Chromotopy

February 13, 2010, by Eric Peterson

I’ve got a bunch of posts that I’d like to eventually write, but in order for an audience to make any kind of sense of them, we need to begin by taking an adult approach to fiber bundles. For a first-year student, a fiber bundle is defined typically while studying relative homotopy, via the two following definitions:

Defintion (Fibration): A continuous map $p: E \to B$ is said to be a (Hurewicz) fibration if, given any homotopy of spaces $H: X \times I \to B$ in the base and an initial lift $X \to E$ of the homotopy into E, there exists a (noncanonical) lift $\tilde H: X \times I \to E$ whose projection to B agrees with H. A Serre fibration is a weakning of the definition of Hurewicz fibration, where we only require the lifting property in the case that X is a simplicial complex.

One quickly checks that the projection $X \times Y \to X$ is a fibration, and moreover a good proof of this never uses the global structure of the product. This leads to the second definition:

Definition (Fiber bundle): A map $p: E \to B$ is said to be a fiber bundle with fiber F if there exists an open cover $U_\alpha$ of B and, for each $\alpha$ , a homeomorphism $f_\alpha: p^{-1}(U_\alpha) \xrightarrow{\simeq} U_\alpha \times F$ .

Motivated by this definition, students are encouraged to think of fiber bundles as homotopically twisted products of F and B. This is reflected in the replacement of the isomorphism $\pi_n(F \times B) = \pi_n F \times \pi_n B$ with a certain long exact sequence of homotopy groups

$\cdots \to \pi_{n+1} B \to \pi_n F \to \pi_n E \to \pi_n B \to \pi_{n-1} F \to \cdots.$

These concepts uncover a lot of information about homotopy theory; for example, every continuous map is homotopic to a fibration, and every homotopy type arises as the iterated twisted product of certain “indecomposable” spaces called Eilenberg-Maclane spaces.

A student’s view of fiber bundles changes when he later learns that life often contains group actions, and given the wild success of fiber bundles so far we ought to produce a definition of fiber bundle compatible with a group action. This gives rise to

Definition (Principal G-bundle): Let G be a topological group. A principal G-bundle is a fibration $p: E \to B$ with fiber G, equipped with a right-action of G on E. We require that the homeomorphisms $f_\alpha$ satisfy the added compatibility condition $f_\alpha(x \cdot g) = (f_\alpha(x), 1) \cdot g$ , where the group action in the latter case is the right-action on the second factor.

This also turns out to be obscenely useful; for example, isomorphism classes of principal G-bundles over a fixed space X turn out to be detected by homotopy classes of maps off X into a particular space BG, and this space of maps can in turn be identified with the first singular cohomology group $H^1(X; G)$ . The basic construction of K-theory arises as the study of vector bundles, where a vector bundle of dimension n is, in this framework, defined to be a principal $U(n)$ -bundle. These objects are extremely rich sources of information!

There is a sheaf-y, Čech-ified riff on this definition that lives in between the above idea of a principal G-bundle and how I want us to start thinking about fiber bundles. Namely, on each $U_\alpha$ we have required that $f_\alpha$ pick out a G-equivariant trivialization of our G-bundle. Of course, being locally trivial doesn’t say anything about the global structure, but it does restrict where the nontrivial twists can be introduced – namely, in the intersections $U_\alpha \cap U_\beta = U_{\alpha \beta}$ . There’s no reason that $f_\alpha$ restricted to ${U_{\alpha \beta}}$ and $f_\beta$ restricted to ${U_{\alpha \beta}}$ should agree, and if we expect any kind of interesting global structure, they won’t. With this in mind, we formulate an equivalent definition of a principal G-bundle:

Definition (Transition functions): Given a space B, a topological group G, and an open covering $U_\alpha$ of B, a set of transition functions on this data is a collection of functions $\{g_{\alpha \beta}: U_{\alpha \beta} \to G\}_{\alpha, \beta}$ satisfying the Čech 1-cocycle condition $g_{\alpha \gamma} = g_{\alpha \beta} \cdot g_{\beta \gamma}$ when restricted to $U_{\alpha \beta \gamma} = U_\alpha \cap U_\beta \cap U_\gamma$ .

It’s an entertaining and easy exercise to verify that every principal G-bundle gives rise to a family of transition functions, every family of transition functions defines a principal G-bundle, and these two are essentially inverse processes.

Other topologists will disagree, but to me the word ‘space’ almost universally means either ‘CW-complex’ or ‘simplicial set’ – let’s not worry about the exact restrictions of this viewpoint for the moment. This shift in perspective to simplicial sets changes a lot of basic notions in topology; one excellent example is connectedness. In general topology, we say that a space is disconnected when it can be written as the union of two disjoint proper open sets. In the context of simplicial sets, we don’t even have ‘open sets’ per se until we pass to geometric realizations, so there’s no way to use this definition directly. Instead, the definition of ‘connected’ that lives on the level of simplicial sets themselves is that of path-connectedness: two points are path-connected if there’s a continuous map into the space from the unit interval, where the left endpoint of the interval is sent to point A and the right endpoint to point B.

The idea of Serre fibrations can be immediately modelled in the language of simplicial sets. Fiber bundles and principal G-bundles, on the other hand, are more difficult to translate, since they both require a sense of ‘open cover’ to work out. However, transition functions can be made to work, given some creativity! Given a point in our favorite principal G-bundle, its image A in the base, and a path connecting A to some other point B, transition functions are supposed to tell us how to alter the fiber as we pass from one open set in the cover to the next, so that locally things always look trivialized. Outright skipping the open set middle-man, we can define a simplicial variant of transition functions by producing a map F from paths $\gamma$ in the base space to endomorphisms $G \to G$ of the fiber. Of course, we need this to fulfill some axioms; for instance, if we compose two paths $\gamma_1$ and $\gamma_2$ , the twist in the fiber by travelling along $\gamma_1$ combined with the twist in the fiber from $\gamma_2$ should be equivalent to the total twist induced by their concatenation. Another key idea is that the action of F on homotopy class of a path $\gamma$ (relative to the endpoints) should preserve some sort of equivalence among the induced maps on the fibers – this is too hard to formalize in a blog post without introducing the language of quasicategories, so let’s just do that.

It Turns Out That (thanks primarily to Joyal and Lurie) a simplicial set plus an axiom corresponding to ‘composition’ can be viewed as a natural generalization of a category, where the 0-simplices correspond to objects, 1-simplices to arrows, 2-simplices to some kind of ‘witness to composition’, 3-simplices to some kind of ‘witness to coherence of compositions’, and so on. This takes some imagination and pain tolerance, so it may not make total sense at first. If you understand ordinary categories, you can begin to get a grip on things: the nerve construction applied to an ordinary category produces one of these simplicial-sets-with-composition, called a “quasicategory.” Anyway, what we were describing above is exactly the same thing as a map of simplicial sets from our base space X to the quasicategory G, which has as its sole object the set G, 1-cells corresponding to endomorphisms of G, and onwards and upwards. To reiterate:

Definition (Quasicategorical principal G-bundles): Given a simplicial set X and a group G, a principal G-bundle can be thought of as a map of simplicial sets from X to the quasicategory of G-torsors.

There are several charming features of this definition which we’ll explore in later posts – among the most striking ones are that we can compose these functors to produce more complicated bundles, we can take colimits to produce (things equivalent to) our original bundles, and we can now build ‘bundles’ out of any quasicategory C by thinking of this functor as identifying a ‘homotopy-coherent network’ parametrized by the space X living inside C. For instance, Serre fibrations are represented by maps of simplicial sets from X to the quasicategory of spaces (so should be thought of as ‘space-bundles’), which in turn gives meaning to an equivalence between Serre fibrations over X and “ $\Omega X$ -bundles.” Fiber bundles occur as the subclass of Serre fibrations whose paths are all mapped to homeomorphisms rather than homotopy equivalences. A vector bundle corresponds to a map from X to the quasicategory of vector spaces. To summarize:

Definition (General bundles): For a quasicategory C and a simplicial set X, a C-bundle over X is exactly a map of simplicial sets (or “quasicategorical functor”) $X \to C$ .

Hopefully future blog posts won’t be nearly so wordy, but this is an essential fact that we all need to be on the same page about. For an excessive amount of further reading, one should look at Jacob Lurie’s Higher Topos Theory and also this paper, by Ando, Blumberg, Gepner, Hopkins, and Rezk, which contains a lot of other information that I’d like to talk about later but also discusses these ideas in some detail. I believe that some of these core ideas are originally due to Peter May, but I don’t know an exact reference, and I might be wrong anyway.