In many places (on MO, elsewhere on the Internet, and perhaps even in some textbooks) one finds a statement of the classical Brown representability theorem that looks something like this:
If $F$ is a contravariant functor from the (weak) homotopy category of topological spaces $\mathrm{Ho}(\mathrm{Top}_\ast)$ to the category $\mathrm{Set}_\ast$ of pointed sets which sends coproducts (i.e. wedges) to products and weak/homotopy pushouts (or CW triads) to weak pullbacks, then $F$ is repesentable.
However, I have been unable to find a convincing proof of this statement. All the proofs I have found fall into one of these classes:
- Proofs of an analogous statement where $\mathrm{Ho}(\mathrm{Top}_*)$ is replaced by its subcategory of pointed connected spaces.
- Proofs which purport to prove the above statment, but which actually implicitly assume all spaces are connected at some point. (It's sometimes hard to tell whether a given proof falls into this class or the previous one, since some classical algebraic topologists seem to use "space" to mean "connected pointed space".)
- Proofs that all cohomology theories are representable by a spectrum. Here the suspension axiom implies that the behavior on connected spaces determines the whole theory.
The issue is that the collection of pointed spheres $\{S^n | n\ge 0\}$ is not a strongly generating set for all of $\mathrm{Ho}(\mathrm{Top}_*)$: a weak equivalence of pointed spaces is still required to induce isomorphisms of homotopy groups with all basepoints, whereas mapping out of pointed spheres detects only homotopy groups at the specified basepoint (and hence at any other basepoint in the same component).
Is the above version of the theorem true, without connectedness hypotheses? If so, where can I find a proof?
Best Answer
I was thinking more about this question and found another paper by Heller which offers an answer (unfortunately a negative one). The paper is
This paper introduces a notion of conjugacy idempotent. It is a triple $(G, g, b)$ consisting of a group $G$, an endomorphism $g \colon G \to G$ and an element $b \in G$ such that for all $x \in G$ we have $g^2(x) = b^{-1} g(x) b$. The theory of conjugacy idempotents can be axiomatized by equations, so there is an initial conjugacy idempotent $(F, f, a)$. The Main Theorem of the paper says (among other things) that $f$ does not split in the quotient of the category of groups by the conjugacy congruence.
Now $f$ induces an endomorphism $B f \colon B F \to B F$ which is an idempotent in $\mathrm{Ho} \mathrm{Top}$ and it follows (by the Main Lemma of the paper) that it doesn't split. It is then easily concluded that $(B f)_{+} \colon (B F)_{+} \to (B F)_{+}$ doesn't split in $\mathrm{Ho} \mathrm{Top}_{*}$.
The map $(B f)_{+}$ induces an idempotent of the representable functor $[-, (B F)_{+}]_{*}$ which does split since this is a $\mathrm{Set}$ valued functor. Let $H \colon \mathrm{Ho} \mathrm{Top}_{*}^\mathrm{op} \to \mathrm{Set}$ be the resulting retract of $[-, (B F)_{+}]_{*}$. It is half-exact (i.e. satisfies the hypotheses of Brown's Representability) as a retract of a half-exact functor. However, it is not representable since a representation would provide a splitting for $(B f)_{+}$.
The same argument with $B f$ in place of $(B f)_{+}$ shows the failure of Brown's Representability in the unbased case.