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1. • CommentRowNumber1.
   • CommentAuthorTim_Porter
   • CommentTimeJul 16th 2013
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexIn a comment on the Café on July 13, talking of a talk by Tom L., Mike said: >Your very nice talk at CT2013 today about this paper has inspired me to wonder about the codensity monads of other similar functors. For instance, the sort of person who believes that ∞-groupoids are better than sets may naturally wonder, what is the codensity (∞,1)-monad of the inclusion of finitely presented ∞-groupoids into all ∞-groupoids? >That may be too much of a mouthful, but what about 1-groupoids? I’m not sure whether we should look at finitely presented groupoids or finitely generated free groupoids. In either case, what sort of structure on a groupoid would allow us to “integrate” a function into any such “finite” groupoid? Is there a connection this profinite groupoids and the profinite completion of groupoids? There was some work in semigroup theory and finite automata/languages by Jorge Almeida? (My reasoning was that the codensity monad / shape theory context has the profinte group example (as in the paper by Gildenhuys and Kennison) so as the Almedia papers had a logical content and language theoretic interpretation...? There is a nice article: PROFINITE METHODS IN AUTOMATA THEORY BY JEAN-ÉRIC PIN, in Symposium on Theoretical Aspects of Computer Science 2009 (Freiburg), pp. 31–50 www.stacs-conf.org . I am mentioning this here rather than initially on the café, as I was out of action for a few days and have not followed all the previous disccsion there.

   In a comment on the Café on July 13, talking of a talk by Tom L., Mike said:

   Your very nice talk at CT2013 today about this paper has inspired me to wonder about the codensity monads of other similar functors. For instance, the sort of person who believes that ∞-groupoids are better than sets may naturally wonder, what is the codensity (∞,1)-monad of the inclusion of finitely presented ∞-groupoids into all ∞-groupoids?

   That may be too much of a mouthful, but what about 1-groupoids? I’m not sure whether we should look at finitely presented groupoids or finitely generated free groupoids. In either case, what sort of structure on a groupoid would allow us to “integrate” a function into any such “finite” groupoid?

   Is there a connection this profinite groupoids and the profinite completion of groupoids? There was some work in semigroup theory and finite automata/languages by Jorge Almeida? (My reasoning was that the codensity monad / shape theory context has the profinte group example (as in the paper by Gildenhuys and Kennison) so as the Almedia papers had a logical content and language theoretic interpretation…? There is a nice article: PROFINITE METHODS IN AUTOMATA THEORY BY JEAN-ÉRIC PIN, in Symposium on Theoretical Aspects of Computer Science 2009 (Freiburg), pp. 31–50 www.stacs-conf.org .

   I am mentioning this here rather than initially on the café, as I was out of action for a few days and have not followed all the previous disccsion there.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJul 16th 2013
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexGood question. The Gildenhuys-Kennison paper is all about codensity monads of functors into Set (right?) but it does seem probably related.

   Good question. The Gildenhuys-Kennison paper is all about codensity monads of functors into Set (right?) but it does seem probably related.