# 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.

**INSERT PICTURES**

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 then map this into X by sending \gamma to \gamma(1). Then A is exactly the space of maps D^1=[0,1]\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, in other words,^{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 many^{7} 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.

**INSERT PICTURES**