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I understand that you are copying material, but let’s do a minimum of adjustment so that it makes sense in the larger context of the nLab.
If you point to the nLab’s entry category then it’s hard for the reader to understand what kind of completion you have in mind. Better to point more specifically to internal category in homotopy type theory – at least for the first occasion.
I am taking the liberty of adding to the References-section the following:
The relation between Segal completeness (now often “Rezk completion”) for internal categories in HoTT and the univalence axiom had been pointed out in:
I have completed the publication data for:
By the way, if you list items (such as under “See also”) without a bullet or numbering in front, then the the parser thinks you keep starting new paragraphs for each item and produces overly large vertical whitespace. Best to use bullet lists markup like this:
* an item
* another
* yet another
I have changed the Theorem from a subsection to a theorem-environment (now here), analogously for the proof. Notice the easily remembered code:
\begin{theorem}...\end{theorem} \begin{proof} ... \end{proof}
By the way, after the proof ends, the text keeps going in a surprising/unclear way. Maybe there is some guiding text missing and/or the proof doesn’t actually end at that point.
Is there any risk of confusion with the localization from the ∞-category of segal spaces in ∞Gpd to the reflective subcategory of rezk-complete segal spaces?
The content of this article seems related, but restricted specifically to the case where the hom-spaces are sets.
The thing this page computes is the same notion as the localization from segal space objects to complete segal space objects, but restricted to segal spaces whose hom-spaces are sets, right?
Is this talked about on the nLab? Would this page be an appropriate place?
The link to “Segal completion” just goes to the complete segal space page and doesn’t talk about forming completions. (and to me that that seems like a weird name for the operation)
There should be a similar Rezk completion higher inductive type which turns Segal types into Rezk types in simplicial type theory.
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