Spectra arise in the wild when studying a collection of functors (indexed by $n$) $$F_n: \{\text{CW-complexes}\} \to \text{AbGrp}$$

Sometimes, there is an object $E_n$ in the source category (in this case, CW-complexes) of $F_n$ such that $F_n(X) \simeq [X, E_n]$. When $F_n(-)$ is a cohomology functor, we call these $E_n$ “spectra.”

Spectra also fill the need for negative dimensional spheres — the need for a category where suspension has an inverse and not just an adjoint!

*If you want a fantastic introduction to the stable category of spectra, and the context of various topological theorems calling for a definition of negative dimensional spheres, this might help.*

Recall that the following functor takes a pointed space $X$ to the suspension spectrum of $X$ (i.e., $\Sigma^\infty$), and that it has a left adjoint named $\Omega^\infty$.

$\text{space}_*\xrightarrow{\Sigma^\infty} \Sigma-\text{spectrum}$

$\text{space}_*\xleftarrow{\Omega^\infty} \Sigma-\text{spectrum}$

The formal definition of $S^{-n}$ is $\Omega^\infty_n S^0$ (that is, the negative dimensional sphere is $S^0$ in the nth slot), but I find it more illuminating to step back and to think of positive dimensional spheres in terms of the natural numbers that index them:

$n \in \mathbb{N}$ $S^n$

$N \times N \to N$ $T \times T \to T$

$n + m \mapsto (n + m)$ $S^n \wedge S^m \mapsto S^{n+m}$

Grothendieck complete $\mathbb{N}$:

$n + -n \mapsto 0$ $S^n \wedge S^{-n} \mapsto S^0$

which can be rewritten as $\Sigma^nS^0 \wedge \Omega^\infty_nS^0 \mapsto S^0$

However, as fun as it is to try to draw mental pictures of negative dimensional spheres, negative dimensional space not a particularly illuminating frame of reference when thinking about spectra. Let us instead take to heart the Erlangen program, which teaches us a powerful message:

Geometry is the study of properties of an object that are invariant under a chosen collection of transformations of that object.

With that in mind, let us look at the suspension functor.

What are the properties of an object that are invariant under repeated application of the suspension functor?

*If you’re interested in a specific answer, here is a great overview (by Peter May) of the theory and calculations that fall out of taking this question seriously. I won’t elaborate here on the implications of this viewpoint.*

Let us restrict our attention to properties invariant under repeated application of the suspension functor.

*Note that $\Sigma$ raises dimension by one; let $X$ be an $n$-dimensional space, $\Sigma X$ is $(n+1)$-dimensional.*

If we are *only looking at stable invariants* (i.e., if we’re thinking about properties invariant under repeated application of the suspension functor), then the map $S^4 \to S^3$ is indistinguishable from, say, the map $\Sigma^{16}S^4 \to \Sigma^{16} S^3$.

The notion of *dimension* is immaterial.

Suspension is functorial and it adds a dimension to both sides…and we can’t tell the difference.

However, we can tell the difference between the map $S^4 \to S^3$ and $S^6 \to S^3$. We can’t apply suspension to both sides of $S^4 \to S^3$ and get $S^6 \to S^3$.

We have a notion of *relative *dimension.

So $S^{-n}$ is obtained by “shifting” the dimension of $S^0$ (relative to $S^n$) by $n$, i.e. moving $S^0$ into the “nth slot.”

Now that we’re starting to get the hang of this point of view, let’s think about a category of spectra.

Let’s look at the classical definition of a spectrum (a collection of indexed topological spaces $\{E_n\}_{n \in \mathbb{Z}}$, equipped with a suspension map $\Sigma E_n \to E_{n+1}$).

A while ago, Akhil mentioned to me that thinking of spectra as an indexed collection of spaces together with the suspension map is like thinking of a real number as a Cauchy sequence that converges. For example, there are multiple sets of spaces which are the same spectrum (just like there are different Cauchy sequences that converge to the same real number).

So what is a “real number” definition of a spectrum?

There are *all* *sorts* of ways to define a model category of spectra (which is confusing as hell when reading the literature and not being aware of this fact), but there’s a comforting theorem that these categories of spectra are Quillen equivalent model categories so it’s not a problem.

We might expect that a category of spectra $S$ would satisfy these perfectly reasonable axioms:

- The category $S$ is a symmetric monoidal category (wrt $\wedge$)
- The functor $\Sigma^\infty$ is left adjoint to the functor $\Omega^\infty$.
- The unit for the smash product in $S$ is the sphere spectrum $\mathbb{S} := \Sigma^\infty S^0$
- There is one of the two following natural transformations:

$\Omega^\infty D \wedge \Omega^\infty E \to \Omega^\infty (D \wedge E)$

$\Sigma^\infty X \wedge \Sigma^\infty Y \to \Sigma^\infty (X \wedge Y)$ - There is a natural weak equivalence

$$\Omega^\infty\Sigma^\infty X \xrightarrow{\delta} QX$$ and the following diagram commutes (in $\text{Top}_*$):

In the paper *Is there a convenient category of spectra?* there’s a no-go theorem. No category of spectra exists which satisfies Axioms 1-5. It is perfectly healthy to rise from your seat and pace about in frustration at this point. When you’ve calmed down, I’ll brush over the proof of his theorem.

Assume that $S$ satisfies 1-4, and $E$ is a strict ring spectrum.

$\Rightarrow$ $\Omega_1^\infty E$ is product of Eilenberg-MacLane spaces. [May, Proposition 3.6]

The unit for the smash product must be a strict ring spectrum and $\Sigma^\infty {S}^0$ is the unit for the monoid operation in $S$ — so $\Sigma^\infty {S}^0$ must be a strict ring spectrum.

*People like to blur the line between saying that a diagram commutes in $C$ or commutes in $hoC$ (i.e., commutes up to homotopy). When we want to specifically say that a diagram commutes in $C$, we say that is satisfies said property “on the nose,” or that the object itself is “strict.”*

$\Sigma^\infty S^0 \to E$ is the unit for the smash product in $S$.

$\Rightarrow$ $\Omega_1^\infty \Sigma^\infty S^0$ must be a product of Eilenberg-MacLane Spaces.

If we assume that $S$ also satisfies 5, then the following diagram commutes:

Apply $\Omega^\infty$ to the unit map $\Sigma^\infty S^0 \to E$ in $S$.

$\Omega^\infty\Sigma^\infty S^0 \to \Omega^\infty E$ is the corresponding unit map in $Top_*$

$\Rightarrow$ The path component of the unit in $QS^0$ is a product of Eilenberg-MacLane spaces. Lewis states that this is false! I’ve not understood why this is false yet, I have to think about it a bit more.

I hate to leave you in disappointment. Worry not! In this same paper that Lewis shatters our hopes and dreams, he lists 4 axioms that are more reasonable.

Afternote: I’m not going to go over the technicalities in defining the smash product here, because I don’t understand them, I’ll just mention that they arise because naive indices are a bother and require arbitrary choices that we have to keep track of. I do want to mention that a common method of indexing arises from using the relative dimension of inner product spaces $V$ and $W$. $$EV \wedge FW \to (E \wedge F)(V \oplus W)$$ There is still choice involved — we choose an isogeny from $\mathbb{R}^\infty \oplus \mathbb{R}^\infty \to \mathbb{R}^\infty$.

*Thank you to Eric Peterson for directing me to Lewis’s paper.*