@EricPeterson Not quite... But it's not your fault or RW's. Turns out that Ω^∞MU is a complicated object :)

(the situation is that we have a sort-of cell structure on an object closely related to Ω^∞MU and we'd like to be able to say things about it )

Reality check: it is true that in a (say, compactly generated) ∞-category, if I have a cofiber sequence x→y→z with x,z compact, then y is compact? I think I have a simple proof of the fact using the fact that if the maps on the base and the fiber of a fiber sequence are equivalences, so is the map on total spaces

(also: can I get away with claiming that this fact is true without proof in a paper or do I have to write down the proof as well?)

"using the fact that if the maps on the base and the fiber of a fiber sequence are equivalences, so is the map on total spaces" is this true? (Don't you need this for all possible base points, I mean?)

Hrmm... good point

Ok, let me reask the question on the top: is it true that compact objects are stable under extensions?

Under stability conditions this is obvious, but the situation I have is very much not stable

If it helps my sequence is in fact a bifiber sequence (i.e. both fiber and cofiber)

@DenisNardin I think it is true that a map of extensions of a space X and a pointed space Y is an equivalence. I mean the problem naturally splits w.r.t. the components of Y so you may assume that Y is connected, and you have two cases: pushout w.r.t. the map from the empty set and the usual case where Y is connected

but as usual I might say nonsense :-)

@S.carmeli The problem is that I have access only to the fiber over the basepoint

and I have no control on what the fibers over the other components are

why? over other points the statement that it is a cofiber sequence tells you that Y is the pushout of the map empty-->Z along the map empty-->empty which is Z no?

Or maybe I miss-interpret the notion cofiber sequence?

Ok, let me give you more details

I have a bifiber sequence x→y→z in a (very nice) ∞-category C, and x and z are compact

Ohhhh I see

Its a fiber sequence not cofiber

I'd like to conclude y is compact

Yeah, Map(-,t) does not turn fiber sequences into cofiber sequences, unfortunately :)

@DenisNardin I seriously doubt that the answer is "yes" in this original form. For instance, I'd think you have to assume that the terminal object is compact at least (assuming that "cofiber sequence" means "pushout along the map to the terminal object").

@TimCampion In my case the category has direct sums (i.e. it is semiadditve)

It is not additive though

The fact that it's a bifiber sequence makes me want to try to replicate the argument from the stable case

Here it is not true though that y is the cofiber of the map Ωz→x

hmm

I'm working essentially in a variant of $\mathrm{Cat}^{ex}$

So, for example, Ωz=0 for every z

Yeah, I think then I'm thinking it's probably not generally true that if $x \to y \to z$ is a bifiber sequence in a semiadditive compactly-generated $\infty$-category, and if $x,z$ are compact then $y$ is compact.

I'm thinking there need to be more input facts

But I don't have a counterexample, so I could be wrong!

The invariant property characterizing flat categorical fibrations of $\infty$-categories $f: X \to Y$ is that the base change functor $f^{\ast} : Cat_{\infty /Y} \to Cat_{\infty /X}$ commutes with colimits. Is the invariant property characterizing flat categorical fibrations of $\infty$-operads $f: \mathcal{O}^{\otimes } \to \mathcal{P}^{\otimes }$ that the base change $f^{\ast} : Op_{\infty /\mathcal{P}^{\otimes }} \to Op_{\infty /\mathcal{O}^{\otimes }} $ commutes with colimits?