Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
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
1 to 5 of 5