What is homotopy coherence?

Today I want to talk about homotopy coherence, which is a very important notion in motivating a lot of the constructions in homotopy theory. Consider complex K-theory, KU. This is represented by the space \mathbf{Z}\times BU, i.e., there is a bijection induced by pullback [X,\mathbf{Z}\times BU]\to KU(X) (see my post for an explanation). Because KU(X) is valued in rings, the space Z is a commutative ring object in the homotopy category of topological spaces. But Z is not a topological commutative ring; instead, the multiplication on Z satisfies homotopy coherence conditions. What do we mean by this?

Suppose X is a topological space. Our goal is to attempt to define what it means for X to be a(n associative, for now) ring in this context. Obviously, we need a map \eta:\mathrm{pt}\to X that is the unit map, and a multiplication \mu:X\times X\to X. This is supposed to satisfy associativity “on the nose”, and this is certainly possible to write down in the homotopy category \mathrm{h}\mathbf{Top}. Ah, but it is also possible to ask for homotopy associativity, in which case we need to specify a homotopy \mu\circ(1\times \mu) and \mu\circ(\mu\times 1) – this is an example of the extra data required by a homotopy coherent diagram. This is simply an isomorphism \eta_{x,y,z}:\mu(x,\mu(y,z))\simeq\mu(\mu(x,y),z). But now, we must ask about what happens with the multiplication X\times X\times X\times X\to X. There are five ways of writing the product of four elements, and they therefore fit into a pentagon. Likewise for more number of elements, we see that the data for the homotopy coherence of a homotopy associative ring object in spaces is specified by a collection of spaces \{\mathcal{A}_n\} with specified compatible maps \mathcal{A}_n\times X^n\to X. The spaces \mathcal{A}_n are called Stasheff associahedra, and I won’t describe them explicitly here. This may seem like too much information, and it may seem like there are only esoteric examples of homotopy associative ring objects in spaces (also called a space with an action of the A_\infty-operad), but this is false. You can also study homotopy commutative objects, etc. Peter May proved the following recognition theorem.

Theorem: A connected space with an action of the A_\infty-operad is homotopy equivalent to a loop space. A connected space with an action of the “E_\infty-operad” (where the n-th space is contractible and has a free action of the symmetric group, i.e., is the space E\Sigma_n) is homotopy equivalent to an infinite loop space.

Consider the loop space \Omega^2 X. This is not necessarily an E_\infty-algebra in spaces, but because it is \Omega(\Omega X), it has more structure than just being an A_\infty-algebra object in spaces. The structure on \Omega^2 X is “homotopy associativity at the second level”, and likewise for \Omega^k X. This means that there can be “homotopy associativity in the k-th level” , and this gives the E_k-operads for 0<k\leq \infty. I do not want to delve into the notion of operads here, so I won’t say more on this.

Let us look at this issue of coherence from a more general context. Let \mathbf{Top} be the category of topological spaces, and consider the homotopy category \mathrm{h}\mathbf{Top}. This is the category of topological spaces and homotopy classes of continuous maps. Consider a category \mathscr{C}, and choose a functor F:\mathscr{C}\to\mathrm{h}\mathbf{Top}. It is natural to ask if there is a functor, denoted G from \mathscr{C} to the category \mathbf{Top} (not the homotopy category!) that induces a functor on the homotopy category that is equivalent to F. This is the problem that homotopy coherence attempts to solve.

Since G induces a functor equivalent to F, and G(f\circ g) = G(f)\circ G(g) for composable morphisms f and g in \mathscr{C}, there must be a specified homotopy between F(f\circ g) and F(f)\circ F(g) (since we are working in the homotopy category). Let us denote this homotopy as H_{f,g}. We may then ask for the relationship between the composite of H_{f,g\circ h} and H_{g,h}, and H_{f\circ g,h} and H_{f,g}. Because of associativity, this should be dictated by some higher homotopy. Now, these higher homotopies should also satisfy associativity, and you can imagine continuing all the way. If F can be lifted to such a G, then F is called a homotopy coherent diagram of topological spaces. So homotopy coherence is not a property, but is rather extra data. The idea is that homotopy coherence is the “right” notion of commutativity in homotopy theory, and so the theory of derived algebraic geometry can be thought of as “homotopy coherent algebraic geometry”. I don’t know nearly enough about derived algebraic geometry to say a lot more about this.

This illustrates the importance of \infty-categories. Since an \infty-category consists of the information of objects, morphisms, homotopies, homotopies between homotopies, and so on, they contain all the information required to study homotopy coherent operations. By extending the notion of an operad to the \infty-categorical context, the notion of a homotopy commutative (associative) ring object can be viewed as an algebra for the \infty-operad in a symmetric monoidal \infty-category, packaged in all its homotopy coherent glory. Work by Lurie and others has made this statement precise. In particular, working with quasicategories makes working with commutative diagrams as hard as working with homotopy coherent diagrams, which is very nice.

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