Let F: C---> Cat_{\inty} be a functor of \infty-categories. Suppose that F factors through the category of stable \infty-categories and exact functors. What can we say about Un_C(F)? Is it itself stable?

@F.Abellan Not in general, no. Think about the case where C=Δ¹, for example. The resulting category is not even pointed

Or, if I am allowed a somewhat elaborated example, the category Cat^h of hermitian ∞-categories is most certainly not stable but it arises exactly from such a construction (it is the unstraightening of the functor Catx^{op}→CATx sending C to Fun^q(C^{op},Sp) )

@DenisNardin of course! Δ¹ in the base forces morphisms to have an specific direction! I guess that example already shows that the bicategory of stable infty-cats has few lax colimits.

@F.Abellan I think Catx has all lax colimits, it's just that they are not preserved by the inclusion Catx->Cat

Ah, probably one needs to stabilize and then it works.

It has all (co)limits and all (co)tensoring by small categories and this should be enough to construct all lax (co)limits

The inclusion Catx→Cat should preserve lax limits though

@DenisNardin Sure! It should preserve al weighted limits. After all given F:C --> Catx and W:C-->Cat one can show that Nat(W,F) (the infty-cat of natural transformations) is stable.

anyway, thanks a lot :)

Always a pleasure :)

@AdrianClough I don't quite understand why the fibers should be ∞-groupoids here (bifibration in HTT only consider pairings into ∞-groupoids)

Oh duh, right. If $K = *$ is, then we just get a coCartesian fibration.

I am mainly trying to understand Notation 4.2.3.1. in HTT, and the resulting construction looks like a cylinder, and I was trying to think of what the corresponding bifibration could be (and then of course forgot to do the necessary quick sanity check). So maybe $\int F \to I \times K$ is a suitably generalised notion of bifibration, and Lurie's construction is suitably generalised cylinder?

Or maybe I'm misunderstanding Lurie's construction all together (-:

Well I guess one thing we know for sure is that $\int F \to I \times K$ is a coCartesian fibration.

@AdrianClough I don't think so... consider the case $I=\ast$

(sorry, I don't have time to look at it more carefully -- just throwing special cases around :))

Hmmm, for $I = \ast$ we get something like a mapping cone of $F(*) \to K$...

I guess Lurie's construction $K_F$ may not have a model independent meaning. Given a map $p: K \to \mathcal{C}$, an extension to $K_F \to \mathcal{C}$ seems to just extend the induced diagram $I \to (\mathbf{Cat}_\infty)_{/\mathcal{C}}$ to a diagram $i \mapsto (K_i^{\rhd} \to \mathcal{C})$.

@SaalHardali Isn't it something like $u_k\mapsto \zeta^{p^k-1}u_k$?

On the Honda formal group $\Gamma$, $\zeta \in \mathbb{F}_{p^n}^\times$ corresponds to the automorphism $x\mapsto \zeta x$. Lift this series to a power series $g$ over $\mathbb{WF}_{p^n}$. Given a deformation $F(x,y)=x+_\Gamma y+_\Gamma u_kC_{p^k}(x,y) + \text{higher degree}$, we can twist it by $g$: so $gF(g^{-1}x,g^{-1}y) = x+_\Gamma y+_\Gamma + \zeta^{1-p^k}u_kC_{p^k}(x,y) + \text{higher degree}$.

I think I have something backwards, but that's the idea.

For $u\in \pi_2E=\widetilde{E}^2(\mathbb{CP}^1)$ a generator, then $\zeta \in \mathbb{F}_{p^n}$ acts on it by $u\mapsto \zeta u$. Thus $v_k := u_k u^{p^k-1}\in \pi_{2(p^k-1)}E$ is invariant under this action, as expected.

@CharlesRezk A down to earth point of view on the map Aut(E_n) -> hAut_*(S^n) is as the composition Aut(E_n) -> hAut(S^{n-1}) -> hAut_*(S^n) where the first map is given by restricting to arity 2 (using E_n(2) ~ S^{n-1}) and the second is suspension.

@archipelago Ah, nice

The "maximal sub-\infty-groupoid" of an ordinary category would just be the nerve of the wide subcategory of isomorphisms, right?

@IanColey yep

(using the quasicategory/sSet model)

Great, that's the analogy I wanted

If by "wide subcategory of isomorphisms" you mean the subcategory of hC with all objects and morphisms the isomorphisms

Yes

I've seen "wide" mean "all objects but fewer morphisms" as opposed to "full" meaning all morphisms but fewer objects. Not sure where "wide" came from

@CharlesRezk That's exactly what I wanted it to be :) thanks for the nice argument as well!

@AdrianClough I was lamenting my inability to understand $K_F$ a few weeks ago. Luckily, Dylan and Rune's arguments in response to this question of mine suggested to me that this bit of HTT might be obselete!

Another question: the fundamental category functor is left adjoint to the nerve functor. Is the fundamental n-category functor left adjoint to some kind of nerve functor from n-categories?

Fundamental n-category / homotopy n-category

@IanColey Yes. There's a place in HTT where Lurie sets that up. Where "n-categories" are a certain subcategory of infinity categories.

Great, thanks. I found some things also about how the n-categorical nerve should look

(there are competing opinions, shockingly enough)