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polished and expanded adjoint (infinity,1)-functor
added some more stuff,
am about to write the section on preservation of limits and colimits, but I am wondering:
in ordinary cat theory, that right(left) adjoints preserve (co)limits follows purely formally from three ingredients:
characterization of adjunction by hom-isomorphism;
preservation of limits by hom-functors in both variables;
the Yoneda lemma.
It would seem that in the (oo,1)-category context we just check these three ingredients and then deduce the statement that left oo-adjoints preserve oo-colimits formally by direct analogy.
Now, instead in HTT, section 5.2.3 a rather tedious simp-set gymnastics is performed to show this.
I am wondering therefore: it seems we essentially do have the three items above in the oo-context:
the first is HTT prop. 5.2.2.8,
the second is essentially HTT A.3.3.12,
and the last one is the combination of statements collected at Yoneda lemma for (infinity,1)-categories .
So one would hope that we can just argue formally on this basis that left adjoints preserves oo-colimits. But there are probably subtleties in the details...
added in the details of construction and proofs on how simplicial and simplicial Quillen adjunctions induce oo-adjunctions in Simplicial and derived adjunctions
Okay, enough (oo,1)-entertainment for today, now I need to focus my thoughts on a talk I will give in Nijmegen tomorrow...
added the proposition on how adjoint $(\infty,1)$-functors pass to slices, at adjoint (oo,1)-functor – on over-(oo,1)-categories
added publication data for
I seem to recall that recently someone asked a question here on the nForum about characterizing adjoint $(\infty,1)$-functors via universal arrows, or equivalently by pointwise representability (that is, if $D(X, G -)$ is representable for all objects $X$ then these representing objects assemble into a functor left adjoint to $G$), and that a reference was given. But after a few minutes of searching I can’t find it, and it doesn’t seem to have made it onto the page adjoint (infinity,1)-functor. Can anyone find the discussion or re-give the reference?
Possibly this? Having an oo-left adjoint
Yes, that’s it; thanks!
That statement from HTT is not exactly this proposition. I can see how it contains essentially the same information, but it’s hidden in the phrase “representable right fibration”. In particular, I’m pretty sure that I glanced past that statement in HTT while looking for this result and didn’t notice it. So I edited the attribution here in an attempt to reduce confusion in the reader.
I wasn’t thinking of credit, but of clarity for the reader. You also have to realize that the pullback of a representable fibration is the relevant comma category, and that this being equivalent to an overcategory over its codomain is the same as having a terminal object.
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