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Brown representability theorem

Idea

The classical Brown representability theorem says that certain contravariant functors on the homotopy category are representable. This is used, for instance, to show that any generalized cohomology theory is representable by a spectrum.

There are various generalizations, such as to homotopy categories of other model categories and to triangulated categories. Note, however, that all of these generalizations require either

• group structure on the values of the functor, as is the case for an (abelian) cohomology theory, and as would be the case for a represented functor in any additive category, OR

• the existence of a strong generator in the homotopy category in question.

In particular, there is no Brown representability theorem for functors from the homotopy category of pointed not-necessarily-connected spaces to pointed sets, or for functors from the homotopy category of unpointed spaces to sets. In fact, there are counterexamples in these two cases.

Classical Brown representability

Let $\mathrm{Ho}({\mathrm{Top}}_{*}^c)$ denote the homotopy category of pointed, connected, topological spaces under weak homotopy equivalence — or equivalently the homotopy category of pointed connected CW complexes under homotopy equivalence. Then $\mathrm{Ho}({\mathrm{Top}}_{*}^c)$ has coproducts (given by wedge sums) and also weak pushouts (namely, homotopy pushouts).

Theorem

(Brown) If a functor $F:\mathrm{Ho}({\mathrm{Top}}_{*}^c{)}^{\mathrm{op}}\to {\mathrm{Set}}_{*}$ takes coproducts to products, and weak pushouts to weak pullbacks, then it is representable. That is, there is a pointed connected CW-complex $(Y,y_0)$ and a universal element $u\in F(Y,y_0)$ such that $T_u:[-,(Y,y_0)]\to F$ is a natural isomorphism.

Note that it is immediate that every representable functor has the given properties; the nontrivial statement is that these properties already characterize representable functors.

When the theorem is stated in terms of CW complexes, the second property (taking weak pushouts to weak pullbacks) is often phrased equivalently as:

• The Mayer-Vietoris axiom: For every triad? $(X;A_1,A_2)$ of CW-spaces (with $A_1\cup A_2=X$) and any elements $x_1\in F(A_1)$, $x_2\in F(A_2)$ such that $x_1\mid A_1\cap A_2=x_2\mid A_1\cap A_2$, there exists $y\in F(X)$ such that $y\mid A_1=x_1$ and $y\mid A_2=x_2$.

Generalizations

Categorical Brown representability

Let $C$ be a category and $C_0$ an essentially small full subcategory such that

• $C$ has coproducts, and $C_0$ is closed under finite coproducts.
• $C$ and $C_0$ have weak pushouts and the inclusion preserves them.
• $C$ has weak colimits of sequences $X_0\to X_1\to X_2\to \dots$, which are taken to actual colimits by $C(Z,-)$ for any $Z\in C_0$ (that is, the objects of $C_0$ are compact in some weak sense).

For example, $C$ could be the homotopy category $\mathrm{Ho}({\mathrm{Top}}_{*})$ of pointed spaces (under weak homotopy equivalence), with $C_0$ the full subcategory of (spaces of the homotopy type of) finite CW complexes. Motivated by this example, we write $[-,-]$ for hom-sets in $C$.

Let $\overline{C_0}\subseteq C$ denote the full subcategory of those $Y\in C$ such that for any $f:Y\to Y\prime$, if the induced map $[X,Y]\to [X,Y\prime ]$ is bijective for all $X\in C_0$, then $f$ is an isomorphism. If $C=\mathrm{Ho}({\mathrm{Top}}_{*})$ as above, then $\overline{C_0}$ is the category of pointed connected spaces.

Theorem

(Brown) With $(C,C_0$ as above, let $F:C^{\mathrm{op}}\to \mathrm{Set}$ be a functor which takes coproducts to products and weak pushouts to weak pullbacks. Then there exists $Y\in C$ and a natural transformation $T:[-,Y]\to F$ such that

• $T_X$ is an isomorphism for all $X\in C_0$.
• If $Y\in \overline{C_0}$, then $T_X$ is an isomorphism for all $X$.

For model categories

Let $C$ be a simplicial model category with a zero object and such that there is a set $S$ of cofibrant compact objects such that the weak equivalences in $C$ are precisely the $S$-local equivalences.

Theorem

(Jardine) Let $F:C^{\mathrm{op}}\to {\mathrm{Set}}_{*}$ be a homotopical functor to the category of pointed sets on $C^{\mathrm{op}}$ such that

1. $F$ preserves small coproducts of cofibrant objects (including preserving the zero object).

2. (Mayer-Vietoris property) $F$ takes any pushout diagram

   $$\begin{array}{ccc}A& \to & X\\ {\downarrow }^i& & \downarrow \\ B& \to & B\coprod _{A}X,\end{array}$$\array{ A &\to& X \\ \downarrow^i && \downarrow \\ B &\to& B \coprod_A X, }

   with all objects cofibrant and $i$ a cofibration, to a weak pullback.

Then $F$ is representable.

This follows essentially immediately from Theorem 2 applied to $\mathrm{Ho}(C)$. Jardine also proves a more refined version (see references).

Counterexamples

In some places, one can find the classical Brown representability stated without the restricted to connected pointed spaces. However, this version is false, as is the analogous statement for unpointed spaces.

In the paper of Freyd-Heller cited below, it is show that if $G$ is Thompson's group F, with $g:G\to G$ its canonical endomorphism, then $g$ does not split in the quotient of Grp by conjugacy. Since the quotient of Grp by conjugacy embeds as the full subcategory of the unpointed homotopy category $\mathrm{Ho}(\mathrm{Top})$ on connected homotopy 1-types, we have an endomorphism $Bg:BG\to BG$ of the classifying space of $G$ which does not split in $\mathrm{Ho}(\mathrm{Top})$.

Thus, if $F:\mathrm{Ho}(\mathrm{Top}{)}^{\mathrm{op}}\to \mathrm{Set}$ splits the idempotent $[-,Bg]$ of $[-,BG]$, then $F$ satisfies the hypotheses of the Brown representability theorem (being a retract of a representable functor), but is not representable. A similar argument using $B{G}_{+}$ applies to $\mathrm{Ho}({\mathrm{Top}}_{*})$.

There is also another example due to Heller, which fais to be representable for cardinality reasons.

• MO questions non-connected spaces, unpointed spaces
• blog post

References

The original theorem was proven in

• Edgar Brown?, Cohomology theories

The categorical generalization was proven in

• Edgar Brown?, Abstract homotopy theory, Trans. AMS 119 no. 1 (1965)

The model-categorical version, with applications to ∞-stacks – or rather to the standard models for ∞-stack (∞,1)-toposes in terms of the standard model structure on simplicial presheaves – is given in

• Jardine, Representability theorem for simplicial presheaves

Warning: this is probably implicitly about reduced cohomology theory, as the functors considered always assign the trivial result to the terminal object (the point in the usual examples).

The counterexamples to nonconnected and unpointed Brown representability are from

• Peter Freyd and Alex Heller, Splitting homotopy idempotents. II. J. Pure Appl. Algebra 89 (1993), no. 1-2, 93–106.