A cohomology theory $E^n: \operatorname{Ho}(\mathbf{Top}_*)^{\mathrm{op}} \to \mathbf{Ab}$ satisfying the Eilenberg--Steenrod axioms (minus the dimension axiom) is representable by a spectrum: $E^n(X) = [X, E_n]$ for a sequence of spaces $E_n$ with $\Omega E_{n+1} \simeq E_n$.

In the $\infty$-categorical setting, this is immediate: the functor $E: \mathcal{S}_*^{\mathrm{op}} \to \mathcal{S}$ defined by the cohomology theory preserves limits and is accessible, so by Lurie's theorem it is representable by an object of $\operatorname{Sp}$ (spectra, the stabilization of $\mathcal{S}_*$).

A moduli problem in derived algebraic geometry is a functor $F: \operatorname{CAlg}_k^{\mathrm{op}} \to \mathcal{S}$ from commutative $k$-algebras to spaces. By Lurie's representability theorem (applied to the appropriate presentable $\infty$-category), $F$ is representable by a derived scheme/stack when it satisfies:

1. $F$ is a sheaf (for the etale topology).
2. $F$ has a cotangent complex (satisfies deformation-theoretic conditions).
3. $F$ is nilcomplete and integrable.

This is the derived version of Artin's representability theorem.

The Yoneda embedding $\mathcal{C} \hookrightarrow \operatorname{Fun}(\mathcal{C}^{\mathrm{op}}, \mathcal{S})$ sends $c \mapsto \operatorname{Map}_{\mathcal{C}}(-, c)$. This is fully faithful:

$\operatorname{Map}_{\mathcal{C}}(c, d) \simeq \operatorname{Map}_{\operatorname{Fun}(\mathcal{C}^{\mathrm{op}}, \mathcal{S})}(\operatorname{Map}(-, c), \operatorname{Map}(-, d))$

The representability theorem tells us exactly which functors $\mathcal{C}^{\mathrm{op}} \to \mathcal{S}$ lie in the essential image of the Yoneda embedding.

To construct an object with a universal property (e.g., a colimit, a free algebra, a classifying object), one defines the representing functor $F(c) = \operatorname{Map}_{\mathcal{C}}(c, ?)$ and verifies it preserves limits and is accessible. The representability theorem then guarantees the existence of the universal object.

For example, the free commutative algebra on a module $M$ represents the functor $A \mapsto \operatorname{Map}_{\operatorname{Mod}_R}(M, A)$ (where $A$ is viewed as a module via the forgetful functor), which preserves limits.

The cotangent complex $L_{A/k}$ of a commutative $k$-algebra $A$ represents the functor of derivations:

$\operatorname{Map}_{\operatorname{Mod}_A}(L_{A/k}, M) \simeq \operatorname{Der}_k(A, M)$

The existence of $L_{A/k}$ follows from representability: the derivation functor preserves limits (it is a limit-preserving functor of $M$) and is accessible.

The classifying space $BG$ of a group object $G$ in an $\infty$-topos $\mathcal{X}$ represents the functor of $G$-torsors:

$\operatorname{Map}_{\mathcal{X}}(X, BG) \simeq \operatorname{Tors}_G(X)$

The right-hand side (the space of $G$-torsors on $X$) preserves limits in $X$ and is accessible, so $BG$ exists by representability.

Brown's classical representability theorem (1962) states that a contravariant functor $F: \operatorname{Ho}(\mathbf{Top}_*)^{\mathrm{op}} \to \mathbf{Set}$ is representable if $F$ takes coproducts to products and weak pushouts to weak pullbacks.

Lurie's theorem is a refinement: it works at the $\infty$-categorical level (not just the homotopy category), provides the full mapping space (not just $\pi_0$), and applies to any presentable $\infty$-category (not just spaces).

The key advantage is that Lurie's theorem produces a representing object in the $\infty$-category, which carries more structure than a representing object in the homotopy category.

The representability theorem is closely related to the adjoint functor theorem: a functor $F: \mathcal{C} \to \mathcal{D}$ between presentable $\infty$-categories has a right adjoint if and only if for each $d \in \mathcal{D}$, the functor $c \mapsto \operatorname{Map}_{\mathcal{D}}(F(c), d)$ is representable. This is automatic when $F$ preserves colimits.

Lurie's representability theorem thus underlies the proof of the adjoint functor theorem.