave for vacation with no time left to dig into it. If you are expert and want to do the community a favour, please feel invited to bring the entry into better shape.)

]]>Added example copied from quotient category.

]]>Thanks.

]]>Fixes typo in the definition of Serre quotient category.

Anonymous

]]>Hm. I think I understand what happened. I have ticked the box as nontrivial and wrote some comments. In later edits the same afternoon I did not want to make new announcements and I ticked a box that those later edits were trivial. But probably 30 minutes did not pass and the previous announcement did not pass through. This was before we discussed the delays extensively and I was not aware that the announcement was already not immediate.

]]>Hi Zoran, I checked the logs, and the tool is correctly finding your nForum user. However, the logs show that all of your edits were marked as ’trivial’: I see edits to invariant differential form, Hausdorff series, and Runge-Kutta method on the 3rd of April, between 13:08 UTC and 15:05 UTC, but all are marked as trivial. Are you sure that you ticked to announce the entries? If you are sure, could you try an edit now (maybe just find a typo somewhere, or make some trivial change) where you click to announce?

For the record, the algorithm that is used to associate an nForum user to an nLab author is described in #26 here. To summarise: two generic methods are attempted to make the association, and if this does not work, then a file which l edit manually to define author-user correspondences is checked. So far I’ve added Todd and David C to that list :-). If there is still no match, then the name of the author is included in the announcement text.

]]>I edited a couple of pages a couple of days ago, ticked the box for announcer and the change were not announced. Maybe my $n$Forum name zskoda is not the same at the $n$Lab contributor signature Zoran Škoda or something ? What happens when the name is not matching, can it be manually related in some database ?

]]>I also see that there are some bugs in Instiki’s mechanism for showing the diff. I can try to look at that at some point.

]]>Thanks! Nice that this person kindly indicated their changes via the new tool; since they are anonymous, they probably would not have left a message at the nForum otherwise. This was the initial motivation for adding the annoucement mechanism, so good to see that it may be beginning to help.

]]>added some more hyperlinks to keywords

]]>I guess it should read as “one defines the (Serre) quotient category $A/T$ as the one having the same objects as $A$ and ….” and not ” … same objects as $T$… ” Similarily in the line following the line mentioned here.

Anonymous

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