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(Don't be afraid about the word "$\infty$-category" here: they're just a convenient framework to do homotopy theory in). I'm going to try with a very naive answer, although I'm not sure I understan …

There is one major topic that is missing from your list: spectra. Spectra (and $E_\infty$-ring spectra) are the basics of modern stable homotopy theory and are not treated, if not very cursorily, in t …

I think that doing algebraic K-theory properly certainly requires a good background on stable homotopy theory, that is to say the homotopy theory of spectra. Unfortunately there are not many textbooks …

Functions $(f_0,f_1,f_2)$ as you requested do in fact exist. Your confusion comes from the fact that you are trying to impose the cocycle condition on the intersection $U_0\cap U_1\cap U_2$, which is …

For cohomology, this is theorem 3.3 in Maunder, C.R.F., The spectral sequence of an extraordinary cohomology theory, Proc. Camb. Philos. Soc. 59, 567-574 (1963). ZBL0116.14603. Theorem 3.3 If $E …

I am not sure if this is going to be a real answer to the question. However I believe these observations might be interesting. Let me briefly sketch a way to describe a $G$-structure in (excessively) …

In general there is an obstruction living in $H^3(X,\pi_2Y)$. Choose a CW structure on $X$ and $Y$ with only one 0-cell. Then you can use $\varphi$ to define a map at the level of 1-skeleta (just by s …

Your first map fits in an action of the Steenrod algebra. In fact $H^*_{ét}(X;\mathbb{F}_2)$ is the homology of $C^*_{ét}(X;\mathbb{F}_2)=R\Gamma(X;\mathbb{F}_2)$, an element of the derived category …

It depends on what you mean by "all results". Of course results regarding manifolds or vector bundles do not admit statements completely internal to the world of simplicial sets (although most of them …

This is a partial answer, but every obstruction theory (in some precise sense) provides you with a spectral sequence (in fact several). Let me clarify what do I mean with obstruction theory. All this …

Let $X$ be a finite complex. Then the functor $$\lim_X:\operatorname{Fun}(X,\operatorname{Sp})\to \operatorname{Sp}$$ sending a local system of spectra $E$ to its limit preserves all colimits. Indeed …

The answer to question (1) is yes and it follows from the following theorem by McCord: Theorem 1. (i) For each finite topological space $X$ there exist a finite simplicial complex $K$ and a weak homo …

Let me assume that $S$ is a regular Noetherian scheme (for example a field). Then algebraic K-theory is a motivic spectrum, and in fact it is represented by the $\mathbb{P}^1$-spectrum that is $BGL_\i …

Just a quick answer to explain the original reason behind the definition and why our modern understanding of Thom spectra vindicates it. Let $X$ be a space and $\xi$ a virtual vector bundle over $X$. …

There is a completely explicit simplicial set realizing to $K(A,n)$ for each $n$, coming from the Dold-Kan correspondence. It is defined as $$\bar{A}[S^n]=\mathrm{ker}(A[S^n]\to A[*])$$ (the kernel …