If you're trying to mimic the idea of truncation in a quasicategory then you probably want to do something like require your mapping complexes to have trivial homology below some degree. But this seems qualitatively different that saying you only want coherence of your composition up to $A_n$. I could be totally wrong though!

In the case of quasicategories, the "truncation" is really about truncating the homotopy types of the mapping spaces. You seem to be asking about the truncating the operad itself however, in your third paragraph (i.e. restricting to $A_n$-operads). Do you mean these to be different notions?

But presumably this functor is not -⊗K...?

Ah thanks very much! Do you have any sense of how we are supposed to think about the relationship between vector spaces and combinatorial geometries? I guess there is a functor from vector spaces over a field 𝔽 to K-modules?

This is only tangentially related, but if you work with Segal's Gamma space approach to spectra then the inclusion Gamma^op→Set_* is often called the sphere spectrum. It's not, of course, because it's not fibrant in the stable model structure. It's not a "spectrum" at all. But its underlying space is indeed S⁰ and it has the simplicial circle as its delooping. This object is what Connes and Consani call F_1. The homology theory it induces, in the sense of Bousfield and Friedlander, is "the identity," basically.

So in this case you've controlled the homology and lost control of the homotopy groups. The other thing to keep in mind is that to build an EM spectrum you attach a ton of cells.

In my experience there's a tension between homology and homotopy. Homology "counts cells" so it's simpler for things with few cells. But this makes the homotopy groups blow up in complexity. On the other hand, if you control the homotopy groups, e.g. Eilenberg-MacLane spectra, then the homology becomes very hard to compute.

Lemme A.1.8 here: arxiv.org/abs/1904.11312 might be relevant.

Although I do agree that this is not quite a research level question. But on the other hand, lots of questions are on MathOverflow which are more of an "important for young researchers to know and be able to search for" nature.

@FernandoMuro I don't see it that way. I see it as someone asking a very common question for newcomers to the field, and doing so with the language they have available. Nothing here is being overcomplicated by categorical decoration, beyond maybe the first sentence where the term "infinity category" was used.

I think it's mostly just vibes.

There are results like this, however: link.springer.com/article/10.1007/s00222-002-0257-4

I think that, indeed, motivic homotopy theory is where you would need to go to even start to get answers to this kind of question. There is a motivic spectrum called MGL which is analogous to MU, but the analogy is not perfect and I think a lot of people would pay good money for a full answer to your question.