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<p>I added at <a href="https://ncatlab.org/nlab/show/Cauchy+complete+category">Cauchy complete category</a> the direct hyperlink to <a href="https://ncatlab.org/nlab/show/free+cocompletion">free cocompletion</a>.</p>
<p>I have a question: why does it say there</p>
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... preserves small coproducts and coequalizers.
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<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="https://ncatlab.org/nlab/show/tiny+object">tiny object</a> the example: every representable in a presheaf category.</p>
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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).
I'll see about getting around to uncompleted business at Cauchy completion.
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.
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 (-: ).
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.
Some edits and a query at tiny object.
[Edit] 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.
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.
I have added a bunch of hyperlinks now to Cauchy complete category.
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).
I wrote connected object, although only for objects of an extensive category, since only this was explained at connected space.
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.
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.
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.
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.
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.
Added link to
Changed “pre-additive category” to “$\mathbf{Ab}$-enriched category” here, since that’s what is meant here, while the article pre-additive category defines a pre-additive category to be an $\mathbf{Ab}$-enriched category with a zero object (while acknowledging that it’s also used to mean an $\mathbf{Ab}$-enriched category).
Am moving the following “Discussion” (originating in revision 19 from 2009) out of the entry to here:
David: Concerning the result that on Set the terminal F-coalgebra is the Cauchy completion of the initial F-algebra, for certain F, I wonder if we have to factor completions through the metric space completion, as Barr does in Terminal coalgebras for endofunctors on sets. Perhaps Adamek’s work on Final Algebras are Ideal Completions of Initial Algebras is more natural.
Does this all tie in with the ideal completion as discussed by Awodey where you sum types/sets in a topos into a universal object?
How many kinds of completion are there for an enriched category? I see some may coincide in certain cases.
If two categories can be Morita equivalent, should this be reflected in the page Morita equivalence?
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