The passage from any small category C to its set-valued functor categoy $\hat{\mathbf C}:=\mathrm{Fun}(C^{\ast},\mathrm{\bf Set})$ i.e. the full Yoneda-embedding $Y\colon \mathrm{\bf C} \to \hat{\mathrm{\bf C}}$ into the presheaf category can be considered as an universal completion-process.

A functor category such as $\mathrm{Fun}(\mathrm{\bf C}^{\ast},\mathrm{\bf Set})$ is a category which is ``almost as good as the target category $\mathrm{\bf Set}$''. In particular such a functor category is a topos and has an injective subobject classifier $\Omega$.

In the simplest case of $\mathrm{\bf Set}$, which can be considered as the special case $\mathrm{\bf C}=\mathrm{\bf 1}$, the subobject classifier is a two-element set $\{0,1\}$ and has the property, that it is a cogenerator. (An object $C$ of a category is called cogenerator, if for any two distinct morphism $f,g\colon X\to Y$ there is a morphism $s\colon Y\to C$ such that $s\circ f\neq s\circ g$). A cogenerator $C$ allows to separate the morphisms in the category and so to ``resolve''
the category, if $C$ happens to be an injective object.

It seem to be natural to ask whether the subobject classifier of any set-valued functor category is a cogenerator. This can not be the case, since in the special case $\mathrm{\bf C}={\mathbb Z}_2$ (category with one object generated by an non-trivial involution) which leads to the functor category $\hat{\mathbb Z}_2=\mbox{Fun}({\mathbb Z}_2^{\ast},\mathrm{\bf Set})$ of sets with a ${\mathbb Z}_2$-action, the subobject classifier $\Omega$ is a two-element set $\{0,1\}$ with the trivial ${\mathbb Z}_2$-action. This object $\Omega$ is not a cogenerator of $\hat{\mathbb Z}_2$ since the the two-element set with the nontrivial ${\mathbb Z}_2$-action gives an object $X$ of $\hat{\mathbb Z}_2$, whose nontrivial automorphism cannot be separated from the identity by any morphism from $X$ to $\Omega$. Hence there are conditions needed before the subobject classifier of a set-valued functor category can be a cogenerator.

Under which precise conditions for the category $\mathrm{\bf C}$ is the subobject classifier $\Omega$ of its free-cocompletion $\hat{\mathrm{\bf C}}$ a cogenerator?

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    $\begingroup$ You may want to look at Borceux's paper "When is $\Omega$ a cogenerator in a topos?", available here: numdam.org/item?id=CTGDC_1975__16_1_3_0 $\endgroup$ – Theo Buehler May 10 '11 at 11:45
  • $\begingroup$ @Theo: Why don't you post this as an answer? I mean this is exactly what was asked here. $\endgroup$ – Martin Brandenburg May 10 '11 at 15:35
  • $\begingroup$ @Theo: Thank you for your link. Unfortunately Borceux's paper does not answer the question. $\endgroup$ – Frank Zenter May 10 '11 at 16:21
  • $\begingroup$ @Martin: Borceux studies in his paper indeed the corresponding question in an arbitrary topos and proves the following result: Theorem: Let $E$ be a topos. If the subobjects of 1 form a set of generators, the $\Omega$ is a cogenerator. As Borceux also remarks, this fact is true for a much wider class of topoi such as the category of graphs. I would like to know for which set-valued functor categories it is true. $\endgroup$ – Frank Zenter May 10 '11 at 16:27
  • $\begingroup$ Another paper that might be of interest is Generating families in a topos by Kenney: ftp.gwdg.de/pub/misc/EMIS/journals/TAC/volumes/16/31/… (I'm not claiming this answers the question, but maybe it provides some insights). $\endgroup$ – Theo Buehler May 10 '11 at 16:41

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