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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeOct 19th 2015
   • (edited Oct 20th 2015)
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexSuppose P is a (colored) [[prop]] in a closed symmetric monoidal locally presentable category C. Is the category Alg_P of algebras over P in C locally presentable? It seems that one can relatively easy establish the accessibility of algebras over P using [[PIE-limits]]. However, the forgetful functor Alg_P→C preserves neither limits nor colimits, so it's unclear how one could establish the (co)completeness of Alg_P (is this even true?).

   Suppose P is a (colored) prop in a closed symmetric monoidal locally presentable category C. Is the category Alg_P of algebras over P in C locally presentable?

   It seems that one can relatively easy establish the accessibility of algebras over P using PIE-limits.

   However, the forgetful functor Alg_P→C preserves neither limits nor colimits, so it’s unclear how one could establish the (co)completeness of Alg_P (is this even true?).

2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTimeOct 20th 2015
   • (edited Oct 20th 2015)
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexNow on MathOverflow: <http://mathoverflow.net/questions/221403/when-are-the-categories-of-algebras-over-props-cocomplete>

   Now on MathOverflow: http://mathoverflow.net/questions/221403/when-are-the-categories-of-algebras-over-props-cocomplete