The sphere spectrum $\mathbb{S}$ has $\mathbb{S}_n = S^n$ with the standard structure maps $\Sigma S^n \to S^{n+1}$ (identity maps). Its homotopy groups are the stable homotopy groups of spheres:

$\pi_n(\mathbb{S}) = \pi_n^s = \operatorname{colim}_k \pi_{n+k}(S^k)$

For example: $\pi_0(\mathbb{S}) = \mathbb{Z}$, $\pi_1(\mathbb{S}) = \mathbb{Z}/2$, $\pi_2(\mathbb{S}) = \mathbb{Z}/2$, $\pi_3(\mathbb{S}) = \mathbb{Z}/24$.

The sphere spectrum is the unit for the smash product monoidal structure on $\operatorname{Sp}$.

For an abelian group $A$, the Eilenberg--MacLane spectrum $HA$ has $(HA)_n = K(A, n)$. Its homotopy groups are:

$\pi_n(HA) = \begin{cases} A & n = 0 \\ 0 & n \neq 0 \end{cases}$

$HA$ represents ordinary cohomology: $H^n(X; A) \cong [X, \Sigma^n HA] = [\Sigma^\infty X, \Sigma^n HA]$.

For a ring $R$, $HR$ is a ring spectrum and $\operatorname{Mod}_{HR} \simeq D(R)$ (the derived $\infty$-category).

For a pointed space $X$, the suspension spectrum $\Sigma^\infty X$ has $(\Sigma^\infty X)_n = \Sigma^n X$. This gives the left adjoint $\Sigma^\infty \dashv \Omega^\infty: \mathcal{S}_* \rightleftarrows \operatorname{Sp}$.

$\Sigma^\infty S^0 = \mathbb{S}$ is the sphere spectrum. The stable homotopy groups $\pi_n^s(X) = \pi_n(\Sigma^\infty X)$ are the "stable limit" of $\pi_{n+k}(\Sigma^k X)$.

The complex K-theory spectrum $KU$ has $KU_{2n} = \mathbb{Z} \times BU$ and $KU_{2n+1} = U$ (Bott periodicity: $\Omega^2 (BU \times \mathbb{Z}) \simeq BU \times \mathbb{Z}$). Its homotopy groups are $2$-periodic:

$\pi_n(KU) = \begin{cases} \mathbb{Z} & n \text{ even} \\ 0 & n \text{ odd} \end{cases}$

$KU$ represents complex topological K-theory: $KU^0(X) = K^0(X)$, the Grothendieck group of complex vector bundles on $X$.

Similarly, $KO$ represents real K-theory with $8$-periodic homotopy groups.

The complex cobordism spectrum $MU$ has $(MU)_n = \operatorname{Thom}(BU(n), \gamma_n)$ (the Thom space of the universal bundle). Its homotopy ring $\pi_*(MU)$ is a polynomial ring:

$\pi_*(MU) \cong \mathbb{Z}[x_1, x_2, x_3, \ldots], \quad |x_i| = 2i$

$MU$ is central to chromatic homotopy theory: the formal group law associated to $MU$ classifies all complex-oriented cohomology theories.

The $\infty$-category $\operatorname{Sp}$ has a symmetric monoidal structure given by the smash product $\wedge$ (or $\otimes$). The unit is the sphere spectrum $\mathbb{S}$.

Key properties:

• $HA \wedge HB \simeq H(A \otimes_{\mathbb{Z}} B)$ for abelian groups.
• $\Sigma^\infty X \wedge \Sigma^\infty Y \simeq \Sigma^\infty(X \wedge Y)$ for pointed spaces.
• $\mathbb{S} \wedge E \simeq E$ for any spectrum.

Ring spectra (also called $E_1$-rings or $A_\infty$-rings) are monoid objects in $(\operatorname{Sp}, \wedge)$. Commutative ring spectra ($E_\infty$-rings) are commutative monoids.

For a ring spectrum $R$, the $\infty$-category $\operatorname{Mod}_R$ of (left) $R$-module spectra is a stable presentable $\infty$-category. When $R = H\mathbb{Z}$:

$\operatorname{Mod}_{H\mathbb{Z}} \simeq D(\mathbb{Z})$

The derived tensor product and derived Hom of chain complexes correspond to the smash product and function spectrum of $H\mathbb{Z}$-modules.

More generally, for any ordinary ring $A$, $\operatorname{Mod}_{HA} \simeq D(A)$.

• $H\mathbb{Z}$ is connective ($\pi_n = 0$ for $n \neq 0$).
• $\mathbb{S}$ is connective ($\pi_n^s = 0$ for $n < 0$).
• $KU$ is NOT connective ($\pi_{-2}(KU) = \mathbb{Z}$).
• The Eilenberg--MacLane spectrum $\Sigma^{-n} HA$ (for $n > 0$) is not connective.

Connective spectra are equivalent to grouplike $E_\infty$-spaces (infinite loop spaces), via $\Omega^\infty$.

The functor $\Omega^\infty: \operatorname{Sp} \to \mathcal{S}_*$ sends a spectrum $E$ to its underlying infinite loop space $\Omega^\infty E = E_0$. The left adjoint $\Sigma^\infty: \mathcal{S}_* \to \operatorname{Sp}$ is the suspension spectrum.

$\Omega^\infty$ does NOT preserve the stable structure: it only sees the "connective part" of a spectrum. But it is an equivalence on connective spectra (up to group completion): $\operatorname{Sp}_{\geq 0} \simeq \operatorname{Mon}_{E_\infty}^{\mathrm{gp}}(\mathcal{S})$ (grouplike $E_\infty$-spaces).

Given a map $f: X \to BGL_1(\mathbb{S})$ (the classifying space for stable spherical fibrations), the Thom spectrum $Mf = X^f$ is the colimit of the induced diagram of suspension spectra. Important examples:

• $MO$, $MSO$, $MU$, $MSp$: cobordism spectra classifying manifolds.
• The Thom isomorphism: $H_*(Mf) \cong H_{*-n}(X)$ (when $f$ is orientable of rank $n$).