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Ok, created adjoint string. I think we should keep adjoint triple and adjoint quadruple as separate pages, but I’ve added interlinks.
I added the 5-tuple for $Set$ and $PSh(Set)$, and the 7-tuple for abelian-ish categories and arrow categories. How general is the setting for the latter? Is the characterisation I found somewhere optimal?
I also recorded the 5-tuple in derivators and 7-tuple in pointed derivators as in Mike’s paper (Generalized stability for abstract homotopy theories, p. 19).
I also added in general examples of odd and even length. Perhaps someone could check I got the conditions right.
I think you got the conditions right. It looks like example 5 is just example 4 hommed into a category $C$. I added another example 6, based on lax idempotent monads, and indicated that example 4 is really the walking version of example 6.
This is quite a nice page now, thanks everyone! I merged the examples of pointed categories and pointed derivators, since they are really the same phenomenon. Note that you shouldn’t write “Example 4” since if someone adds more examples or reorganizes the list then the numbering will change.
I think something is off with the examples 5 and 6 (in current numbering); example 5 says it is a $(2n+1)$-tuple but example 6 is a $(2n+2)$-tuple, so if one is just homming out of the other then one of them is wrong. Is the extra adjoint in example 6 obtained from the assumed terminal object?
Re #6: yes, it must be. Technically it’s not just homming into $C$ because of the extra degeneracy, but that degeneracy must be due to the terminal (which doesn’t exist in all categories). But the homming remark seems to me to explain what is essentially going on in that example. (It might even be better to leave out the terminal business, or perhaps even saying what happens if you have an initial.)
I borrowed from Zhen Lin’s Stack Exchange answer, presumably designed to supply strings of even and odd length.
I added the example of ambidextrous adjunction to the list.
Personally, I like that example where it is; maybe it needs less sophistication in some sense, but it’s also more artificial, and I like having more naturally occurring examples first. But I don’t feel especially strongly.
Why don’t you dip your toe in the water of editing the nLab by adding a citation to Street yourself? (-: You can just copy the style of the other citations on the page.
Actually, I don’t think it’s more artificial, as it is the walking lax idempotent monoid (enriching the sense in which the augmented simplex category is the walking monoid). But I don’t feel too strongly either. :-)
Some of the examples that come before have rich semantic content, which I also like. The adjoint characterization of $Set$ is kind of sweet.
The adjoint characterization of $Set$ is kind of sweet.
Is there something more to be said about why $Set$ behaves like this? Is similar behaviour found for the (∞,1)-Yoneda embedding as a way to characterise ∞Grpd?
Is there anything that can be told about a functor from the fact that beyond an adjoint on one side there are further adjoints?
If I were to try and condense the Wood-Rosebrugh result in just a few words, it’s a meditation on adjoint strings as imposing tighter and tighter exactness conditions on a category $C$. We have $C$ is a total category if $Y = yoneda_C$ has a left adjoint $X$, lex-total (same as being a Grothendieck topos under a mild size restriction) if that left adjoint $X$ preserves finite limits, a totally distributive category if $X$ has itself a left adjoint $W$. Totally distributive comes close to being a presheaf topos; the candidate for the small site would be the inverter $i: A \to C$ (akin to a 2-categorical equalizer) of the canonical 2-cell $W \to Y$ that is mated to $1 \cong X Y$, which wants to be the category $A$ of atoms of $C$.
The candidate actually works (i.e., $C$ is an atomic category on top of being totally distributive – this forces $C$ to be a presheaf topos) if $W$ has itself a left adjoint. Then the situation is this: $C = P(A) = Set^{A^{op}}$, the category of presheaves on $A$, and the adjoint string looks like
$X = Set^{y_A^{op}}: Set^{(Set^{A^{op}})^{op}} \to Set^{A^{op}},$and $W = P(y_A)$, the left Kan extension along $y_A^{op}$ that is left adjoint to $Set^{y_A^{op}}$. (It’s almost as if the presheaf construction $P$ wants to be a lax idempotent monad, except there are size restrictions getting in the way. The multiplication for the monad would be $m_A = Set^{y_A^{op}}$.) But now we have a further left adjoint $V \dashv P(y_A)$, and now the argument is that $V$ must look like $P(\xi)$ where $\xi \dashv y_A$. In other words, now the (small) category $A$ is itself a total category!
We are near the end: a small totally cocomplete category $A$ must be a poset (a sup-lattice), by a famous argument of Freyd. If further $V = P(\xi)$ has a left adjoint $U$, then $U$ is of the form $P(\eta)$ where $\eta \dashv \xi$. In other words, now the small category $A$ is now a small posetal totally distributive category, forcing it in particular to be a posetal Grothendieck topos, and the only candidate there would be the degenerate topos $A = Set^0 \simeq 1$. So then $C \simeq Set^1$.
I’m not sure how much of the argument carries over to the $(\infty, 1)$-world, but I expect a lot, and maybe even all of it. I’m inclined to think that’s true.
Note that you shouldn’t write “Example 4” since if someone adds more examples or reorganizes the list then the numbering will change.
To nevertheless refer to examples by numbers, as is good practice, enclose each example in
+-- {: .num_example #ExampleName}
###### Example
(Example text)
=--
and then refer to it from elsewhere by
see example \ref{ExampleName}
This then makes sure that correct numbering is automatically taken care of.
(Most of you know this, of course.)
Re #16: of course, that doesn’t work if the examples are an enumerated list, and it would be weird to have one example in an environment but the rest in a list. One could break them all out into Example
environments, but that seems overkill and visually distracting when they are all fairly short. It would be nice if we had a way to refer to the items in an enumerated list by number.
Thanks Keith! I fixed up your link a little: before linking to a subsection of another page, it’s better to give that section an anchor name manually that is more stable in case of renaming.
Re: #15, it’s interesting that Freyd’s non-constructive theorem comes in there. I wonder whether constructively there can be categories other than $Set$ admitting such an adjoint string.
that doesn’t work if the examples are an enumerated list
It would be better not to have them in an enumerated list. Giving them each their environment, whether they are short or not, is more gentle on the reader and is more robust against future edits.
We ought to find a home for what Todd writes at #15. Perhaps at the end of this page, or a page for itself.
It probably needs a little cleaning up, but sure. The apparent reference to a nonconstructive result might be spurious, or my recollection unnecessarily coarse.
It would be better not to have them in an enumerated list.
I guess we have to agree to disagree about that.
Perhaps a logical place for #15 would be at Set? Or maybe totally distributive category?
Is there a name for totally distributive categories with one further left adjoint? “Atomically distributive”?
How do you do those quotes?
Select the radio button “Format comments as Markdown+Itex”. (I thought that was supposed to be the default these days.) That way you also get linking, math syntax, etc.; I don’t think there is any good reason to use “Text”.
is there any way to get a preview of changes to the pages before they are submitted?
No, instiki’s philosophy is that “save is preview”. If something comes out wrong in your edit, just edit again; multiple edits from the same person within a short span of time are recorded in the database as only one edit.
And you need to preface the block of text with a greater than character, with a blank like after the quoted text.
Re #28, typo: …blank line…
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