Spectra in classical algebraic topology#
Let’s first recall what spectra are. The definition we gave before was the following.
Definition 1. A spectrum is an ∞-functor $F \colon \mathcal{S}_ \ast^\mathrm{fin} \to \mathcal{S}_ \ast$ such that
- $F$ sends homotopy pushouts to homotopy pullbacks,
- $F(\ast) = \ast$.
From this, we see that $$ \begin{CD} F(S^n) @>>> \ast \br @VVV @VVV \br \ast @>>> F(S^{n+1}) \end{CD} $$ is a pullback, so that $F(S^n) \simeq \Omega F(S^{n+1})$. The data of $F$ turns out to be equivalent up to homotopy to the data of $\lbrace F(S^n) \rbrace$ together with these isomorphisms $F(S^n) \simeq \Omega F(S^{n+1})$. Classically, these are called $\Omega$-spectra. But these are hard to write down, so we use the following definition.
Definition 2. A spectrum is a sequence of pointed spaces $\lbrace X_n \rbrace_ {n \ge 0}$ along with maps $\Sigma X_n \to X_{n+1}$ (which is the same as $X_n \to \Omega X_{n+1}$).
Here are some examples.
- The sphere spectrum is defined as $\mathbb{S} = \lbrace S^n \rbrace_{n \ge 0}$ with $\Sigma S^n \cong S^{n+1}$.
- For $A$ is an abelian group, we can define $HA = \lbrace K(A, n) \rbrace_{n \ge 0}$ and this is an $\Omega$-spectrum.
- If $\lbrace Y_n \rbrace_{n \ge 0}$ is a sequence of spaces, we can define a spectrum inductively by setting $X_0 = Y_0$ and $X_{n+1} = \Sigma X_n \vee Y_n$.
We want to make this into a category, and this is actually quite annoying to do. If $X, Y$ are $\Omega$-spectra, then we define $$ \Hom_{\mathrm{h}\mathsf{Sp}}(X, Y) = \lbrace f_n \colon X_n \to Y_n \rbrace $$ where we only consider those family of $f_n$ such that $f_n$ and $\Omega f_{n+1}$ fit in the natural commutative diagram. When $X$ and $Y$ are not $\Omega$-spectra, we define $$ \Hom_{\mathrm{h}\mathsf{Sp}}(X, Y) = \varinjlim_{X^\prime \subseteq X} \Hom(X^\prime, Y) $$ where $X^\prime \subseteq X$ ranges over all weak homotopy equivalences. This is defined in the following way.
Definition 3. If $X$ is a spectrum and $n \in \mathbb{Z}$, we define $$ \pi_n(X) = \varinjlim_k \pi_{n+k}(X_k). $$
Example 4. We have $\pi_n(\mathbb{S}) = \pi_n^s(S^0)$. The homotopy groups of the Eilenberg–Mac Lane spectrum is $$ \pi_n(HA) = \begin{cases} A & n = 0 \br 0 & \text{otherwise.} \end{cases} $$
Example 5. If we set $X_n = S^n \vee S^{n-1} \vee \dotsb \vee S^1$, then $\pi_k(X) \neq 0$ for all $k \in \mathbb{Z}$.
We can also define the “$\Omega$-fication” of a spectrum $X$ by $$ (\Omega^\infty X)_ n = \varinjlim_k \Omega^n X_{n+k}. $$
Proposition 6. The inclusion $N(\mathsf{Ab}) \to \mathsf{Sp}$ defined by $A \mapsto HA$ is fully faithful.
Proof.
The point is to compute $[K(A, n), K(B, n)]$. This is $$ H^n(K(A, n), B) \cong \Hom(H_n(K(A, n), \mathbb{Z}), B) = \Hom(A,B) $$ by Hurewicz.
There is a natural $t$-structure on $\mathsf{Sp}$, given by $$ \mathsf{Sp}^{\ge 0} = \lbrace X : \pi_n X = 0 \text{ for } n \lt 0 \rbrace $$ and similarly $$ \mathsf{Sp}^{\le 0} = \lbrace X : \pi_n X = 0 \text{ for } n \gt 0 \rbrace $$
Proposition 7. This is a $t$-structure, and moreover $\mathsf{Sp}^\heartsuit = N(\mathsf{Ab})$. In particular, if $X \in \mathsf{Sp}^\heartsuit$ then $X \cong H \pi_0(X)$.
Smash product#
When $X$ and $Y$ are pointed topological spaces, we define their smash product as $$ X \wedge Y = (X \times Y) / (X \vee Y). $$ For example, $S^0 \wedge X \cong X$ and $S^1 \wedge X = \Sigma X$. We also have $$ S^m \wedge S^n \cong S^{m+n}, $$ but there is a sign issue we need to be careful with. If we look at $$ S^2 \cong S^1 \wedge S^1 \xrightarrow{\sigma} S^1 \wedge S^1 \cong S^2 $$ where $\sigma$ just flips the two components, this is $-1$.
We have these universal properties $$ \Hom_{\mathsf{Top}_ \ast}(X \wedge Y, Z) \cong \Hom_{\mathsf{Top}_ \ast}(X, \Hom_{\mathsf{Top}_ \ast}(Y, Z)). $$ We also have $$ X \wedge Y \cong Y \wedge X, \quad (X \wedge Y) \wedge Z \cong X \wedge (Y \wedge Z). $$
Definition 8. If $X$ and $Y$ are spectra, define $X \wedge Y$ as follows. Let $p_n$ and $q_n$ be an increasing sequence of positive integers such that $p_n + q_n = n$. We define $$ (X \wedge_{p,q} Y)_ n = X_{p_n} \wedge Y_{p_n}. $$
Remark 9. There is a choiceless way of defining this, but it is complicated.
Theorem 10. If $p, q \to \infty$ then $X \wedge_{p,q} Y$ does not depend on the choice. If $\Sigma X_n \to X_{n+1}$ is an equivalence for $n \ge d$, and $p_N \ge d$ for large $N$, then it also doesn’t depend on the choice.
Proof.
Assume $X, Y$ are CW-spectra. Given different $(p, q)$ and $(r, s)$, define a new spectrum by $$ Z_n = X_{\min \lbrace p_n, r_n \rbrace} \wedge S^{\lvert p_n - r_n \rvert} \wedge Y_{\min \lbrace q_n, s_n \rbrace}. $$ Then there are maps $$ \lbrace X_{p_n} \wedge Y_{q_n} \rbrace \leftarrow \lbrace Z_n \rbrace \to \lbrace X_{r_n} \wedge Y_{s_n} \rbrace, $$ and the claim is that these are weak equivalences.
When $p, q \to \infty$, then this is true because we get all cells eventually. Similarly, if $\Sigma X_n \to X_{n+1}$ are isomorphisms for all $n \ge d$, then $X_d$ see all the cells.
Corollary 11. We have $\mathbb{S} \wedge X = X$, since $S^0 \wedge X = X$.
Now let me state the ∞-categorical fact.
Theorem 12. There exists a symmetric monoidal structure $\otimes \colon \mathsf{Sp} \times \mathsf{Sp} \to \mathsf{Sp}$ with unit $\mathbb{S}$ which commutes with small colimits in both variables. This moreover has the following property: if $\mathcal{C}$ is a symmetric monoidal ∞-category which is stable and presentable, and $\otimes \colon \mathcal{C} \times \mathcal{C} \to \mathcal{C}$ preserves small colimits, then there exists a unique symmetric monoidal functor $\mathsf{Sp}^\otimes \to \mathcal{C}^\otimes$ such that the underlying $\mathsf{Sp} \to \mathcal{C}$ preserves small colimits.
Ring spectra#
Definition 13. An $\mathbb{E}_ k$-ring is a $\mathbb{E}_ k$-algebra object of $\mathsf{Sp}$. The category of such thing is $\mathsf{Alg}_ {\mathbb{E}_ k}$.
It’s not true that $HA \otimes HB = H(A \otimes B)$ for $A, B$ ordinary abelian groups. But algebras stay algebras.
Proposition 14. There is a functor $$ N(\mathsf{CAlg}) \to \mathsf{CAlg}(\mathsf{Sp}); \quad A \mapsto HA. $$
Proof.
To define an algebra structure on $HA$, we can construct a map $$ K(A,m) \wedge K(A,n) \to K(A, m+n). $$ This is constructing a $H^{m+n}$ class of $K(A,m) \wedge K(A,n)$. Because this $K(A,m) \wedge K(A,n)$ is $m+n-1$-connected, we can construct a class on $\pi_{m+n}$ instead. At this point we can just use what we know about homotopy groups of $K(A, m)$ and $K(A, n)$.
Example 15. We have $\mathbb{S} \wedge \mathbb{S} = \mathbb{S}$, so this is a commutative ring, at least at the level of $\mathrm{h}\mathsf{Sp}$.
If $R$ is an $\mathbb{E}_ 1$-ring, then $\pi_\ast R = \bigoplus_ {n \in \mathbb{Z}} \pi_n R$ is a graded ring. The multiplication is constructed by $$ [\mathbb{S}[n], R] \times [\mathbb{S}[m], R] \to [\mathbb{S}[n] \otimes \mathbb{S}[m], R \otimes R] \to [\mathbb{S}[m+n], R]. $$ If $R$ is an $\mathbb{E}_ \infty$-ring, then $\pi_\ast$ is graded-commutative. The sign comes the fact mentioned above.
Modules over a ring#
There are are notions of left/right modules and bi-modules. These are certain algebra objects in $\mathsf{Sp}$. Let me do left modules. There is a colored operad $\mathrm{LM}$ with two colors $a, m$, defined by $$ \operatorname{Mult}(\lbrace X_i \rbrace, a) = \begin{cases} \operatorname{ord}(I) & X_i = a \text{ for all } i \br \emptyset & \text{otherwise} \end{cases} $$ and $$ \operatorname{Mult}(\lbrace X_i \rbrace, m) = \begin{cases} \operatorname{ord}(I)_ {m \text{ largest}} & X_i = a \text{ for all but exactly one } i \br \emptyset & \text{otherwise.} \end{cases} $$
A left module is an $\mathrm{LM}$-module object of $\mathsf{Sp}$. Then we can take the fiber product $$ \begin{CD} \mathsf{LMod}_ R @>>> \mathsf{LMod} \br @VVV @VVV \br \lbrace R \rbrace @>>> \mathsf{Alg}_ {\mathbb{E}_ 1}(\mathsf{Sp}) \end{CD} $$ to define the category $\mathsf{LMod}_ R$ of left $R$-modules. Let me mention some facts.
- $\mathsf{LMod}_ R$ is stable, and has a natural $t$-structure given by looking at homotopy groups.
- If $\pi_n R = 0$ for all $n \neq 0$, then $\mathsf{LMod}_ R \simeq D(\mathsf{LMod}_ {\pi_0(R)})$ preserving $t$-structures.
If $R$ is a commutative ring, then $\mathsf{Alg}^\mathrm{dg}(R)$ the category of dg-algebras over $R$ has a model structures where
- weak equivalences are quasi-isomorphisms,
- fibrations are levelwise surjections.
Proposition 16. We have $$ N(\mathsf{Alg}^\mathrm{dg}(R)^c) [W^{-1}] \simeq \mathsf{Alg}_ {\mathbb{E}_ 1}(R). $$ If $\mathbb{Q} \subseteq R$, then $$ N(\mathsf{CAlg}^\mathrm{dg}(R)^c) [W^{-1}] \simeq \mathsf{Alg}_ {\mathbb{E}_ \infty}(R). $$