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1. • CommentRowNumber1.
   • CommentAuthorKarol Szumiło
   • CommentTimeOct 12th 2012
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   Author: Karol Szumiło
   Format: MarkdownItexThis is a terminology question. Is there a standard name for a diagram in a relative category with a property that its colimit exists and is already a homotopy colimit? Those of course include diagrams which are cofibrant with respect to some compatible model structure on diagrams (if one happens to exist). However, in some categories there are such diagrams which don't seem to come from any notion of cofibration, so it would be convenient to have a nice name for them.

   This is a terminology question. Is there a standard name for a diagram in a relative category with a property that its colimit exists and is already a homotopy colimit? Those of course include diagrams which are cofibrant with respect to some compatible model structure on diagrams (if one happens to exist). However, in some categories there are such diagrams which don’t seem to come from any notion of cofibration, so it would be convenient to have a nice name for them.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeOct 12th 2012
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   Author: Urs
   Format: MarkdownItexSometimes people say that some 1-categorical structure is "homotopy-good" if some universal construction on it yields the correct homotopy-theoretic result.

   Sometimes people say that some 1-categorical structure is “homotopy-good” if some universal construction on it yields the correct homotopy-theoretic result.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeOct 12th 2012
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   Author: Mike Shulman
   Format: MarkdownItexOr perhaps more specifically "colim-good" to specify the functor in question, or maybe "colim-cofibrant".

   Or perhaps more specifically “colim-good” to specify the functor in question, or maybe “colim-cofibrant”.