it could definitely be true. part of why stable homotopy theory has kept chugging along through the years is all the successful conceptual / organizational work that's accompanied its grittier parts. meanwhile, there's a great deal of "raw" unstable work out there, but afaict much less (successful) organizing principles, hence it's harder to learn about and (i hope no one finds this assessment too untoward) often feels like it's groaning under its own weight

imo it'd be great if the "community average interest" pulled more in that direction—EHP sequences, the billion things mahowald did, the various goofy forms of "surgery" on homotopy types, etc etc. one can't expect conceptual work to happen overnight, especially if we aren't regularly revisiting this stuff, and so just being more exposed to / redigesting the body of literature would already be great step

or: i hope you write notes :P one doesn't have to improve on toda to make an impact here

i also hope someone comes through who has a better immediate answer for you, but i'd still wager that that answer indeed points to material from the 70s, 80s

what i'm cognizant of is surely very biased, but i'm the most modern unstable excitement i know of is around the goodwillie tower. behrens has an AMS memoir which covers its interaction with the EHP system, so you could check his bibliography—but, at a glance, his references are from the era you name, and it's hard to tell if that's bc he's responsible about citing original work or bc those are (also) the unique references available

The inclusion of $\Delta^{inj} \hookrightarrow \Delta$ is famously initial (a.k.a. cofinal), and the inclusion $\Delta^{surj} \hookrightarrow \Delta$ is final (simply because both have a final object, and the inclusion preserves these). Is it true in general, that if I have a Reedy category $R$, that the inclusions $R^+ \hookrightarrow R$ and $R^- \hookrightarrow R$ are initial and final respectively?

(If so, it is probably not so hard to check, but this seems like something that should already be well known)

(Here, when interpreting "initial" and "final" I view all ordinary categories as $\infty$-categories).

Oh god, @AdrianClough please please, think of the children (me!) when you use 'initial'

Cofinal = final, not initial, also. Cofinal comes from order theory, and it means 'mutually final'

I mean, initial and final, maybe, but never use cofinal to mean dual-to-final!

I haven't slept for 2 days sorry

So are you saying cofinal = final, coinitial = initial. Also, I apologise for my iconoclastic tendencies.

@HarryGindi im totally confused, and actually would like to have a clarification of terminology. What is called final/cofinal/initial? I probably used it wrong like 100 times or so.

Cofinal=final=colimit-cofinal=right-cofinal

limit-cofinal=initial=coinitial=left-cofinal

I personally use "cofinal" for "final" and "limit-cofinal" for "initial", but other people have their own opinions

I don't personally like left/right cofinal bc it can be disorienting. The others are fine though

I, being a faithful category theorist, use "final" and "initial", and encourage others to do the same.

I rather like the 'co' it makes it more lively =]

I like whatever convention makes the inclusion of an initial object (co)initial, and the inclusion of a terminal object (co)final

Somebody suggested using "initial" and "terminal" functors, specializing to initial and terminal objects when the source is a point, which seems quite logical, but I haven't tried to use it in practice myself

I like to reserve 'terminal functor' for 'the terminal functor'.

C→✱

I thought of "left acyclic" and "right acyclic", the idea being that the criterion is that the "left fiber" is contractible

So the original question seems to have gone unanswered, and it's nontrivial to actually chase down the proof of the original fact. It's definitely not for the reason that Adrian stated because $\Delta^+$ and $\Delta^-$ have neither an initial nor final object. You can find the proof at HTT 6.5.3.7

Surely $0$ is the final object of both $\Delta$ and $Delta^{surj}$.

Err, hang on

Okay sorry, was thinking with ops; [0] is still not final in $\Delta^+$ because there are no maps into [0]. It's also not initial because there are two maps [0] -> [1] which are not the same

Moreover the inclusion $\Delta^-\to\Delta$ is the one where you'd like to preserve limits along the pullback so that the final object is sent to the final object doesn't mean anything afaik

Anyway you prove a functor is cofinal by proving that all the comma categories are weakly contractible; this is what Lurie does at the proposition I cited. I don't see immediately that Lurie's proof works for an arbitrary Reedy category since I don't know how to make sense of the map $G$ without some fiddling

@IanColey doesn't Δ^+ have an augmentation attached?

so it does have a terminal object

or initial object or sth

anyway one way to prove cofinality is by Quillen's theorem A

does he do that?

oh lol

that's quillen's theorem A

sorry just woke up

Sorry I was using Reedy notation: $\Delta^+ = \Delta^\text{inj}$

If you write down the actual comma categories involved you can see they don't admit initial or final objects either

e.g. $[n]/\Delta^\text{inj}$ has objects of the form $[n]\to[m]$ and morphisms $[m_1]\leftarrow [n]\rightarrow [m_2]$ with the last leg of the commuting triangle $[m_1]\to[m_2]$ being injective

At least I don't think this admits an initial object; you certainly wouldn't expect it to for arbitrary Reedy categories

You can try to pick $[n]\to[n]$ the identity but not everything maps to or from that object