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1. • CommentRowNumber1.
   • CommentAuthorceciliaflori
   • CommentTimeAug 14th 2013
   • PermaLink
   Author: ceciliaflori
   Format: MarkdownItexLet <img src="http://latex.codecogs.com/gif.latex?\xymatrix{C_1\ar[r]^{L_1}\ar[d]^F%26D_1\ar[d]^G\\C_2\ar[r]^{L_2}%26D_2}" alt="diagram"/> be a commutative square of categories and functors. Assume that $L_1$ and $L_2$ have right adjoints $R_1$ and $R_2$, respectively. Under which conditions do we have $FR_1\cong R_2G$? The thing is that we have a concrete situation in which this does seem to be the case, but we would like to have an easy-to-check criterion which implies it. In our case, all four categories are actually functor categories and the right adjoints correspond to taking Kan extensions.

   Let

   

   be a commutative square of categories and functors. Assume that $L_1$ and $L_2$ have right adjoints $R_1$ and $R_2$, respectively. Under which conditions do we have $FR_1\cong R_2G$?

   The thing is that we have a concrete situation in which this does seem to be the case, but we would like to have an easy-to-check criterion which implies it.

   In our case, all four categories are actually functor categories and the right adjoints correspond to taking Kan extensions.

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeAug 14th 2013
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   Author: DavidRoberts
   Format: MarkdownItexHi Cecillia, the phrase [[Beck-Chevalley condition]] is what you are looking for. Proposition 1 at that page is an example similar to your case. (BTW, nice to meet you in Sydney)

   Hi Cecillia,

   the phrase Beck-Chevalley condition is what you are looking for. Proposition 1 at that page is an example similar to your case. (BTW, nice to meet you in Sydney)

3. • CommentRowNumber3.
   • CommentAuthorceciliaflori
   • CommentTimeAug 15th 2013
   • PermaLink
   Author: ceciliaflori
   Format: MarkdownItexHi :) thanks for the link, it is right to the point! Looks like the pullbacks of opfibrations example in the nLab might apply to the case we are considering.

   Hi :)

   thanks for the link, it is right to the point!

   Looks like the pullbacks of opfibrations example in the nLab might apply to the case we are considering.