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• CommentRowNumber1.
• CommentAuthorDavid_Corfield
• CommentTimeOct 19th 2009
• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeOct 21st 2009
This comment is invalid XHTML+MathML+SVG; displaying source. <div> <p>I added at <a href="http://ncatlab.org/nlab/show/Cauchy+complete+category">Cauchy complete category</a> the direct hyperlink to <a href="http://ncatlab.org/nlab/show/free+cocompletion">free cocompletion</a>.</p> <p>I have a question: why does it say there</p> <blockquote> ... preserves small coproducts and coequalizers. </blockquote> <p>and not just "preserves small colimits". Do you mean "...preserves small coproducts and ALL coequalizers"?</p> <p>I am also not sure if I understand the "all presheaves that are connected and projective". What does "connected" mean here? I think that should have a hyperlink.</p> <p>I'll add to <a href="http://ncatlab.org/nlab/show/tiny+object">tiny object</a> the example: every representable in a presheaf category.</p> </div>
• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeOct 21st 2009

Please have a look at the example section I wrote at tiny object, pointing to Cauchy completion.

I haven't really looked into this Cauchy completion business. I'd be interested in seeing at Cauchy completion one or two examples spelled out, or at least indicated. I understand and appreciate how metric spaces are enriched categories, but I am not sure yet about the Cauchy completion part (of couse I know that i can read about it in Lawvere's article, if you force me to).

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeOct 21st 2009

An object X is connected if hom(X, -) preserves coproducts (as explained at connected space). An object is projective if hom(X, -) preserves coequalizers of parallel pairs of maps.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeOct 21st 2009

I have now added material to Cauchy complete category, with particular detailed attention to the classical case of metric spaces. I think it might be time to remove the old discussion I had with Toby, since we apparently came to a consensus (and I'm tired of rereading my snippy response (-: ).

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeOct 21st 2009

Are you sure you want to use "projective" that way? The usual meaning of projective object is that hom(X,-) preserves some kind of epimorphism. But I think even preserving regular epimorphisms doesn't mean that it preserves coequalizers.

• CommentRowNumber7.
• CommentAuthorTodd_Trimble
• CommentTimeOct 21st 2009

Some edits and a query at tiny object.

• CommentRowNumber8.
• CommentAuthorTodd_Trimble
• CommentTimeOct 21st 2009
• (edited Oct 21st 2009)

 I think there's every likelihood that Mike is right about preservation of coequalizers not meaning the same thing as preservation of regular epis (off hand not even sure it's true if pullbacks are preserved). So we might want to reconsider using "projective" that way, although I thought I had a vague memory that it was sometimes used that way.

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeOct 21st 2009
That's wonderfully written. Thanks, Todd.
• CommentRowNumber10.
• CommentAuthorDavid_Corfield
• CommentTimeOct 22nd 2009
Yet another question about the relation between Cauchy completion and ideal completion.
• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeOct 22nd 2009

I also wanted to say: Thanks, Todd, that's an impressive entry. I need to keep this stuff in mind, it clearly points towards something big and important.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeOct 22nd 2009
• (edited Oct 22nd 2009)

I have added a bunch of hyperlinks now to Cauchy complete category.

• CommentRowNumber13.
• CommentAuthorTobyBartels
• CommentTimeOct 22nd 2009

I changed references to 'contractive maps' into 'short maps'. The former term seems to vary in the literature; for example, Wikipedia defines it (presumably following the reference that it gives) as any Lipschitz map, while the next Google hit defines it as a map with a Lipschitz constant strictly less than $1$ (and which therefore has a fixed point).

• CommentRowNumber14.
• CommentAuthorTobyBartels
• CommentTimeOct 22nd 2009

I wrote connected object, although only for objects of an extensive category, since only this was explained at connected space.

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeJul 26th 2011

I have added to Cauchy complete category an Example-subsection on Cauchy-completion of prosets and posets in regular and exact categories, citing an article by Rosolini.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeJul 27th 2011

I have added to Cauchy complete category a fairly extensive list of explicit definitions, propositions and theorems from Borceux-Dejean (linked to there).

Nothing that was not already indicated in the Idea-section, but I found it worthwhile to spell it all out in a bit more detail.

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeJul 28th 2011

I have added to Cauchy complete category a further subsection: In terms of essential geometric morphisms.

This is supposed to contain the statements that Todd was pointing out in another thread here.

• CommentRowNumber18.
• CommentAuthorTim Campion
• CommentTimeMar 2nd 2014

I’ve added a few examples based on this nCafe discussion. They’re not worked out at all, but I wanted to highlight the sheer variety of different things that Cauchy completion can mean. One thing this list could use is some references – I really wasn’t sure about these.

• CommentRowNumber19.
• CommentAuthorTim_Porter
• CommentTimeMar 2nd 2014

There is an early paper by Street (1974) mentioned in Kelly’s enriched monograph. Kelly discusses Cauchy completeness to some extent, so perhaps that could be put as a starting point. I do not know that stuff well enough to be sure.

• CommentRowNumber20.
• CommentAuthorMike Shulman
• CommentTimeMar 19th 2021