In Popescu's textbook Abelian Categories with Applications to Rings and Modules, Theorem $3.3$ of section $4.1$ says:

Let $\mathscr A$ be a dense subcategory of a locally small abelian category $\mathscr C$ and let $\Sigma = \Sigma_{\mathscr A}$. Then the category $\mathscr C_\Sigma$ of additive fractions of $\mathscr C$ relative to $\Sigma$ exists.

Where $\Sigma_\mathscr A$ is the system of all morphisms $s$ of $\mathscr C$ such that $\ker s$ and $\text{coker } s$ are in $\mathscr A$.

Now, in the exercises for this section, there is a question that says:

Let $\mathscr A$ be a dense subcategory of $\mathscr C$. Give sufficient conditions (more general than in Theorem $3.3$) for the existence of the category $\mathscr C/\mathscr A$. (In this case $\mathscr A$ is called a quotifiable subcategory). Then the category $\mathscr C/\mathscr A$ is abelian and the canonical functor $T : \mathscr C\rightarrow \mathscr C/\mathscr A$ is exact.

My question is about what he means by the more general than in Theorem $3.3$. I don't understand what is being asked here, and I can't find any references anywhere to a quotifiable subcategory. The google search "quotifiable subcategory" returns no results.

Can anyone shed any light on this? The reason I'm interested is due to another of the exercises here.

Let $H : \mathscr C\rightarrow \mathscr C'$ be an exact functor. Let $\ker H$ be the full subcategory containing all $X\in\mathscr C$ such that $HX = 0$. Then $\ker H$ is a dense subcategory. Furthermore, let us assume that $\ker H$ is quotifiable. Then the functor $\bar{H} : \mathscr C/\ker H \rightarrow \mathscr C'$ induced by the universal property of the quotient category is faithful.

I paraphrased that slightly, but I'm interested in it because I would like to know sufficient and necessary conditions for that induced functor to be faithful, and if quotifiable is necessary I'd like to understand what it means.

Thanks in advance.

  • $\begingroup$ How is $\mathscr C/\mathscr A$ defined? $\endgroup$ – Berci Sep 6 '17 at 21:41
  • $\begingroup$ It's defined to be the category of additive fractions $\mathscr C_\Sigma$ where $\Sigma$ is defined as above. $\endgroup$ – IAlreadyHaveAKey Sep 7 '17 at 0:55
  • $\begingroup$ $\mathscr C_\Sigma$ is constructed by leaving the objects the same as in $\mathscr C$ and $\Hom(X,Y)$ is the colimit of $\Hom(X', Y/Y')$ where $X/X'$ and $Y'$ are both in $\mathscr A$ $\endgroup$ – IAlreadyHaveAKey Sep 7 '17 at 0:57
  • $\begingroup$ ncatlab.org/nlab/show/localizing+subcategory perhaps? $\endgroup$ – Dap Sep 7 '17 at 6:51
  • $\begingroup$ I don't think so because the next section of the textbook discusses localizing subcategories. $\endgroup$ – IAlreadyHaveAKey Sep 7 '17 at 6:55

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