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I don't think it has any prerequisites per se, since all used notions are explained, however without familiarity with category theory and classical algebraic topology it can be too much to swallow. I would suggest starting with Hatcher's book on algebraic topology and first 4 chapters of Maclane's "Categories for...". It definitely can't serve as an introduction to topology.
How about a manifold of constant negative curvature? E.g. any hyperbolic plane. It cannot be imbedded as a whole but some bounded (closed or open) subset can be.
@AchimKrause , I may be missing something obvious, but how do you map $Aut (K (A,n)) $ on $Aut (A) $? This essentially means that you associate basepoint preserving map to a non-preserving one. Or that a shift conjugated by any map is still a shift, which looks even stranger.
Actually, the exact statement depends on how you consider $B$. I was assuming you mean $B$ as a homotopy type. Then on one side there is an $\infty$-category of locally trivial $\infty$-sheaves of $R$-module chain complexes and on the other side $HR\times B$ modules in fibrations of spectra over $B$. Similarly if you don't want locally trivial fibrations, but general sheaves.
Your parametrized chain complexes and ring spectra are just categories of $(\infty,1)$-functors from $B$ to $Ch(R)$ or $HR-Mod$. So of course these two categories are also equivalent.
I'd say they are easier in the same way as p-adic numbers are much easier than any localization of Z. Not only do you kill all other primes but you can also consider infinite sums and metric effects. This is tricky to define in topology, but let's say you get such structure on relevant homotopy/homology groups and this makes life better.
I think Helemskii's book "Lectures and Exercises on Functional Analysis" contains a very nice intro into category theory. It may be a bit light on the algebraic side, but it certainly does contain a lot of analytic examples and motivations, and once you get the ball rolling and have some favourite examples in mind you can study any classical text on category theory at your leisure.
@MartinBrandenburg , the problem is multiplication. As stated by you the $a^{-1}b $ braiding is trivial, since $(a, b) \equiv (a, 1)(1, b) \equiv (1, b)(a,1 ) $ on the nose with your definition, thus Thomason's counterexample. Symmetric still means you must keep track of coherence isos. Same problem with your definition of morphisms, you sweep all permutations under the rug which the gods of homotopy forbid.
Key issue: nontrivial braiding $b a^{-1} = a^{-1} b$ which comes from $a b = b a$ via left and right action by $\otimes a^{-1}$. This is the defect dropped out in Quillen's $S{-1}S$ and pointed out by Thomason. Such isos should be added universally. They are also precisely what makes the naturally stated abelian group structure only homotopy equivalent to the true one. P.S. I assume "abelian group" means fro you isomorphisms $1 = a^{-1} a$ for all $a$ in group completion. One could also produce a version of this construction for noninvertible $1 \to a^{-1} a$ with whatever dualizability.
