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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

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeJun 18th 2019

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?

• CommentRowNumber7.
• CommentAuthorjweinberger
• CommentTimeJun 18th 2019

Possibly this? Having an oo-left adjoint

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeJun 18th 2019

Yes, that’s it; thanks!

• CommentRowNumber9.
• CommentAuthorMike Shulman
• CommentTimeJun 19th 2019

Added characterization in terms of universal arrows.

Dylan

• CommentRowNumber11.
• CommentAuthorMike Shulman
• CommentTimeJun 20th 2019

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.

• CommentRowNumber12.
• CommentAuthorGuest
• CommentTimeJun 20th 2019
“With some work” feels even more misleading though. ‘representable fibration’ is synonymous with “equivalent to an overcategory” so the work required is “know the definition of representable fibration, or guess it” (and also dualize between right and left adjoints, but that seems harmless). It also seems strange to prioritize credit to Rigel-Verity just because they say over/undercategory instead of “something equivalent to an over/undercategory”, doesn’t it?
• CommentRowNumber13.
• CommentAuthorGuest
• CommentTimeJun 20th 2019
(Sorry the above and this are from Dylan- my phone forgot my login).
• CommentRowNumber14.
• CommentAuthorMike Shulman
• CommentTimeJun 21st 2019

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.