Well, I meant that it still has homology in degree $0$ (the original sequence is exact everywhere, including degree $0$, if and only if $M\to N$ was an isomorphism). But I now realize that the question probably asked for "exact in positive degrees".

If you just take $M$ and all the $P_i$ to be zero, your map is a monomorphism, but the sequence isn't necessarily exact.

Does the $n$-ary product of hypermatrices really satisfy the version of associativity required for $n$-groups? Maybe I don't understand what multiplication you have in mind.

The two answers given explain nicely what goes wrong in specific examples. I want to add the observation that an exact functor between stable $\infty$ categories is faithful if and only if it is actually fully faithful. In your case, this happens precisely if $R$ is a spectrum with $R\otimes_{\mathbb{S}} R = R$, i.e. some kind of localisation.

Ok, yeah, maybe that's a better way to put it. There IS such a notion, but it's far too strong since it requires isomorphisms on all higher homotopy groups of mapping spaces.

There isn't even a good notion of faithfullness for $\infty$-categories. A good replacement is to ask for maps on mapping spaces to be injective on $\pi_0$ and isomorphisms on all higher $\pi_i$. But that's basically never satisfied in this case.

For groupoids, something akin to your first condition should indeed work, namely that the functors are injective on $\pi_0$ and faithful. This should reduce to the corresponding statement for classifying spaces of groups. For general categories, condition 1 is definitely not enough. You can consider a pushout where the upper right corner is an arbitrary category, the upper left corner is a disjoint union of multiple $\Delta^1$, and the l.l. corner is the localisation of the u.l. corner. Then the pushout is a localisation of the category you started with, and not generally a $1$-category.

You can just consider geometric realisation of simplicial spaces as an endofunctor of simplicial spaces: It takes a simplicial space to the constant simplicial space given by the geometric realisation. Is this what you had in mind?

I think you can even get a hands-on $0$-dimensional counterexample: Let $X = \operatorname{Spec}\mathbb{R}[x]/(x^2 + 1)x$ and $Y=\operatorname{Spec}\mathbb{R}[x]/(x^2-1)x$. Then I'd think that over $\mathbb{C}$ both have automorphism group $\Sigma_3$, but the action is trivial for $Y$ and conjugation with a transposition for $X$. (Am I messing anything up here?)

The general principle that governs the dimensions (and other properties) of irreps is definitely given by character theory. I recommend looking into Serre's book!

I'm still not sure if I quite understand the terminology. When you say "under which conditions degeneracy can occur in a group", do you mean "under which conditions does a group have irreps of dimension $n>1$"? In that case, that happens precisely if your group is nonabelian (or, I guess, if you're working with real irreps, if your group is not of the form $C_2^{\times n}$).

There aren't that many subgroups of $O(3)$, so if that's all you're ultimately interested in, an explicit list is probably the most helpful. For more general finite groups I don't think it's easy to deduce $n_{\mathrm{max}}$ directly from, say, a presentation, but representation theory says a lot (for example, the character table directly gives some inequalities). For an introduction, I recommend Serre's "Linear representations of finite groups".

By "generators", do you mean an additive basis (as $\mathbb{F} _p$ vector space) or something else?

Even though OOP is the first place where one encounters the concept of types conciously (say, as a beginning CS student), and so this is totally fine for the scope of the question, I still want to point out that having types does not require being object-oriented. In fact, I would argue that the programming languages which most cleanly embrace the concept of types are typically not object-oriented (Haskell is probably the best example). In a way, the object-oriented paradigm forces you to lump together the types (i.e. what objects are) with all the things you can do with them.