Who is making these little hamster pictures? They're beautiful.

That's Opus design. I only draw Riemann surfaces, badly.

There's more than the ones you see now by the way.

@HarryGindi yes. Take any strict 2-category that isn't (2,1), then throw away the non-invertible 1-arrows.

That is, take the full sub-hom-categories on the invertible arrows (this will often be empty)

OK, take an ordered group, which gives a category with objects the elements of the group, and at most one arrow between two elements, and then suspend to a 2-category with one object.

Is there a version of Straightening-unstraightening (ref. HTT 3.2.0.1) that is invariant wrt. equivalence of simplicial categories? To be precise, I have a simplicial category $C$ with an equivalence $C\rightarrow \text{Fin}_*$. I have a simplicial functor $C\rightarrow \text{sSet}^+$ and want to pull it back to a symmetric monoidal $\infty$-category.

I guess I can pull it back to a coCart. fib. over $N_\Delta C$ and then compose with $N_\Delta C\rightarrow N\text{Fin}_*$. Just need to check that it's still a coCart. fibration.

What does Lurie mean by a weakly fibrant diagram?

@EspenNielsen What is the context?

@CharlesRezk The context is Section 3.2.5 of HTT, where $F:C\rightarrow \text{sSet}^+$ is to be a (weakly) fibrant diagram.

Here C is a simplicial category.

And $\text{sSet}^+$ is the simplicial category of marked simplicial sets.

No idea.

@EspenNielsen A guess is that it means projectively fibrant

injecgibely fibrant would be strongly fibrant

*injectively

And I believe that is what makes sense in context

@HarryGindi That's the closest thing I could find to an answer too. Weakly fibrant is mentioned in the index, and points to some part of the section in the appendix where projective/injective (co)fibrations are defined.

=]

I really wish Jacob would pick one name for things and stick to it...

Could also mean simplicially homotopic to a fibrant diagram?

but I think it is the girst thing

Does that make sense in the context of (un)straightening?

Weakly fibrant is supposed to give you a cartesian fibration

proj fibrant is what you want for thT obviously

Yeah, I thought so.

Thanks for the help. :)

that means that thr diagram is a diagram of spaces (resp infty cats)

(for marked case)

Do the morphisms need to be inner fibrations?

no why should they be

proj fibrant means pointwise fibrant

proj cofibrant is the one that is complicated

Ah, okay.

inj fib (strongly) is for taking jolims

*holims

@EspenNielsen If C is fibrant, then the weak equivalence of quasicategories NC -> NFin_* has a homotopy inverse...

@EspenNielsen Btw, \text isn't how to typeset that. should be \mathrm or \operatorname

\text is for actual text in line with mathematics display

like when you are doing \cases etc

err, \begin{cases...

how do y'all feel about the terminology "exact sequence" of spectra? i don't like "cofiber" or "fiber" sequence, because it privileges one handedness or the other. and "distinguished triangle" is alright, but i like "sequence" better than "triangle". any other suggestions...?