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From: Michael Barr <barr@math.mcgill.ca>
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Subject: Re: Question on choosing subobjects consistently
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Here is what Grothendieck actually said (in our translation):

"Thus a subobject of $A$ is not simply an object of \C, but an
object $B$, together with a monomorphism $u:B\to A$ called the
\emph{canonical injection} of $B$ into $A$.  (Nonetheless, by abuse of
language, we will often designate a subobject of $A$ by the name $B$
of the corresponding object of \C.)  The containment relation defines an
\emph{order} relation (not merely a preorder relation) on the class of
subobjects of $A$.  It follows from the above that the subobjects of
$A$ that are contained in a subobject $B$ are identified with the
subobjects of $B$, this correspondence respecting the natural
order."

It all depends on what he meant by "are identified with".  Maybe I am
making too much of this.  He regularly uses "=" for isomorphism (which I
have mostly changed in the translation).

Michael


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