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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeJul 24th 2010
   • (edited Jul 24th 2010)
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   Author: Harry Gindi
   Format: MarkdownItexLet f:C->C' be a functor. Let A be a complete and cocomplete category. Then f*:A^C'->A^C admits both a right and left adjoint. By the AFT, it suffices to show that f* is continuous and cocontinuous. Why is it true that f* is continuous and cocontinuous?

   Let f:C->C’ be a functor. Let A be a complete and cocomplete category. Then f:A^C’->A^C admits both a right and left adjoint. By the AFT, it suffices to show that f is continuous and cocontinuous.

   Why is it true that f* is continuous and cocontinuous?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJul 24th 2010
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   Author: Mike Shulman
   Format: MarkdownItexBecause limits and colimits of functors valued in a complete and cocomplete category are computed pointwise. Of course, in general you need some size condition on your categories in order to apply AFT. For instance, $f^*$ is still continuous and cocontinuous even when $C$ and $C'$ are large, but in that case $f^*$ need not have a left and right adjoint.

   Because limits and colimits of functors valued in a complete and cocomplete category are computed pointwise.

   Of course, in general you need some size condition on your categories in order to apply AFT. For instance, $f^*$ is still continuous and cocontinuous even when $C$ and $C'$ are large, but in that case $f^*$ need not have a left and right adjoint.

3. • CommentRowNumber3.
   • CommentAuthorHarry Gindi
   • CommentTimeJul 24th 2010
   • (edited Jul 24th 2010)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexCould you sketch the proof really quickly? I feel like an idiot because it's not clear to me how I should do it. Alright, I got it. I think I should probably take a nap before continuing... This was a bit embarrassing...

   Could you sketch the proof really quickly? I feel like an idiot because it’s not clear to me how I should do it.

   Alright, I got it. I think I should probably take a nap before continuing… This was a bit embarrassing…