Michael Hong wrote:

> The left adjoint seems to tell you the buckets that contains only the balls in A and right adjoint seems to tell you the buckets that contain at least one of the balls in A. Tell you the truth they both sound like they are saying the same thing to me...

The left adjoint \\(f_{!} (A) \\) is the set of buckets such that _some_ ball in that bucket comes from \\(A\\).

The right adjoint \\(f_{\ast}(A) \\) is the set of buckets such that _all_ balls in that bucket come from \\(A\\).

These are quite different. For example, suppose a bucket has two balls in it: one from \\(A\\) and one not from \\(A\\). Then this bucket is in \\(f_{!}(A)\\) but not in \\(f_{\ast}(A)\\). _Some_ ball in that bucket is from \\(A\\), but not _all_.

Or suppose a bucket has no balls in it. Then this bucket is in \\(f_{\ast}(A)\\) but not in \\(f_{!}(A)\\). _All_ balls in that bucket are from \\(A\\), but not _some_. If this seems surprising, read my previous comment. Since there are no balls in this bucket, it's *vacuously* true that all balls in this bucket come from \\(A\\).

If you give me a specific example where you seem to be getting a different answer, we can talk about it. Your pictures are actually pictures of many choices of \\(A\\) - too much to talk about.

> The left adjoint seems to tell you the buckets that contains only the balls in A and right adjoint seems to tell you the buckets that contain at least one of the balls in A. Tell you the truth they both sound like they are saying the same thing to me...

The left adjoint \\(f_{!} (A) \\) is the set of buckets such that _some_ ball in that bucket comes from \\(A\\).

The right adjoint \\(f_{\ast}(A) \\) is the set of buckets such that _all_ balls in that bucket come from \\(A\\).

These are quite different. For example, suppose a bucket has two balls in it: one from \\(A\\) and one not from \\(A\\). Then this bucket is in \\(f_{!}(A)\\) but not in \\(f_{\ast}(A)\\). _Some_ ball in that bucket is from \\(A\\), but not _all_.

Or suppose a bucket has no balls in it. Then this bucket is in \\(f_{\ast}(A)\\) but not in \\(f_{!}(A)\\). _All_ balls in that bucket are from \\(A\\), but not _some_. If this seems surprising, read my previous comment. Since there are no balls in this bucket, it's *vacuously* true that all balls in this bucket come from \\(A\\).

If you give me a specific example where you seem to be getting a different answer, we can talk about it. Your pictures are actually pictures of many choices of \\(A\\) - too much to talk about.