Topology of categories, very basic facts surrounding Quillen's Higher Algebraic K-Theory I

Question

In his paper Higher Algebraic K-Theory I (see [here][1]), Quillen introduces a topological space $BC$, called the classifying space of $C$, and tries to relate its topology to the categorical structure of $C$. I am curious as to what is the importance of 1. adjoint functors inducing homotopy equivalences, and 2. computing the fundamental group of $BC$ in terms of purely categorical information. Could anyone tell me? I guess when I say "importance", I mean "having shown these facts, how might have stuff later in the paper arose naturally to Quillen?"

Answer 1

N.B.: I have reread your question and it occured to me that you a probably asking something entirely different. However since I'm unclear what exactly is your question and since I don't want to delete this wall of text, here it is.

Firstly, both categories and (homotopy types of) topological spaces admit a description as simplicial sets which satisfy certain lifting properties. This explains why we would want to consider them on an equal footing. Specifically, a topological space gives a simplicial set for which any n-horn is fillable. Here an n-horn is the boundary of an n-simplex with one face omitted, and fillability means that any map from it can be extended to a map from the whole n-simplex. This defines an equivalence of homotopy categories. Categories can also be descibed via filling of horns, only now we 1) only consider inner horns (a combinatorial simplex has its vertices numbered; an inner horn is the one for which we drop any face other than those lying against the first or last vertex) and 2) demand that the lifting is unique. It is a nice exercise to verify that such data indeed defines a category.

In particular, there is an intermediary notion of $(\infty,1)$-categories (also called quasicategories in this specific case). A quasicategory is a simplicial set, for which any inner horn is fillable (unlike classical categories, we don't require uniqueness). It is easy to see that both categories and spaces are specific kinds of quasicategories. Morally a quasicategory is a category enriched in topological spaces, but there are complications (see e.g. introduction to J. Lurie's "Higher topos theory" for details). Quasicategories themselves form a quasicategory, just like there is a category of categories. Also there is a natural quasicategory of spaces, for which the hom-space of morphisms between two spaces is just the mapping space (especially easy to define for simplicial sets).

Now we have an inclusion of the quasicategory $Top$ of spaces into the quasicategory $Cat_\infty$ of quasicategory and this inclusion has a left adjoint. Generally it can be described as Kan fibrant replacement, but for categories this is just the functor $BC$ defined in Quillen's paper. Essentially this is the answer to your question. The classifying space of the category is the universal space in which your category maps, a truely universal way to invert all arrows of your category. Since this is a functor, it also applies to hom-categories. This means that any natural transformation between functors become invertible as natural transformation between classifying spaces, i.e. a homotopy equivalence.

This is certainly not the only way to gain topological information from a category. For example for a category $C$ we can consider its core groupoid $C^{core}$, which has the same objects but only invertible morphisms. It can also be considered as a space and thus carries much more interesting information. In particular, Quillen's Q-construction associates a K-theory space to core groupoids (with extra data). Under the equivalence stated above, the core groupoid functor is a right adjoint to the inclusion of spaces into quasicategories, so it is also functorial and universal.

Regarding your second question, any covering space $E \stackrel{f}{\to} B$ can be considered as a functor $B \to Set$ under the equivalence between spaces and certain quasicategories stated above. Geometrically you associate to any point $x: B$ the fiber $f^{-1}(x): Set$, to any path $p: \mathrm{path}(x, y)$ the map $f^{-1}(x) \to f^{-1}(y)$ obtained by the unique lifting of $p$ to a path in $E$, and more generally to any n-simplex $\Delta : B$ its lift to an n-simplex in $Set$ constructed similarly. This means that we have a chain of equivalences $$\begin{eqnarray} \mathrm{Covers}(BC) & = & BC \to Set \\ & = & BC \to Set^{core} \\ & = & C \to Set^{core} \end{eqnarray}$$

This is precisely the theorem you stated.