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1. • CommentRowNumber1.
   • CommentAuthorAli Caglayan
   • CommentTimeSep 21st 2018
   • PermaLink
   Author: Ali Caglayan
   Format: MarkdownItexAdded the second proof of the Rezk completion. What would the eliminator for this HIT type look like? I cannot find anything in the literature apart from the HoTT book. It seems this was originally crafted up by Mike so perhaps he has an eliminator written down somewhere? Specifically the third and fourth constructors are giving me a headache. Because there is a higher transport involved, which even for simple spaces like a torus is a bit difficult to wrap your head around. <a href="https://ncatlab.org/homotopytypetheory/revision/diff/Rezk+completion/4">diff</a>, <a href="https://ncatlab.org/homotopytypetheory/revision/Rezk+completion/4">v4</a>, <a href="https://ncatlab.org/homotopytypetheory/show/Rezk+completion">current</a>

   Added the second proof of the Rezk completion.

   What would the eliminator for this HIT type look like? I cannot find anything in the literature apart from the HoTT book. It seems this was originally crafted up by Mike so perhaps he has an eliminator written down somewhere?

   Specifically the third and fourth constructors are giving me a headache. Because there is a higher transport involved, which even for simple spaces like a torus is a bit difficult to wrap your head around.

   diff, v4, current

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeSep 21st 2018
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI don't think I ever wrote out the full induction principle in gory generality, only what was necessary for the proof of Theorem 9.9.5. One related thing in the literature is the [Licata-Finster construction](https://homotopytypetheory.org/2014/04/15/eilenberg-maclane-spaces-in-hott/) of $K(G,1)$, which is the special case of the Rezk completion when $A_0 = 1$. Egbert Rijke's thesis may also contain a general "groupoid quotient" construction which includes the Rezk completion, but I don't think it is publically available yet.

   I don’t think I ever wrote out the full induction principle in gory generality, only what was necessary for the proof of Theorem 9.9.5. One related thing in the literature is the Licata-Finster construction of $K(G,1)$, which is the special case of the Rezk completion when $A_0 = 1$. Egbert Rijke’s thesis may also contain a general “groupoid quotient” construction which includes the Rezk completion, but I don’t think it is publically available yet.