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Brown representability theorem

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Idea

The classical Brown representability theorem (Brown 62) 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 (Brown 65) 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 Ho(Top * c)Ho(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 Ho(Top * c)Ho(Top_*^c) has coproducts (given by wedge sums) and also weak pushouts (namely, homotopy pushouts).

Theorem

(Brown) If a functor F:Ho(Top * c) opSet *F:Ho(Top_*^c)^{op} \to 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)(Y,y_0) and a universal element uF(Y,y 0)u\in F(Y,y_0) such that T u:[,(Y,y 0)]FT_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)(X; A_1, A_2) of CW-spaces (with A 1A 2=XA_1\cup A_2 = X) and any elements x 1F(A 1)x_1\in F(A_1), x 2F(A 2)x_2\in F(A_2) such that x 1|A 1A 2=x 2|A 1A 2x_1|A_1\cap A_2 = x_2|A_1\cap A_2, there exists yF(X)y\in F(X) such that y|A 1=x 1y|A_1 = x_1 and y|A 2=x 2y|A_2 = x_2.

Generalizations

Categorical Brown representability

Let CC be a category and C 0C_0 an essentially small full subcategory such that

  • CC has coproducts, and C 0C_0 is closed under finite coproducts.
  • CC and C 0C_0 have weak pushouts and the inclusion preserves them.
  • CC has weak colimits of sequences X 0X 1X 2X_0 \to X_1 \to X_2 \to\dots, which are taken to actual colimits by C(Z,)C(Z,-) for any ZC 0Z\in C_0 (that is, the objects of C 0C_0 are compact in some weak sense).

For example, CC could be the homotopy category Ho(Top *)Ho(Top_*) of pointed spaces (under weak homotopy equivalence), with C 0C_0 the full subcategory of (spaces of the homotopy type of) finite CW complexes. Motivated by this example, we write [,][-,-] for hom-sets in CC.

Let C 0¯C\bar{C_0}\subseteq C denote the full subcategory of those YCY\in C such that for any f:YYf:Y\to Y', if the induced map [X,Y][X,Y][X,Y] \to [X,Y'] is bijective for all XC 0X\in C_0, then ff is an isomorphism. If C=Ho(Top *)C=Ho(Top_*) as above, then C 0¯\bar{C_0} is the category of pointed connected spaces.

Theorem

(Brown) With (C,C 0(C,C_0 as above, let F:C opSetF:C^{op}\to Set be a functor which takes coproducts to products and weak pushouts to weak pullbacks. Then there exists YCY\in C and a natural transformation T:[,Y]FT:[-,Y] \to F such that * T XT_X is an isomorphism for all XC 0X\in C_0. * If YC 0¯Y\in \bar{C_0}, then T XT_X is an isomorphism for all XX.

For model categories

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

Theorem

(Jardine) Let F:C opSet *F : C^{op} \to Set_{*} be a homotopical functor to the category of pointed sets on C opC^{op} such that

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

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

    A X i B B AX, \array{ A &\to& X \\ \downarrow^i && \downarrow \\ B &\to& B \coprod_A X, }

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

Then FF is representable.

This follows essentially immediately from Theorem 2 applied to Ho(C)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 GG is Thompson's group F, with g:GGg:G\to G its canonical endomorphism, then gg 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 Ho(Top)Ho(Top) on connected homotopy 1-types, we have an endomorphism Bg:BGBGB g:B G \to B G of the classifying space of GG which does not split in Ho(Top)Ho(Top).

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

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

References

The original theorem was proven in

  • Edgar Brown, Cohomology theories, Annals of Mathematics, Second Series 75: 467–484 (1962)

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

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.

A review in the context of chromatic homotopy theory is in

The relation to Grothendieck contexts (six operations) is highlighted in

  • Amnon Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), 205-236 (web)

Revised on May 5, 2014 06:10:58 by Urs Schreiber (89.204.154.42)