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I'd have taken "excision" to mean that the map $H_n(A,A\cap B)\rightarrow H_n(X,B)$ is an isomorphism. I'd have taken "Mayer-Vietoris" to mean not just the exactness of some sequence $$\cdots\rightar …

This (elementary and perfectly standard) example might help show the power of sheaves with non-constant coefficients: First, think about the circle $S^1$. Suppose you want to understand (real) line …

Quillen's book on Homotopical Algebra is a great pleasure to read, and likely to appeal to a geometer.

For a manifold $X$, define $H_0(X)$ to be the direct sum of all the tangent spaces to $X$. This extends in the obvious way to a functor on the category of manifolds and smooth maps. For a pair $(X,A …

Re your first question: If $\pi_1(X)=H_1(X,{\mathbb Z})=0$, then any $m$-th symmetric product of $X$ is simply connected; this is a special case of Theorem 1.1 in this paper by Kallel and Taamallah.

As an appendix to Tom Harris's nice answer, it might be worth mentioning that the idea of defining algebraic K-groups as the homotopy groups of something was certainly in the air before Quillen, e.g. …

First: The identity map on $X$ factors through the inclusion from $X$ to $X\times {\mathbb A}^1$. Therefore $Pic(X)$ is a direct summand of $Pic(X\times{\mathbb A}^1)$. This is the same argument yo …

I will use capital letters to denote $R[t]$ modules and capital letters subscripted with $0$ to denote $R$ modules. All rings are assumed noetherian and all modules are assumed finitely generated. T …

If your homology theory is of the form $H_n(X) = H_n(S(X)) $ where $S$ is some functor from your original category to non-negative chain complexes, then the Dold-Kan correspondence gives you a corr …

Check out Vic Snaith's work on Explicit Brauer Induction.

This is idle noodling, and I'm prepared to learn that it's foolish as well as idle. But.... Let $M$ be an $n\times n$ matrix over, oh, let's say an algebraically closed field for now. There have be …

Re your second question, I don't have it in front of me but I believe you'll find this in Artin and Mazur's book "Etale Homotopy Theory", near the very beginning of the first section.

Kleiman's paper in the "Dix Exposes sur la cohomologie des schemas" volume might be what you're looking for.

For the Zariski topology, one has cohomological descent if $R$ is regular. (This yields the Brown-Gersten spectral sequence.) For the etale topology, still assuming $R$ regular, descent fails for …

I'm not sure I completely understand your question, but: There is a double complex with $\Gamma(C^pF^q)$ in the $(p,q)$ place. By taking cohomology first vertically and then horizontally, or vice ve …