Section 15.3 Brown’s Representability Theorem

The wedge and gluing conditions also control coequalizers and increasing unions. The following observations motivate the hypotheses of Brown’s theorem.

Proposition 15.8.

Let \(F\) be a contravariant functor from the based homotopy category to pointed sets. Suppose it sends wedges to products and pushouts along cofibrations to weak pullbacks. Then:

If \(A \rightrightarrows B \to C\) is a coequalizer in \(\mathbf{Top}_*\) and \(A \vee A \to B\) is a cofibration, then \(FC \to FB \rightrightarrows FA\) is a weak equalizer in \(\mathbf{Set}_*\text{.}\)
If \(X_0 \subseteq X_1 \subseteq X_2 \subseteq \cdots\) is a sequence of based CW subcomplexes with union \(X\text{,}\) then \(FX\) is a weak limit of the diagram \(FX_0 \leftarrow FX_1 \leftarrow FX_2 \leftarrow \cdots\text{.}\)

For (i), write the coequalizer as the pushout of \(A\vee A\to B\) and the fold map \(A\vee A\to A\text{.}\) Applying \(F\) gives a surjection onto the pairs \((b,a)\) with \(F(f)(b)=a=F(g)(b)\text{,}\) which are precisely the elements of the equalizer. For (ii), use the mapping telescope of the inclusions. It is homotopy equivalent to \(X\text{;}\) its decomposition into alternating strips is a pushout along cofibrations, with each family of strips homotopy equivalent to a wedge of the \(X_i\text{.}\) The wedge and gluing conditions therefore lift every compatible sequence of classes to a class on the telescope, and hence on \(X\text{.}\)

Proposition 15.9.

Let \(F \colon \mathrm{Ho}(\mathbf{Top}_*)^{\mathrm{op}} \to \mathbf{Set}_*\text{.}\) If \(F\) sends every pushout of two cofibrations to a weak pullback in \(\mathbf{Set}_*\text{,}\) then it sends pushouts along a single cofibration to weak pullbacks.

To see this, replace the other map \(f\colon A\to B\) by the inclusion of \(A\) into its reduced mapping cylinder \(M_f\text{.}\) Both maps out of \(A\) are now cofibrations. Since the original map \(A\to C\) is a cofibration, collapsing \(M_f\) onto \(B\) induces a homotopy equivalence \(M_f\cup_A C\to B\cup_A C\text{.}\) Applying \(F\) gives the assertion.

We can now state the two conditions. All wedges and pushouts below are formed in based spaces:

\(F\bigl(\bigvee_\alpha X_\alpha\bigr) \to \prod_\alpha F(X_\alpha)\) is an isomorphism;
\(F(B \cup_A C) \to F(B) \times_{F(A)} F(C)\) is surjective whenever \(A \to B\) and \(A \to C\) are cofibrations.

Theorem 15.10. Brown’s Representability.

Let \(F \colon \mathrm{Ho}(\mathbf{CW}_{*,\mathrm{conn}})^{\mathrm{op}} \to \mathbf{Set}_*\) be a functor satisfying (B1) and (B2). Then \(F\) is representable: there exist a pointed connected CW complex \(X\) and a generic element \(\alpha \in F(X)\) such that

\begin{equation*} \varphi_Y \colon [Y, X]_* \to F(Y), \qquad [f] \mapsto f^*(\alpha), \end{equation*}

is a bijection for every based connected CW complex \(Y\text{.}\) We call the pair \((X, \alpha)\) a classifying space for \(F\text{.}\)

We use Brown’s theorem without proof; see Hatcher, Theorem 4E.2. The conclusion includes uniqueness up to based homotopy equivalence, as we will deduce from the Yoneda lemma.

Remark 15.11.

Every generalized cohomology theory \(E^*\) satisfies (B1) and (B2):

(B1) is the additivity axiom.
(B2) follows from the exact Mayer–Vietoris sequence

\begin{equation*} E^n(B\cup_A C)\longrightarrow E^n(B)\oplus E^n(C) \xrightarrow{\;i_B^*-i_C^*\;} E^n(A). \end{equation*}

A pair of classes agrees on \(A\) exactly when it lies in the kernel of the second map, so exactness supplies a class on the pushout.

For example, ordinary cohomology with coefficients in \(G\) recovers the classification \([Y, K(G, n)]_* \cong \widetilde H^n(Y; G)\) from Theorem 14.9.

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