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I have created final lift, and added to adjoint triple a proof that in a fully faithful adjoint triple between cocomplete categories, the middle functor admits final lifts of small structured sinks (and dually). This means that it is kind of like a topological concrete category, except that the forgetful functor need not be faithful.
I find this interesting because it means that in the situation of axiomatic cohesion, where the forgetful functor from “spaces” is not necessarily faithful, we can still construct such “spaces” in “initial” and “final” ways, as long as we restrict to small sources and sinks.
I’ve replaced the $U$s in the proof with $G$s, and swapped $C$ and $D$, to match the notation used on the rest of the page, and also added a few words that I hope make the whole thing clearer (but do check that I’ve understood it right).
Thanks! I caught a couple of $U$s that you missed. That notation mismatch was because I couldn’t decide for a while what page to put that proof on, and $U$ is used on some other pages.
I added to final lift the example that (semi-)final lifts along the map to the terminal category are just colimits, and the observation that in HTT (strictly) final lifts for $(\infty,1)$-categories are called “$U$-colimits”.
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