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Nov 21, 2013 at 17:54 comment added Gerrit Begher Interesting: $f_!(\{x\})$ is allways the complement of the image $f(X)$ except when $x$ is the only element in the respective fiber of $f$. So in general $f_!$ should maybe say something about the triviality of fibers. Let's see where this leads for general quasiorders aka 2-enriched categories. Here $\{x\}$ needs to be replaced with the respective down-set.
Nov 21, 2013 at 12:05 history answered Fujita Tomomi CC BY-SA 3.0