Loop spaces, spectra, and operads (Part 1)

8th December, 2017

I have been reading recently about spectra and their use in defining cohomology theories. Something that came up quite a lot was the idea of E_\infty-algebras, which I knew roughly corresponded to some commutative version of A_\infty-algebras, but beyond that I knew nothing. After some enlightening discussions with one of my supervisors, I feel like I’m starting to see how the ideas of spectra, E_\infty-algebras, and operads all fit together. In an attempt to solidify this understanding and pinpoint any difficulties, I’m going to try to write up what I ‘understand’ so far.


The rough idea is to study loop spaces and consider their ‘generalisation’ as homotopy pullbacks, then to define spectrum objects in stable (\infty,1)-categories. We also consider why we can call A_\infty-algebras ‘generalised loop spaces’. Then we will have a look at the E_\infty and A_\infty operads and try to describe the algebras over them, before seeing how this is linked to the idea of spectra (spectrum objects in the category of topological spaces).

After this, I might do a series on the uses of spectra, like ‘nice’ cohomology theories in (\infty,1)-topoi, and ‘brave new geometry’ in the framework of Toën and Vezzosi’s homotopical algebraic geometry.

There probably won’t be many references or pictures to begin with, but I will try to come back and add some at some point in the hopefully not-too-distant future.

Loop spaces

Classical approach

Given some topological space X, we have many good reasons to be interested in its fundamental group. That is, we are interested in what happens to loops (maps from S^1 to X) in our space when we consider everything up to homotopy. With this in mind, we define the (pointed) loop space \Omega(X,x) of X at some point x\in X as the mapping space

\Omega(X,x):=\mathrm{Map}_*\big(S^1,(X,x)\big)

which has a base point given by the constant map to x, and where we consider S^1 as a pointed space with some (fixed) base point s\in S^1 (e.g. embed S^1 into \mathbb{R}^2 and let s=(1,0)).

Now, by definition, we have an isomorphism of groups \pi_0\Omega(X,x)\cong\pi_1(X,x), since elements of the latter are classes of homotopy-equivalent paths, which are the same as classes of points of the former, where we identify points if there is a path between them. This suggestive isomorphism makes us wonder if there are ‘higher’ loop spaces \Omega^i(X,x) such that \pi_i(X,x)\cong\pi_0\Omega^i(X,x). The answer is, as you would hope, yes. In fact, we can just iterate loop spaces to obtain such objects: we define the i-fold loop space as

\Omega^i(X,x):=\Omega\big(\Omega^{i-1}(X,x)\big).

Suspensions and smashing

This definition of higher loop spaces might be a tad surprising in so far as one might have expected ‘higher loops’ to be defined as maps from ‘higher spheres’, and thus to have defined the i-fold loop space as

\Omega^i(X,x):=\mathrm{Map}_*\big(S^i,(X,x)\big).

It is a beautiful convenience that these two proposed definitions agree, and the story goes as follows.

The category \mathsf{Top}_* of pointed topological spaces admits the structure of a symmetric monoidal category,1 where the product is the smash product, given by

X\wedge Y:=\frac{X\times Y}{X\vee Y}

and where X\vee Y is the wedge sum, given by identifying base points. It would be tantamount to a crime to not put in any pictures here, so below are some examples.

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In fact, \mathsf{Top}_* is a closed symmetric monoidal category:2 we have the adjunction (\wedge\dashv\mathsf{Top}_*), and thus the natural isomorphism

\mathsf{Top}_*(X\wedge Y, Z)\cong\mathsf{Top}_*(X,\mathsf{Top}_*(Y,Z)).

In particular then, taking Y=S^1, we see that

\mathsf{Top}_*(X\wedge S^1, Z)\cong\mathsf{Top}_*(X,\Omega(Z,z))

naturally in both X and Z, so defining the (reduced)3 suspension of X as \Sigma X:=X\wedge S^1 gives us an adjunction (\Sigma\dashv\Omega). But now we notice that if we can show that \Sigma S^i\simeq S^{i+1} then

\begin{aligned} \mathsf{Top}_*(S^i,(X,x)) &\cong \mathsf{Top}_*(S^{i-1},\Omega(X,x))\\ &\cong \ldots\\ &\cong \mathsf{Top}_*(S^1,\Omega^i(X,x)) \end{aligned}

whence the two possible definitions of higher loop spaces agree.

We can actually prove a stronger statement: S^i\wedge S^j\simeq S^{i+j}. To ‘see’ this, we consider S^i as a cell complex with cells e^0\sqcup e^i. Then

S^i\wedge S^j\simeq\frac{e^0\sqcup e^j\sqcup e^i\sqcup e^{i+j}}{e^0\sqcup e^i\sqcup e^j}\simeq e^0\sqcup e^{i+j}.

Homotopy pullbacks

We claim that the loop space construction can also be described as the homotopy pullback of X along the base point x\colon*\to X. That is, we want to realise

\begin{array}{ccc} A &\to &* \\ \downarrow & \lrcorner&\downarrow \\ * &\to &X \end{array}

as the loop space \Omega(X,x) we defined previously. But now we use a ‘simple’ trick:4 replace the point with the contractible space \\\{\gamma\colon[0,1]\to X \mid \gamma(0)=x\\\}, and map this into X by sending \gamma to \gamma(1). Then A is exactly the space of maps D^1\to X based at x (i.e with \gamma(0)=x) such that \gamma(1)=x as well, and this is exactly \Omega(X,x).

Dually, the (reduced) suspension can be realised as the homotopy pushout

\begin{array}{ccc} X &\to &* \\ \downarrow & \ulcorner&\downarrow \\ * &\to &B \end{array}

or, said in a different way,5 as the mapping cone of the terminal map X\to *.

Sneak peek at A_\infty-algebras

In one of Stasheff’s theses, in the 60s, he formalised the idea of algebras that behave ‘like loop spaces’ in that they have a multiplication structure that is only associative up to some higher homotopies. These are called A_\infty-algebras, and there are plenty of classic introductions to them in their own right. The ‘most classic’ is probably Keller, Bernhard (2001). Introduction to A-infinity algebras and modules, which also describes (in §2.2) what I’m going to briefly sketch out below (and will cover in more detail in the future): how loop spaces are ‘almost associative algebras’.

Given two loops, \gamma_1 and \gamma_2 in X, we can define their ‘product’ \gamma_1\cdot\gamma_2 as their composition: we ‘do/follow/trace out’ the loop \gamma_1 followed immediately by the loop \gamma_2. Clearly we have some identity element under this product, as well as inverses, and associativity is ‘obvious’ because following the first loop then the second, and then the third is the same as following the first, and then the second and the third, right?

I mean, obviously here there is some caveat so that the actual answer is no, but I think that making this mistake is in some way a good thing, because it means that we’re thinking homotopically without maybe even realising it.6 The problem with associativity comes from the fact that we have picked some model of S^1, namely [0,1], and so our composed loops have to respect this. That is, we can’t just define \gamma_1\cdot\gamma_2 as the obvious map [0,2]\to X; we have to scale the paths so that \gamma_1\cdot\gamma_2\colon[0,1]\to X. This is easy enough: we do \gamma_1(t/2) for 0\leqslant t\leqslant\frac12 and \gamma_2((t+1)/2) for \frac12\leqslant t\leqslant1. But the problem then arises that (\gamma_1\cdot\gamma_2)\cdot\gamma_3\neq\gamma_1\cdot(\gamma_2\cdot\gamma_2) (this might be better explained in the pictures below), although they are clearly homotopic. Then, however, we have a similar problem when we want to look at \gamma_1\cdot\gamma_2\cdot\gamma_3\cdot\gamma_4, but now we have many7 ways of bracketing this. All possible ways will be homotopic, of course, but there is a lot of structure contained in this information of how the different choices are homotopic, and if we can somehow remember all of this then maybe we will gain something from it.

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  1. Careful here: we need to be working with ‘nice enough’ spaces, otherwise this product isn’t associative (not even weakly).↩︎

  2. This needs proving, but I’m not gonna do that here because half of the fun of writing these posts is that I can just talk about the ideas themselves, and not repeat proofs that I don’t want to have to type out. It’s also nice to escape from the usual ‘definition, theorem, proof’ format. But for a proof of this particular claim see e.g. the nLab (although they also refer elsewhere for a proof so I feel less bad for doing so).↩︎

  3. There is also the notion of a non-reduced suspension, but in a lot of cases these two constructions give objects that are weakly homotopic to one another.↩︎

  4. This trick will turn up again in a future part of this series, when we talk about the little discs operad.↩︎

  5. See here for a nice picture as well as some clever words↩︎

  6. You always have to find some positive way of saying ‘I got that wrong’.↩︎

  7. Sometimes 5 can seem like a really big number.↩︎