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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 7th 2010

    polished and expanded adjoint (infinity,1)-functor

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 7th 2010
    • (edited Apr 7th 2010)

    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:

    1. characterization of adjunction by hom-isomorphism;

    2. preservation of limits by hom-functors in both variables;

    3. 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:

    1. the first is HTT prop. 5.2.2.8,

    2. the second is essentially HTT A.3.3.12,

    3. 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...

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 7th 2010

    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...

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeNov 17th 2010
    • (edited Nov 17th 2010)

    added the proposition on how adjoint (,1)(\infty,1)-functors pass to slices, at adjoint (oo,1)-functor – on over-(oo,1)-categories

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJan 7th 2019

    added publication data for

    diff, v32, current

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