I feel like a lot of the stuff I consider to justifiably be called obvious are statements that are "checkable" - not in the strict sense of checkable by a computer but more along the lines of "verifable" by a (say masters) student in the relevant area.

For that reason I always liked "it is straight forward to check"

Straight forward is more accurate description of a step that doesn't require an extra idea etc... And I also agree that it's useful to separate the deep from the not so deep

Just for me almost nothing is ever obvious so I never liked that word...

I mean the original meaning of this the word is something like "self evident" or "requires no explanations". If we ever get in the position of explaining why something requires no explanations maybe this word doesn't mean the same thing anymore for us...

Which could also be fine I guess

@AaronMazel-Gee yeah that makes sense. regarding the actual content of your comment, I see what you're saying. I think I was stuck just thinking formally about the concept of being fully faithful, rather than, as you say, just thinking about simplicial sets (although I seem to recall you being rather serious about proving things with universal properties rather than models)

As far as words like "obvious" go, I've already pissed enough people off saying what I think about that kind of thing. In general, things are rarely obvious to me. I'm willing to accept at this point that this is more specific to me than the homotopy theory community, since people so frequently tell me things are "obvious" or say "Doesn't that just follow from....?"

But it may nonetheless make the homotopy theory community less inviting.

(I will say however that many people, including you Aaron, have been immensely generous with their time and energy in helping me understand all this stuff, despite my habit of asking the same question over and over again, so I know I can get sassy sometimes but I don't want to seem like I think anyone owes the expenditure of their expertise. I just think it'd be good if, when we DO explain things to people, we avoid language that can seem belittling.)

At least in my head I tend to mark a lot of stuff as "obvious" and I find it useful to do so. But there's an element of self-delusion in it. This is evidenced by how often I still compulsively check such things and then feel "oh, phew, it really was 'obvious' ". Anyway, I think "obvious" is very much a personal marker.

(just musing out loud, not trying to make any real point)

@AaronMazel-Gee a sort of nice geometrically natural example is the vanishing locus of a section of a vector bundle. ring of functions is the koszul complex associated to the section. an example i'm fond of is the derived version of the space of arcs into a variety (which can be seen as a dg intersection), u get something subtle for a fat point already.

Hi all, happy new year! I have a question which might be a bit silly but nonetheless it has been bothering me. In several classical references (including Ginzburg) a Calabi-Yau algebra of dimension d is defined as a homologically smooth dg algebra A "such that there is a quasi-isomorphism A \to A^![d]"... instead of "equipped with a quasi-isomorphism."

I have often heard in talks and discussions people say "...the algebra A is Calabi Yau". Hence it sounds like if this was a property, when it is certainly structure! Is this a sloppy use of language or am I missing some uniqueness issue?

@ManuelRivera i think the terminology is analogous to the situation in complex geometry, where a variety is called calabi-yau if it has a nonvanishing holomorphic n-form/trivial canonical bundle (but such a differential form/trivialization isn't generally part of the data)

@S.carmeli @AaronMazel-Gee The reference is mathematik.ur.de/hoyois/papers/efimov.pdf. Following Aaron's comment, and the reference above, perhaps it means that the right adjoints are also required to preserve small colimits.

Does anyone know why Lurie calls the things he defines in Definition 2.2.6.1 of HA "norms" (i.e. one operad is a norm of another)?

I can only guess but in some contexts when a right adjoint to a restriction functor exists it is called a norm. Like in unstable equivariant homotopy theory.

There are possibly other examples that i'm not aware of and/or not remembering right now...

I have kind of a funny question. Unless i'm misunderstanding something we know that for all $n$ the operations on the homology of $E_n$-$\mathbb{F}_p$-algebras are generated by the $H_{\ast}(E_n)$-algebra structure and the power operations of weight p. Is there an a priori justification for this?

Can we formulate a reasonable condition on an operad $\mathcal{O}$ which would imply that for all primes $p$, all the operations on the homotopy groups of $\mathcal{O}$-algebras in $Mod_{H \mathbb{F}_p}$ would be generated from $H_{\ast}(\mathcal{O})$-algebra structure and the weight p power operations?

I have an even funnier question: given a bicomsimplicial object in an $\infty$-category $\Delta\times\Delta\to C$, is there a natural transformation between the two restrictions $\Delta\times [0]$ and $[0]\times\Delta$ which goes something like $(n,0)\to (n,1)\to\ldots\to (n,n)\to (n-1,n)\to\ldots (0,n)$ at level n?

And I guess to be more specific, I want to do something like repeatedly apply the bottom-most coface map in the right coordinate n times, followed by applying the bottom-most codegeneracy in the left coordinate n times.

Yeah that must be true right? Basically by alternatively thinking about functors $\Delta\times\Delta\to C$ and $Fun(\Delta,Fun(\Delta,C))$

So if I've got a k-fold loop space X then I can take a bar construction and get BX=|Bar(1,X,1)|, a k-1-fold loop space. What if I've got a module over X, say Z. Can I apply a bar construction BZ=|Bar(Z,X,1)| and get a module over BX? I know I get a comodule over BX at least.

I guess I feel like I want this to be true, because of some kind of idea about "the bar construction eats one degree of monoidality" but.... maybe it's not?