I'm looking for a big-picture treatment of algebraic K-theory and why it's important. I've seen various abstract definitions (Quillen's plus and Q constructions, some spectral constructions like Waldhausen's) and a lot of work devoted to calculation in special cases, e.g., extracting information about K-theory from Hochschild and cyclic homology. As far as I can tell, K-theory is extremely difficult to compute, it yields deep information about a category, and in some cases, this produces highly nontrivial results in arithmetic or manifold topology. I've been unable to piece these results into a coherent picture of why one would think K-theory is the right tool to use, or why someone would want to know that, e.g., K22(Z) has an element of order 691. Explanations and pointers to readable literature would be greatly appreciated.
[Math] Motivation for algebraic K-theory
algebraic-k-theoryat.algebraic-topologykt.k-theory-and-homologymotivation
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I can't resist pointing out that while the EH-argument shows that a group object in the category of groups is an abelian group, this does not apply to a group object in the category of groupoids, which is equivalent instead to a crossed module, which represents a pointed, connected homotopy 2-type.
Higher groupoids are in some sense ``more nonabelian'' than groups, and provide a route to some nonabelian calculations in higher homotopy theory.
So circumstances in which the EH-argument fails (for example a group object in the category of semigroups) are maybe more interesting.
$\newcommand\Z{\mathbf{Z}}$ $\newcommand\Q{\mathbf{Q}}$
I'm a number theorist who already thinks of the algebraic $K$-theory of $\Z$ as part of number theory anyway, but let me make some general remarks.
A narrow answer: Since (following work of Voevodsky, Rost, and many others) the $K$-groups of $\Z$ may be identified with Galois cohomology groups (with controlled ramification) of certain Tate twists $\Z_p(n)$, the answer is literally "the information contained in the $K$-groups is the same as the information contained in the appropriate Galois cohomology groups." To make this more specific, one can look at the rank and the torsion part of these groups.
The ranks (of the odd $K$-groups) are related to $H^1(\Q,\Q_p(n))$ (the Galois groups will be modified by local conditions which I will suppress), which is related to the group of extensions of (the Galois modules) $\Q_p$ by $\Q_p(n)$. A formula of Tate computes the Euler characteristic of $\Q_p(n)$, but the cohomological dimension of $\Q$ is $2$, so there is also an $H^2$ term. The computation of the rational $K$-groups by Borel, together with the construction of surjective Chern classes by Soulé allows one to compute these groups explicitly for positive integers $n$. There is no other proof of this result, as far as I know (of course it is trivial in the case when $p$ is regular).
The (interesting) torsion classes in $K$-groups are directly related to the class groups of cyclotomic extensions. For example, let $\chi: \mathrm{Gal}(\overline{\Q}/\Q) \rightarrow \mathbf{F}^{\times}_p$ be the mod-$p$ cyclotomic character. Then one can ask whether there exist extensions of Galois modules:
$$0 \rightarrow \mathbf{F}_p(\chi^{2n}) \rightarrow V \rightarrow \mathbf{F}_p \rightarrow 0$$
which are unramified everywhere. Such classes (warning: possible sign error/indexing disaster alert) are the same as giving $p$-torsion classes in $K_{4n}(\Z)$. The non-existence of such classes for all $n$ and $p$ is Vandiver's conjecture. Now we see that: The finiteness of $K$-groups implies that, for any fixed $n$, there are only finitely many $p$ such that an extension exists. An, for example, an explicit computation of $K_8(\Z)$ will determine explicitly all such primes (namely, the primes dividing the order of $K_8(\Z)$). As a number theorist, I think that Vandiver's conjecture is a little silly --- its natural generalization is false and there's no compelling reason for it to be true. The "true" statement which is always correct is that $K_{2n}(\mathcal{O}_F)$ is finite.
Regulators. Also important is that $K_*(\Z)$ admits natural maps to real vector spaces whose image is (in many cases) a lattice whose volume can be given in terms of zeta functions (Borel). So $K$-theory is directly related to problems concerning zeta values, which are surely of interest to number theorists. The natural generalization of this conjecture is one of the fundamental problems of number theory (and includes as special cases the Birch--Swinnerton-Dyer conjecture, etc.). There are also $p$-adic versions of these constructions which also immediately lead to open problems, even for $K_1$ (specifically, Leopoldt's conjecture and its generalizations.)
A broader answer: A lot of number theorists are interested in the Langlands programme, and in particular with automorphic representations for $\mathrm{GL}(n)/\Q$. There is a special subclass of such representations (regular, algebraic, and cuspidal) which on the one hand give rise to regular $n$-dimensional geometric Galois representations (which should be irreducible and motivic), and on the other hand correspond to rational cohomology classes in the symmetric space for $\mathrm{GL}(n)/\Q$, which (as it is essentially a $K(\pi,1)$) is the same as the rational cohomology of congruence subgroups of $\mathrm{GL}_n(\Z)$. Recent experience suggests that in order to prove reciprocity conjectures it will also be necessary to understand the integral cohomology of these groups. Now the cohomology classes corresponding to these cuspidal forms are unstable classes, but one can imagine a square with four corners as follows:
stable cohomology over $\mathbf{R}$: the trivial representation.
unstable cohomology over $\mathbf{R}$: regular algebraic automorphic forms for $\mathrm{GL}(n)/\Q$.
stable cohomology over $\mathbf{Z}$: algebraic $K$-theory.
unstable cohomology over $\mathbf{Z}$: ?"torsion automorphic forms"?, or at the very least, something interesting and important but not well understood.
From this optic, algebraic $K$-theory of (say) rings of integers of number fields is very naturally part of the Langlands programme, broadly construed.
Final Remark: algebraic K-theory is a (beautiful) language invented by Quillen to explain certain phenomena; I think it is a little dangerous to think of it as being an application of "homotopy theory". Progress in the problems above required harmonic analysis and representation theory (understanding automorphic forms), Galois cohomology, as well as homotopy theory and many other ingredients. Progress in open questions (such as Leopoldt's conjecture) will also presumably require completely new methods.
Best Answer
Algebraic K-theory originated in classical materials that connected class groups, unit groups and determinants, Brauer groups, and related things for rings of integers, fields, etc, and includes a lot of local-to-global principles. But that's the original motivation and not the way the work in the field is currently going - from your question it seems like you're asking about a motivation for "higher" algebraic K-theory.
From the perspective of homotopy theory, algebraic K-theory has a certain universality. A category with a symmetric monoidal structure has a classifying space, or nerve, that precisely inherits a "coherent" multiplication (an E_oo-space structure, to be exact), and such an object has a naturally associated group completion. This is the K-theory object of the category, and K-theory is in some sense the universal functor that takes a category with a symmetric monoidal structure and turns it into an additive structure. The K-theory of the category of finite sets captures stable homotopy groups of spheres. The K-theory of the category of vector spaces (with appropriately topologized spaces of endomorphisms) captures complex or real topological K-theory. The K-theory of certain categories associated to manifolds yields very sensitive information about differentiable structures.
One perspective on rings is that you should study them via their module categories, and algebraic K-theory is a universal thing that does this. The Q-construction and Waldhausen's S.-construction are souped up to include extra structure like universally turning a family of maps into equivalences, or universally splitting certain notions of exact sequence. But these are extra.
It's also applicable to dg-rings or structured ring spectra, and is one of the few ways we have to extract arithmetic data out of some of those.
And yes, it's very hard to compute, in some sense because it is universal. But it generalizes a lot of the phenomena that were useful in extracting arithmetic information from rings in the lower algebraic K-groups and so I think it's generally accepted as the "right" generalization.
This is all vague stuff but I hope I can at least make you feel that some of us study it not just because "it's there".