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I have added a section about unary sites to regular and exact completions, and a mention of the fact that there are higher-ary versions (also to pretopos completion. I think this is about all I’m going to do right now in the way of adding content to the nLab from my exact completions paper.
Below I copy an old discussion from regular and exact completions.
begin old discussion
Todd Trimble: Hi Mike. Thanks very much for writing this article. Could you clarify for me your use of parentheticals below in the case of reg/lex completion? I am guessing that all parentheticals like “(weak)” can be safely ignored in the case where $C$ is actually finitely complete (as opposed to weakly finitely complete), so that, for example, the objects of $C_{reg/lex}$ can in that case be taken to be pairs $(c, e)$ where $e$ is a kernel pair of a morphism $f: c \to d$, and morphisms $(c, e) \to (c', e')$ are morphisms $c \to c'$ which respect the equivalence relations.
If my guess is correct, I think I might have found it easier to read if the finitely complete case were run through first, followed by discussion of what modifications are to be made in the case of weak finite completeness, rather than discussing the two cases simultaneously using parentheticals.
Also, I wasn’t able to find under Properties something about the reg/lex completion I was expecting or hoping to find. (My immediate interest in this is the example of equilogical spaces as regular completion of $Top$.) Is there supposed to be a property along the lines that the regular completion is cartesian closed if $C$ has weak exponentials, and locally cartesian closed if $C$ has weak dependent products? (I don’t have ready access to the literature, so these should be treated as guesses.)
Mike Shulman: You’re exactly right about the meaning of the parentheticals. Sorry for writing that in kind of a confusing way. Your guess is exactly right, and your suggestion is a good one; see if you think it’s clearer now.
I’m surprised to realize that I omitted the conditions for (local) cartesian closure of the reg/lex completion, since one needs them to conclude that it preserves lextensive quasitoposes. The only conditions I can find in the literature require somewhat more than mere weak (local) cartesian closure of $C$. The point being that the set of function from a quotient of $X$ to a quotient of $Y$ may, in general, be only a quotient of the set of functions from $X$ to $Y$, so we need some way to ensure that that quotient exists in $C_{reg/lex}$ since it doesn’t have all quotients. But $Top$ does satisfy those conditions, so equilogical spaces are locally cartesian closed.
Todd: Thanks again; I think the section on regular completion does read more clearly now, and the material you added to Properties is very helpful. I think what you wrote on equilogical spaces might fit well in Examples, as would realizability toposes as examples of exact completions. If I have time soon I might write down something stubby here.
end old discussion
Partly on the principle of extracting a punchline from a removed discussion, I added the example of realizability toposes to the Examples section of regular and exact completions. Hopefully I can get to equilogical spaces as well, as another example.
Great, thanks!
Nit-picky question: in the Properties section you (I think it was you, Mike) wrote, “Note that $Top_{reg/lex}$ is called the category of equilogical spaces.” This $Top$ is the category $Top_0$ of T$_0$ spaces, yes?
Trying to fit some things together here: $Equ$ is also the result of splitting partial equivalence relations in the category of algebraic lattices. So I’m guessing we have a chain
$AlgLat \stackrel{Cor_{split}}{\to} Top_0 \stackrel{reg/lex}{\to} Equ$based on the idea that we can split PERs in two stages: first split coreflexive maps (that’s $Cor_{split}$), and from there split total equivalence relations (that’s the reg/lex completion, at least if equivalence relations coincide with kernel pairs, which I guess is true for T$_0$-spaces (?)).
Does that sound accurate?
Erm, maybe (on all points). It’s been a long time since I looked at equilogical spaces, and I don’t have the time right now to re-read up, sorry.
To regular and exact completions I added several observations with supporting references: (1) that the ex/lex completion of a presheaf topos $Set^{C^{op}}$ is a topos only if $C$ is a groupoid, (2) that the ex/lex completion of a $\Pi W$-pretopos is a $\Pi W$-pretopos, and (3) that the free small cocompletion of a (possibly large) finitely complete category is the ex/lex completion of its free coproduct completion.
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