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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Added a reference for Cauchy complete 2-categories.

]]>Added more information about Lack and Tendas’ result on when an enriched category is Cauchy complete.

]]>Added reference to more discussion about Cauchy completion of suplattice-enriched categories.

]]>Added description of Cauchy completion for $R Mod$-enriched categories, and a reference where you can get a proof.

]]>Explained what Cauchy completion has to do with Karoubi envelope in the case of Ab-enriched categories.

]]>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).

Added note about smallness vs essential smallness.

]]>Added link to

- {#NST2020} Branko Nikolić, Ross Street, Giacomo Tendas,
*Cauchy completeness for DG-categories*, arxiv, 2020

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.

]]>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.

]]>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 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 an Example-subsection on Cauchy-completion of prosets and posets in regular and exact categories, citing an article by Rosolini.

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

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

]]>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.

]]>[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.

]]>Some edits and a query at tiny object.

]]>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.

]]>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 (-: ).

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