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nLab > Latest Changes: Higher local Kan extensions

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    • CommentRowNumber1.
    • CommentAuthorfpaugam
    • CommentTimeJul 10th 2011
    Do you have some ideas on how to define a general/higher notion of local Kan extension in an n-category, that gives back the usual notion in a 2-category? I am talking of local kan extension Lan_F G, with F and G two morphisms, that is given by a 2-cell with F and G on the boundary that is universal among such 2-cells.

    I would define it using as in the nlab page, the corepresentation of the functor Hom(G,F^*_) but this does not make sense in a weak n-category. I don't want of a Lurie type kan extension given by adjoint to F^*. Want something weaker.

    One could also use simply truncation to a 2-category, but is there something finer than that?

    The applications i have in mind are related to higher doctrines and theories, derived algebras and their universal properties.

    Is there in the litterature something finer than that and useful?
    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJul 13th 2011

    I think the “obvious” extension to an n-category would be to ask for a 2-cell of the same shape which is universal in an appropriate n-categorical sense. That is, there is some functor between (n-2)-categories of 2-cells which we would ask to be an equivalence.

    • CommentRowNumber3.
    • CommentAuthorfpaugam
    • CommentTimeSep 8th 2011
    Thanks a lot Mike, this looks like the right thing for me!

    Can you be more specific?

    Could you add this to the nlab page on local kan extension (or make a new page)?
    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeSep 9th 2011

    Sure.

    • CommentRowNumber5.
    • CommentAuthorfpaugam
    • CommentTimeSep 20th 2011
    Great! I am impressed by the apparent symplicity of this definition... :-)

1 to 5 of 5

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