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at model structure for Segal categories I have very briefly added the basic definition, some basic properties, and added references.
Silly question: Is a weak equivalence between fibrant objects in this model structure a categorical equivalence in the sense of the article, i.e. without applying the precategory-to-category completion functor? I can’t find any statement one way or the other…
Yes, I think so.
This follows under 2-out-of-3 from the characterizing def. 2.1 in Hirschowitz-Simpson (which says that completion is homotopy-idempotentent, in particular that the completion of a pre-Segal category which already is a Segal category is a weak equivalence).
I haved aded a remark on this to the entry model structure for Segal categories now.
(Maybe you feel inspired to fill in some more of the missing text in the entry.)
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