# Loop spaces, spectra, and operads

## Part 2

###### 11th December, 2017

In the previous post of this series, I talked a bit about basic loop
space stuff and how this gave birth to the idea of
‘homotopically-associative algebras’. I’m going to detour slightly from
what I was going to delve into next and speak about
**delooping** for a bit first. Then I’ll introduce
**spectra** as sort of a generalisation of infinite
deloopings. I’ll probably leave the stuff about E_\infty-algebras for another post, but will
definitely at least mention about how it ties in to all this stuff.

As a warning: I am far (oh so very far) from being an expert in this stuff, so it’s very possible that there are mistakes. If you spot any then please do give me a shout.

**Edit.** This
paper by Clark Barwick starts off with a really nice explanation of
all this spectra/cohomology stuff.

# Delooping

## The basic idea

The basic idea of delooping is exactly what the name suggests: given
some object A, if we can find another
object B such that A\simeq\Omega B then we say that B is the **delooping of A**. We often write \mathbb{B}A or \mathcal{B}A to mean the delooping, depending
on the context (‘groupoidal’ or ‘topological’), though these are, of
course, linked.

As an important note, by looking at the homotopy pullback diagram of a delooping, we see that in some sense a delooping is the same as a suspension. We will come back to this later.

## Groupoids and nerves and geometric realisation

If A is some topological group then we have (at least) two ways of describing it:

- ‘topological’: we can study principal A-bundles, and then we have a classifying space \mathcal{B}A\in\mathsf{Top};
- ‘groupoidal’: we can consider A as the one-object groupoid \mathbb{B}A\in\infty\mathsf{Grpd} whose morphisms are given by A.

As the choice of notation suggests, these two ‘associated objects’
are (we claim) both deloopings of A in
the relevant category, and, in fact, under the homotopy
hypothesis, these two are equivalent via geometric realisation (of
the nerve):^{1}

\vert N(\mathbb{B}A)\vert\simeq\mathcal{B}A.

This is all closely linked to the bar
construction, but I’m still trying to get my head around the ‘good’
order in which to understand these things, so maybe I’ll come back to
this point in a future post.^{2}

Before going any further, we are actually going to look at a concrete example (something that I often forget to do when working in this more abstract setting).

## An example

Let A be some group, *which we
think of as a discrete groupoid*, and define \mathbb{B}A to be the one-object groupoid
with morphisms given by A, as above.
We’re going to show that this really *is* a delooping of A, in that the homotopy square below
homotopy-commutes and is homotopy-universal:

\begin{array}{ccc} A &\to &*\\ \downarrow & &\downarrow\\ * &\to &\mathbb{B}A\tag{1} \end{array}

Now the map A\to *\to\mathbb{B}A is the constant functor \Lambda to the single object of \mathbb{B}A, and there is a natural transformation \eta from this functor to itself given by the identity. Let’s unwrap what that says.

First of all, since A is a discrete groupoid, the only morphisms are of the form \mathrm{id}\colon x\to x, so any natural transformation \eta\colon\Lambda\implies\Lambda is composed of morphisms \eta_x\colon\Lambda(x)\to\Lambda(x), for all x\in A, such that

\begin{array}{ccc} \Lambda(x) &\xrightarrow{\Lambda(\mathrm{id})} &\Lambda(x)\\ \eta_x\downarrow & &\eta_x\downarrow\\ \Lambda(x) &\xrightarrow{\Lambda(\mathrm{id})} &\Lambda(x) \end{array}

commutes. But by the definition of \Lambda, this just means that \eta_x is any morphism in \mathbb{B}A, so we can pick the morphism x (we’re really just using that \mathrm{Ob}(A)=\mathrm{Mor}(\mathbb{B}A)).

Now we see that \eta (defined by the
identity) is, as one would hope, a natural isomorphism, which tells us
that the square (1) commutes up to homotopy.^{3}

As for showing universality, we cheat:^{4} if
we have some other A' with a
natural isomorphism \eta'\colon\Lambda\implies\Lambda then it
must map objects in A' to
isomorphisms (i.e. morphisms) in \mathbb{B}A. But say that it maps an object
x' in A' to the morphism x in \mathbb{B}A. Then we can factor through A by sending the object x' in A' to the object x in A.

## Infinite delooping

What we showed above is really that every *group* has a
delooping (at least, in the groupoidal sense). A natural question to
pose now is, *can we always deloop things*? There is a theorem
that states that, when working in the (\infty,1)-category of topological spaces, we
can deloop a space whenever it is an A_\infty-space, and thus homotopy equivalent
to loop space (in the classical sense).

As I’ll ‘explain’ in a bit, if our category is a stable (\infty,1)-category then all deloopings exist
and are given by suspension. Before I do this though, I’m going to give
one important example of objects that are
**infinitely-deloopable**, i.e. their deloopings can be
delooped, and so can *their* deloopings, and etc., so we have an
infinite sequence of objects

A, \mathbb{B}A, \mathbb{B}^2A, \ldots.

## Singular cohomology and the associated infinite delooping

I claim that any *abelian* group can be infinitely delooped by
setting \mathbb{B}^nA to be the (strict
omega) groupoid that has one object; only identity k-morphisms for k\neq n; and \mathrm{Mor}_n(\mathbb{B}^nA)=A. So we just
take our usual construction of \mathbb{B}A and ‘shift the morphisms up in
degree’.

Again, under the homotopy hypothesis, this sequence of deloopings maps to the sequence of Eilenberg-Mac Lane spaces:

\vert \mathbb{B}^nA\vert \simeq K(A,n).

This sequence ‘is’ (under the identification of connective spectra and strict abelian infinity-groups) the Eilenberg-Mac Lane spectrum, whatever that might mean (considering that I’ve yet to even define spectra). Now comes the magic bit.

Let X be a pointed topological
space^{5} and A an abelian group. It is a classical fact^{6} that the group H^n(X,A) can be identified with \pi_0\mathsf{Top}_*(X,K(A,n)), and so the
fact that \vert\mathbb{B}^nA\vert \simeq
K(A,n) tells us that we could just as well *define*
singular cohomology as

H^n(X,A) := \pi_0\mathbb{H}(X,\mathbb{B}^nA)

where \mathbb{H} is some ‘nice’ category in which we have infinite deloopings of A, e.g. \mathsf{Top}_* or \infty\mathsf{Groupoid}. Even better, these two example choices of \mathbb{H} give us homotopically equivalent objects, by the homotopy hypothesis.

This is really nice for many reasons, but one particularly nice one
is that singular cohomology with coefficients in A is *representable*, and it is pretty
much represented exactly by A. There is
a theorem
that tells us that, in particular, any sufficiently well-behaved
cohomology theory is representable in this way.

# Spectra

↟So with all this talk around spectra, it’s time to actually define
them. Like many concepts, there are a bunch of different models and
settings, and we’re not going to explore most of them, but instead focus
on **sequential pre-spectra** and **\Omega-spectra**.

## The setting (infinity categories)

First of all we work in \mathsf{Top}_* and keep our example of K(A,n) in mind. Here, we define a
**sequential pre-spectrum \mathcal{E}** to be a sequence (\mathcal{E}_n)_{n\in\mathbb{N}} of pointed
spaces, along with the data of **structure maps**, which
are continuous maps \Sigma \mathcal{E}_n\to
\mathcal{E}_{n+1}. Then, using the adjunction (\Sigma\dashv\Omega), we can turn these
structure maps into maps \mathcal{E}_n\to\Omega \mathcal{E}_{n+1}. If
these are isomorphisms^{7} then we say that \mathcal{E} is an **\Omega-spectrum**.

Our previous example of \mathcal{E}_n=K(A,n) then gives us an \Omega-spectrum. And we see that spectra can be viewed as a generalisation of infinitely-deloopable objects.

In the bigger picture, every sequential pre-spectrum has a
**spectrification**, which is an equivalent \Omega-spectrum given by a certain
filtered colimit, and these \Omega-spectra are exactly the fibrant
replacements in some model structure.

Generally, we can define \Omega-spectrum objects in any (\infty,1)-category, but when people talk about ‘spectra’ with no reference to a specific category, it’s usually implicit that they are working in \mathsf{Top}_* with weak homotopy equivalences.

But the ‘honest’ setting for spectra is in a **stable (\infty,1)-category**: pointed^{8} (\infty,1)-categories \mathcal{C} with finite limits, where the
loop (\infty,1)-functor \Omega\colon\mathcal{C}\to\mathcal{C} is an
*equivalence*, with the inverse given by suspension \Sigma. Given some not-necessarily-stable
pointed (\infty,1)-category \mathcal{C} with finite limits, we can form
its **stabilisation** by taking the limit^{9} of
iterating the loop space functor on the category of **pointed
objects of \mathcal{C}**:

\mathsf{Stab}(\mathcal{C}) := \lim_{\leftarrow}\big(\ldots\to(*\downarrow\mathcal{C})\xrightarrow{\Omega}(*\downarrow\mathcal{C})\xrightarrow{\Omega}(*\downarrow\mathcal{C})\big).

The prototypical example of such a category is exactly what we’ve
been looking at: *\mathsf{Stab}(\mathsf{Top}) is the category
of \Omega-spectra in \mathsf{Top}_**. In general, we can
construct \mathsf{Stab}(\mathcal{C}) exactly by taking the category
of spectrum objects in (*\downarrow\mathcal{C}).

A really good introduction to all of this, and where I actually saw
most of this for the first time, is Moritz Groth’s *A short course
on \infty-categories*.

This is all understood much better when we bring operads into the picture. In the next post in this series I’ll try to show how certain spectra are ‘just commutative monoids’, and that commutative monoids in an (\infty,1)-category are ‘just E_\infty-algebras’.

## Nice properties

Before wrapping this post up, I’m just going to mention briefly something that always confused me about spectra that represent cohomology theories.

We can define ‘cohomology’ in any (\infty,1)-category \mathbb{H} by just setting H(X,A):=\pi_0\mathbb{H}(X,A), but if we want
this to behave like ‘usual’ cohomology then we have to impose some
restrictions. First of all, if we want our ‘cocycles’ to classify
something (namely principal bundles over X) then we have to be working in an *(\infty,1)-topos*. Secondly, if we want
to obtain a \mathbb{Z}-graded
cohomology then the simplest way^{10} of doing so is to ask
that our coefficient object A be a
*component of a spectrum object*, i.e. A=\mathcal{E}_n for some spectrum object
\mathcal{E}. Thirdly, if we want our
cohomology to have a group structure then we need^{11}
A to be a *group object*, which
is always the case if A is a component
of a spectrum object.^{12} Note that this is abelian if A is at least an E_2-object. Finally, if we want a ring
structure then we can ask that the spectrum object \mathcal{E} of which A is a component be an E_\infty-ring, or a *ring
spectra*.

The point here is that, cohomology can be defined very generally, but
if we want to recover something that we really recognise from it then we
can basically restrict ourselves to looking at **ring spectra in
an (\infty,1)-topos**.

# That ‘classical fact’

↟A full and good proof (of the fact in question but also much more
besides) can be found in Chapter 4.3 of Hatcher’s
*Algebraic Topology*. I’ll be content here to give a small
sketch, as found in e.g. these
lecture notes.

By the universal coefficient theorem, there is a short exact sequence

\begin{aligned} 0 \longrightarrow& \mathrm{Ext}^1_\mathbb{Z}(H_{n-1}(K(A,n),A)) \longrightarrow H^n(K(A,n),A)\\ &\longrightarrow \mathrm{Hom}_\mathbb{Z}(H_n(K(A,n),A),A) \longrightarrow 0. \end{aligned}

Then, by the Hurewicz theorem, we have an isomorphism

H^n(K(A,n),A)\xrightarrow{\sim}\mathrm{Hom}_\mathbb{Z}(A,A).

Now take the class of \mathrm{id}_A in \mathrm{Hom}_\mathbb{Z}(A,A) and look at its image [\iota_n] in H^n(K(A,n),A). This gives us a natural (in X) transformation

\begin{aligned} \eta_X\colon\pi_0\mathsf{Top}_*(X,K(A,n))&\to H^n(X,A)\\ [\gamma]&\mapsto \gamma^*([\iota_n]) \end{aligned}

and we claim that it is a natural isomorphism when we restrict to the subcategory of pointed CW complexes.

**Edit.** This ‘classical fact’ is really a key example
of the so-called Eckmann-Hilton
duality.