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• CommentRowNumber1.
• CommentAuthorZhen Lin
• CommentTimeAug 30th 2012

Started pure subobject. I wish someone would tell me the intuitive reason for their importance though!

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeAug 30th 2012
• (edited Aug 30th 2012)

I have added the commutative diagram that you describe. This made me realize that there seems to be a typo: it must be

$a = \bar a \circ f'$

$a = \bar a \circ f$

no? Maybe I am mixed up. And there is no additional conidition that $b = f \circ \bar a$? That would have implied that pure subobjects are those monos which have the right lifting property against morphisms between $\kappa$-compact objects – which might help understand their meaning. But if not, maybe one should amplify this in the text (“Note that the other evident equations involving $\bar a$ is not required.”)

I also did a bit of formatting:

• I made sure that pure subobject and subobject point to each other. In particular at pure subobject I added a remark right in the first sentence telling the reader what this has to do with subobjects.

• I changed

 .num_proposition


to

.num_prop


Unfortunately the naming of these environments in unsystematic, but only the right usage of names will bring out the numbering, as expected. See at Instiki Theorems for the required syntax.

• I made your pointers to the reference LPAC appear as actual hyperlinks. I see from your code that you did intend to have code that makes this happen. I may not know the code you were trying to use, but maybe your forgot something, because it didn’t produce hyperlinks. I made it a hyperlink by pointing to the reference’s anchor.

• CommentRowNumber3.
• CommentAuthorZhen Lin
• CommentTimeAug 30th 2012

Sorry, yes, $a = \bar{a} \circ f'$. There is no condition involving $b$ – intuitively, this is because we are only saying the equation (encoded by $a$ and $f'$) can be solved in $A$, but not necessarily so that the solution agrees with the provided solution in $B$.

• CommentRowNumber4.
• CommentAuthorMarc
• CommentTimeAug 30th 2012
I added a reference to the article pure morphism which already exists for quite a while.

Perhaps the two entries should be merged?
• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeAug 30th 2012

In pure morphism, where unpublished work of Joyal and Tierney is mentioned, there is the phrase “For a local case of commutative rings”, whose meaning is unclear to me.

• CommentRowNumber6.
• CommentAuthorZhen Lin
• CommentTimeAug 31st 2012
• (edited Aug 31st 2012)

The definition in the pure morphism article says there is a right lifting property when there isn’t (by what Urs said). Merging wouldn’t be a bad idea.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeAug 31st 2012

I made the two entries at least point to each other. I wouldn’t merge them to a single entry: after all also morphism and subobject are not a single entry. But certainly the definitions need to be harmonized.

I don’t have time to look into this, and I guess some of you may have the references available much more easily. So I just added a Warning remark at pure morphism, indicating that the definition needs to be at least checked.

Anybody who knows for sure should fix whatever statement is currently wrong in the entry. (Let’s just make sure that there are not by any chance two different definitions of “pure” in the literature?!)

• CommentRowNumber8.
• CommentAuthorMarc
• CommentTimeAug 31st 2012
> Anybody who knows for sure should fix whatever statement
> is currently wrong in the entry. (Let's just make sure that
> there are not by any chance two different definitions
> of "pure" in the literature?!)

I corrected the definition. But in fact there is already an older
definition of pure submodule, which is not equivalent to the modern one.
The story is described somewhere in Kaplansky's "Infinite Abelian Groups".
• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeAug 31st 2012
• (edited Aug 31st 2012)

One trivial issue: at the very very end, after the last reference, there is an out-of-place “Mike Prest”. Either this needs to be deleted or it needs to be moved to the beginning of that line, or something.

• CommentRowNumber10.
• CommentAuthorzskoda
• CommentTimeSep 4th 2012
• (edited Sep 4th 2012)

I also think it might be good to merge eventually. Namely, pure morphism is automatically monomorphism (at least in generality in which I studied the concept, I do not know in general). We do indeed have subobject separately from monomorphism but not the variants like regular subobject in addition to regular monomorphism. The difference between monomorphism and subobject does not need more than one morphism dedicated. I very recently wrote a stub Ziegler spectrum which is involving pure injectives.

• CommentRowNumber11.
• CommentAuthorTim_Porter
• CommentTimeSep 4th 2012

@9 I checked back. It was an accidental addition some time ago so I deleted it. The paper mentioned on the same line seems to have been published so I updated the reference. I also created an entry for Mike Prest.

• CommentRowNumber12.
• CommentAuthorZhen Lin
• CommentTimeSep 4th 2012

According to [LPAC, Examples 2.28], every non-constant morphism in the category of complete semilattices is pure… more or less vacuously, because the only $\kappa$-compact complete semilattice (for any $\kappa$) is the trivial one.

• CommentRowNumber13.
• CommentAuthorzskoda
• CommentTimeSep 5th 2012
• (edited Sep 5th 2012)

Mike Prest is the author of several references at Ziegler spectrum which is about indecomposable pure injectives. I added a link at Mike Prest.

1. Proposition 2.31 in the book of Adámek and Rosický is about locally presentable categories (and that pure morphisms are regular mono there), for accessible categories this is an open problem according to the book (unless this has been proved by now, but then a reference to that source should be given).

Mark Kamsma