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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeApr 7th 2010

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

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

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 $(\infty,1)$-functors pass to slices, at adjoint (oo,1)-functor – on over-(oo,1)-categories

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJan 7th 2019