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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.
I have added the definition (here) of adjoint $(\infty,1)$-functors as an adjunction in the homotopy 2-category of (infinity,1)-categories.
I had wanted to next add pointers to theorems proving the equivalence of this to other definitions, but I am getting a bit lost in the text. Maybe somebody can help:
In Riehl-Verity 16, p. 3 it says that the equivalence to Lurie’s definition is
in one direction: 4.4.5 in Riehl-Verity 15
in the other direction: Riehl-Verity 17
Now 4.4.5 in Riehl-Verity 15 is a Remark, and it mainly points to the previous Observation 4.4.4. The relevant bit of Observation 4.4.4, in turn, seems to say that another Observation 3.4.4 implies that the homotopy 2-categorical definition of adjunction implies the expected hom-equivalence.
This is all very plausible from a broad-brush perspective, of course, but it remains unclear to me if/where a claim is made concerning the crucial fine-print (of which there is a fair bit, already notation-wise). Observation 4.4.4 speaks of an “equivalence of hom-spaces” which is probably really “natural equivalence”. The next sentence has “This should be thought of as $[$the expected statement$]$” . I can see how it is probably all clear to the authors at this point, but it does make it hard to cite a result here.
Then, Remark 4.4.5 speaks of a definition of Lurie’s “more complicated” than the authors “prefer to recall”. This probably alludes to Lurie’s definition via Cartesian fibrations. But their pointer is actually to Lurie’s Proposition 5.2.2.8, which is just the proposition where Lurie proves that this is equivalent to the expected natural hom-equivalence, hence to what, presumably, the previous Observation 4.4.4 did mean to recall.
Similarly, I am not sure yet where exactly in Riehl-Verity 17 “the converse” is proven, and in which form.
Anyway, that just to explain where I get lost. If anyone here can state with authority a crisp form of the expected statement, such as
An adjunction in the homotopy 2-category of $(\infty,1)Cat$ is equivalently a natural hom-equivalence as in …
then it would be great to add that to our entry here.
I’ve included the category of adjunctions (analogous to Lurie’s category of presentable categories) mainly because I found it useful to organize various properties, in particular to properly state the $ladj$ and $radj$ operations taking arrowwise adjoints.
But for the sake of completeness… are there any useful general properties we should expect it to have as an ∞-category? I’ve found very little material on this construction even in the case of ordinary 1-category theory. Is there simply not much that one can say?
Hi Hurkyl,
I don’t know the answer to your question, but it’s great to see you do all these edits!
Maybe if you could make it a habit to include key technical terms in double square brackets? Such as [[Grothendieck construction]]
.
It costs essentially no effort, while it considerably increases the general value of edits here on the wiki, in the long run.
(For that’s one big advantage of a wiki over other sources, that the reader is always just one click away from learning about that one Definition/Proposition which they were missing. Imagine yourself like a teacher talking to students. You may take the Grothendieck construction for granted, but they may be grateful to be handed a link where they can remind themselves.)
I (think I) usually do. I confess to being mentally exhausted at the time and glossing over polish, but for some reason I thought the previous link to it in the article was somewhat closer. I’m a little shy about overlinking since I’m used to style guides saying only the first instance should be made into a link. While that’s surely too few, I’m still hesitant about overdoing it.
I only now see the reply #23, sorry. Allow me to give a belated reaction to:
I’m a little shy about overlinking since I’m used to style guides saying only the first instance should be made into a link.
Yes, I am aware of this reasoning, but it should not apply here:
We don’t know what “the first instance” is for any given reader, who may have jumped to a Proposition in the middle of the entry. The idea of “first instances” of a term comes from linear texts and does not properly apply to hypertext.
The only good reason not to have every single keyword hyperlinked is the potential optical effect. For that very reason the $n$Lab has (following similar discussions we had here years ago) chosen a pale color for hyperlinks. I’d prefer to even remove the underline (as on my personal web) but this proposal didn’t win a majority when we last discussed this, many years ago. Maybe we could discuss it again.
I have beautified the typesetting of the slice adjoints a little (here).
I think I’ll give this a little stand-alone entry to be !include
-ed here, since the same paragraph is at slice category, hence should also be at slice (infinity,1)-category (but isn’t yet) and should really also be at adjoint functor (but isn’t yet).
I’d like to add more detail to the section “In the homotopy 2-category” (here), but I find myself needing to invest some work to extract this:
What actually is the strongest understood statement regarding the relation between adjoint $(\infty,1)$-functors and adjunctions in $2Ho\big( Cat_{(\infty,1)} \big)$?
My problem in extracting this is that the canonical reference, namely 4.4.5 in Verity & Riehl’s p. 50 is but a “Remark” which mainly points to a previous “Observation” (4.4.4) which gives hints on how to interpret a previous Proposition (3.3.18) – and when I start trying to unwind these hints I get further drawn into unravelling definitions.
I guess I could drill to the bottom of this and extract a concise Theorem-like statement, but I am hoping I could instead just read up on it.
I’d like to see a statement like this:
Given a pair $(\mathcal{C}, \mathcal{D})$ of $(\infty,1)$-categories, there is
the homotopy 2-category $2Ho\Big( Adj_{Cat_{(\infty,1)}}(\mathcal{C}, \mathcal{D}) \Big)$ of adjoint $(\infty,1)$-functors
the 2-category of adjunctions $Adj_{2Ho(Cat_{(\infty,1)})}(\mathcal{C}, \mathcal{D})$ in the homotopy 2-category of (infinity,1)-categories
between them. And there is a comparison 2-functor from the former to the latter.
Now which properties does this comparison 2-functor have? Has this been cleanly stated anywhere? In parts?
There is a paper by Riehl–Verity “Homotopy coherent adjunctions and the formal theory of monads”, which shows that at least on homotopy (2,1)-categories, the comparison functor should be an equivalence since an adjunction in the homotopy 2-category of (∞,1)-categories has a contractible space of liftings to an adjunction of (∞,1)-categories.
Thanks. I have been scanning through that article. I certainly get the impression that what you say should be the case. But is it a theorem? I’d still need to translate the notion of homotopy coherent adjunction used there to the quasi-categorical definition.
Re #30: There is no statement that looks exactly like this, but since you already have a functor, you can invoke the Whitehead theorem for 2-categories.
Objects are the same in both cases.
Essential surjectivity on 1-morphisms: this is Theorem 4.3.11 in Riehl–Verity “Homotopy coherent adjunctions…”.
Fully faithfulness on 2-morphisms: this is Theorem 4.4.11 in Riehl–Verity.
Right, but that gives the equivalence between two of their own definitions (which are certainly good definitions, in themselves).
But for the equivalence of either definition to Lurie’s definition they still point (on p. 3) to that Remark 4.4.5 in arXiv:1306.5144 (which is reference “I” = [25] here, by p. 2).
In #17 and #28 above I wished that Remark I.4.4.5 were stated more like a theorem.
I guess a minimum theorem could say:
Let $\mathcal{C} \overset{\;\;\;}{\leftrightarrows} \mathcal{D}$ be a pair of antiparallel morphisms in $Cat_{(\infty,1)}$. Then these admit the structure of a pair of adjoint $\infty$-functors in the sense of Lurie HTT if and only if their image in $2Ho\big(Cat_{(\infty,1)}\big)$ carries the structure of an adjunction in the classical sense of adjunctions in 2-categories.
I do think this is the case and that this is what Remark 4.4.5 is ultimately meaning to say. And uniqueness would not be an issue for this simple form of the theorem (in #28 I mentioned something more ambitious) since we already know both of these adjunction structures are essentially unique on both sides. So I am not worried that it doesn’t hold. But I wish authors who had dug through all the nitty-gritty details had stated a confident claim as above more explicitly.
@Urs
Emily is not inactive on Twitter, she might respond to a mention to come here and clear things up…
I have added pointer to Joyal 2008 p. 159 (11 of 348) for the first (brief) mentioning of the idea of defining adjoint $\infty$-functors simply as adjoints in the homotopy 2-category of $\infty$-categories. Also slightly adjusted the commentary around the following references in order to reflect how the idea developed.
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