# Loop spaces, spectra, and operads

## Part 3

###### 12th March, 2018

This post is a weird one: it’s not really aimed at any one audience, but is more of a dump of a bunch of information that I’m trying to process.

Now would be a great time to talk about operads, but I really could
not do a better job than Tai-Danae
Bradley has already done over at her blog in the post *What is an
Operad*, part
1 and part
2. So throughout this post I’m going to assume that you’ve read
this, and I’ll freely reference bits of it without necessarily saying
so.

One of the really important points that Tai-Danae makes is talking about the theorem by May that says that

- 1-fold loop spaces are (weakly homotopy) equivalent to algebras over the associahedra operad; and
- k-fold loop spaces (for k\geqslant2) are (weakly homotopy) equivalent to algebras over the little k-cubes operad.

But as well as algebras over an operad, we also have the notion of
**spaces** over an operad, where we obtain similar looking
equivalences. This idea is more in the language of **brave new
algebra** – a phrase introduced by Waldhausen for topological
spaces that have some algebraic structures up to coherent homotopies.
There is a paper, *Introduction to Algebra over “Brave New
Rings”*, by R.M. Vogt (Rendiconti del Circolo Matematico di Palermo,
Serie II, Suppl. 59 (1999) pp. 49-82) that is a really great read, if
you keep in mind the proviso of slightly different approach: he often
works with **non-\Sigma-operads**, which are operads
without necessarily having the datum of an action by the symmetric
group. Just to save you from finding his exact definition, I paraphrase
it here.

**Definition.** A **non-\Sigma-operad** is a \mathsf{Top}-enriched category \mathcal{A} with objects in bijection with
\mathbb{N}=\{0,1,\ldots\}, along with
an associative continuous bifunctor \oplus\colon\mathcal{A}\times\mathcal{A}\to\mathcal{A}
given on objects by m\oplus n=m+n, and
such that, for all k and n, the morphism

\begin{aligned} \coprod_{n=\Sigma r_i} \mathcal{A}(r_1,1)\times\ldots\times\mathcal{A}(r_k,1) &\to \mathcal{A}(n,k)\\ (f_1,\ldots,f_k) &\mapsto f_1\oplus\ldots\oplus f_k \end{aligned}

is a homeomorphism.

There is also the technical assumption that \mathrm{id}_1\in\mathcal{A}(1,1) is a closed fibration, which we will pass over.

An important thing to note is how this corresponds to May’s definition of an operad. The property satisfied by \oplus on coproducts means that \mathcal{A} is actually uniquely determined by composition and the morphism spaces \mathcal{A}(n,1) for n\geqslant0, which is the data that May uses to define (non-\Sigma-)operads.

The very first page of Vogt’s paper has a beautiful ‘dictionary’ that I’ll reproduce (in full, even though we won’t talk about some of the more complicated rows) here, as well as adding another column with references to the places where Vogt proves these equivalences.

classical algebra | ref. | brave new algebra | spectra |
---|---|---|---|

monoid | 4.3 | A_\infty-space | |

group | 4.4 | loop space | |

abelian monoid | 5.5 | E_\infty-space | |

abelian group | 6.5, 7.5 | infinite loop space | \mathbb{S}-module spectrum |

ring | A_\infty-ring space | monoid in \mathsf{Mod}_\mathbb{S} | |

commutative ring | E_\infty-ring space | commutative monoid in \mathsf{Mod}_\mathbb{S} | |

ground ring \mathbb{Z} | E_\infty-ring space Q(S^0) | sphere spectrum \mathbb{S} | |

tensor product \otimes_\mathbb{Z} | ? | smash product \wedge_\mathbb{S} |

Now, first of all, we need to define *what* exactly e.g. an
A_\infty-space is, which means we need
to define two things:

- the operad \mathcal{A}_\infty, and
- the notion of a space over an operad.

For the second point, given some non-\Sigma-operad \mathcal{A}, we define an **\mathcal{A}-space** to be a continuous
functor X\colon\mathcal{A}\to\mathsf{Top} that sends
\oplus to \times. In particular then, X(n)=X(1)^n, and so we often abuse notation
and identify X with the space X(1).

For the first point, we let \mathrm{Assoc} be the terminal non-\Sigma-operad, so that each \mathrm{Assoc}(n,1) consists of just a single point which we will call {\mu_n}. This forces the composition to satisfy

\mu_k\circ(\mu_{r_1}\oplus\ldots\oplus\mu_{r_k}) = \mu_{r_1+\ldots+r_k}.

It can be seen that an \mathrm{Assoc}-space is exactly a monoid (this is example 2.4 in Vogt).

**Definition.** Let \mathcal{B} be a non-\Sigma-operad. We say that it is an
**A_\infty-operad** if the
unique functor \mathcal{B}\to\mathcal{A}_\infty is a
homotopy equivalence of the underlying morphism spaces. If this is the
case, then we call a \mathcal{B}-space
an **A_\infty-space**.

**Warning.** *What follows is much less even than a
sketch of an elephant by a blind man.*

Again, there is an important note about names and notation. What is
commonly known as the **associative operad** \mathrm{Assoc} is referred to as \mathcal{A} in Vogt, and what is now called
\mathcal{A}_\infty is called \mathcal{W}\mathcal{A} in Vogt, because he
constructs it as a ‘homotopy universal A_\infty operad’ from taking some free operad
and quotienting by certain relations (sections 2.6, 2.7). In fact, there
is the idea of Boardman-Vogt
resolution, which is a specific choice of cofibrant resolution of
operads, and the associahedra operad gives such a resolution of \mathrm{Assoc}, i.e. a cofibrant
replacement:

\mathcal{A}_\infty\to\mathrm{Assoc}.

As for the commutative case, section 3 of Vogt explains in detail how the little cubes operad (in one dimension) is an A_\infty-operad that acts on loop spaces. This generalises in section 6 to showing that n-fold loop spaces can be acted on by the little n-cubes operad, and taking a suitable colimit we find that infinite loop spaces are acted on by some ‘little \infty-cubes operad’. That is, we define E_n to be the little n-discs operad, and E_\infty to be the colimit of the E_n, which is weakly equivalent to the commutative operad \mathrm{Comm}. As an interesting fact, we can see that E_n is generally homotopically non-trivial (tiny picture below), but E_\infty is contractible.

I haven’t really talked much about spectra here, but the definition
of an **infinite loop space** in Vogt is the same as the
definition of an \Omega-spectrum in the
previous post. Vogt mentions (5.2) that if X is an infinite loop space then Y\mapsto[Y,X_n]^\bullet defines a cohomology
theory, and that any cohomology theory arises in this way. This all
seems really nice, but I’m not entirely certain how complicated a proof
of this is, or how trivial this statement is.