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• CommentRowNumber1.
• CommentAuthorceciliaflori
• CommentTimeAug 14th 2013

Let

$\xymatrix{C_1\ar[r]^{L_1}\ar[d]^F&D_1\ar[d]^G\\C_2\ar[r]^{L_2}&D_2}$

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.

• CommentRowNumber2.
• CommentAuthorDavidRoberts
• CommentTimeAug 14th 2013

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)

• CommentRowNumber3.
• CommentAuthorceciliaflori
• CommentTimeAug 15th 2013

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.