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Brown representability theorem

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Idea

The classical Brown representability theorem (Brown 62, Adams 71) says contravariant functors on the (pointed) classical homotopy category satisfying two conditions (“Brown functors”) are representable.

This is used notably to show that every additive reduced generalized (Eilenberg-Steenrod) cohomology theory E E^\bullet is representable by a spectrum EE as

E n(X)[X,E n]. E^n(X) \simeq [X,E_n] \,.

(But beware that the cohomology theory in general contains less information than the spectrum, due to phantom maps (see also this MO discussion).)

The Brown representability theorem as such, with the two conditions on a Brown functor understood, only applies to contravariant functors, not to covariant functors. But by way of Spanier-Whitehead duality it implies at least over finite CW-complexes that dually every additive generalized homology theory E E_\bullet is representable by a spectrum EE via

E n(X)π n(E nX). E_n(X) \simeq \pi_n(E_n \wedge X) \,.

There are various generalizations, such as to triangulated categories (Neeman 96), to homotopy categories of various model categories (Jardine 09) and homotopy categories of (∞,1)-categories (Lurie, theorem 1.4.1.2). But in any case there are some crucial conditions for the theorem to apply, such as

  • 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

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 (Freyd-Heller 93, see remark 1.

Classical formulation for homotopy functors on topological spaces

Let Ho(Top * c)Ho(Top_*^c) denote the homotopy category of connected pointed 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-Adams)

A functor F:Ho(Top * c) opSet *F:Ho(Top_*^c)^{op} \to Set_* is representable precisely if

  1. it takes coproducts to products,

  2. it takes weak pushouts to weak pullbacks.

(e.g. Aguilar-Gitler-Prito 02, theorem 12.2.18)

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 CW-triple (X;A 1,A 2)(X; A_1, A_2) (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.
Remark

(counterexamples)

The statement of theorem 1 without the restriction to connected pointed spaces is false, as is the analogous statement for unpointed spaces.

In (Freyd-Heller 93), 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.

For homotopy functors on presentable (,1)(\infty,1)-categories

The classical formulation of Brown representability is only superficially concerned with topological spaces. Instead, via the classical model structure on topological spaces, one finds that the statement only concerns the simplicial localization of Top at the weak homotopy equivalences, hence the (∞,1)-category L wheTopL_{whe} Top\simeq ∞Grpd.

We discuss now the natural formulation of the Brown representability theorem for functors out of homotopy categories of (∞,1)-categories following (Lurie, section 1.4.1). See also the exposition in (Mathew 11).

Definition

Let 𝒞\mathcal{C} be a locally presentable (∞,1)-category. A functor

F:Ho(𝒞) opSet F \;\colon\; Ho(\mathcal{C})^{op} \longrightarrow Set

(from the opposite of the homotopy category of 𝒞\mathcal{C} to Set)

is called a Brown functor if

  1. it sends small coproducts to products;

  2. it sends (∞,1)-pushouts in 𝒞Ho(𝒞)\mathcal{C}\to Ho(\mathcal{C}) to weak pullbacks in Set.

Remark

A weak pullback is a diagram that satisfies the existence clause of a pullback, but not necessarily the uniqueness condition. Hence the second clause in def. 1 says that for a (∞,1)-pushout square

Z X Y XZY \array{ Z &\longrightarrow& X \\ \downarrow &\swArrow& \downarrow \\ Y &\longrightarrow& X \underset{Z}{\sqcup}Y }

in 𝒞\mathcal{C}, then the induced universal morphism

F(XZY)epiF(X)×F(Z)F(Y) F\left(X \underset{Z}{\sqcup}Y\right) \stackrel{epi}{\longrightarrow} F(X) \underset{F(Z)}{\times} F(Y)

into the actual pullback is an epimorphism.

Definition

Say that a locally presentable (∞,1)-category 𝒞\mathcal{C} is compactly generated by cogroup objects closed under suspensions if

  1. 𝒞\mathcal{C} is generated by a set

    {S i𝒞} iI \{S_i \in \mathcal{C}\}_{i \in I}

    of compact objects (i.e. every object of 𝒞\mathcal{C} is an (∞,1)-colimit of the objects S iS_i.)

  2. each S iS_i admits the structure of a cogroup object in the homotopy category Ho(𝒞)Ho(\mathcal{C});

  3. the set {S i}\{S_i\} is closed under forming reduced suspensions.

Example

(suspensions are H-cogroup objects)

Let 𝒞\mathcal{C} be an (∞,1)-category with finite (∞,1)-colimits and with a zero object. Write Σ:X0X0\Sigma \colon X \mapsto 0 \underset{X}{\coprod} 0 for the reduced suspension functor.

Then the fold map

ΣXΣX0X0X00XXX00X0ΣX \Sigma X \coprod \Sigma X \simeq 0 \underset{X}{\sqcup} 0 \underset{X}{\sqcup} 0 \longrightarrow 0 \underset{X}{\sqcup} X \underset{X}{\sqcup} 0 \simeq 0 \underset{X}{\sqcup} 0 \simeq \Sigma X

exhibits cogroup structure on the image of any suspension object ΣX\Sigma X in the homotopy category.

This is equivalently the group-structure of the first (fundamental) homotopy group of the values of functor co-represented by ΣX\Sigma X:

Ho(𝒞)(ΣX,):YHo(𝒞)(ΣX,Y)Ho(𝒞)(X,ΩY)π 1Ho(𝒞)(X,Y). Ho(\mathcal{C})(\Sigma X, -) \;\colon\; Y \mapsto Ho(\mathcal{C})(\Sigma X, Y) \simeq Ho(\mathcal{C})(X, \Omega Y) \simeq \pi_1 Ho(\mathcal{C})(X, Y) \,.
Example

In bare pointed homotopy types 𝒞=\mathcal{C} = ∞Grpd */{}^{\ast/}, the (homotopy types of) n-spheres S nS^n are cogroup objects for n1n \geq 1, but not for n=0n = 0, by example 1. And of course they are compact objects.

So while {S n} n\{S^n\}_{n \in \mathbb{N}} generates all of Grpd */\infty Grpd^{\ast/}, the latter is not an example of def. 2 due to the failure of S 0S^0 to have cogroup structure.

Removing that generator, the (,1)(\infty,1)-category generated by {S n} nn1\{S^n\}_{{n \in \mathbb{N}} \atop {n \geq 1}} is Grpd 1 */\infty Grpd^{\ast/}_{\geq 1}, that of connected pointed homotopy types. This is one way to see how the connectedness condition in the classical version of Brown representability theorem arises.

See also (Lurie, example 1.4.1.4)

In \infty-categories compactly generated by cogroup objects closed under forming suspensions, the following strenghtening of the Whitehead theorem holds.

Proposition

In an (,1)(\infty,1)-category compactly generated by cogroup objects {S i} iI\{S_i\}_{i \in I} closed under forming suspensions, according to def. 2, a morphism f:XYf\colon X \longrightarrow Y is an equivalence precisely if for each iIi \in I the induced function of maps in the homotopy category

Ho(𝒞)(S i,f):Ho(𝒞)(S i,X)Ho(𝒞)(S i,Y) Ho(\mathcal{C})(S_i,f) \;\colon\; Ho(\mathcal{C})(S_i,X) \longrightarrow Ho(\mathcal{C})(S_i,Y)

is an isomorphism (a bijection).

(Lurie, p. 114, Lemma star)

Proof

By the (∞,1)-Yoneda lemma, the morphism ff is an equivalence precisely if for all objects A𝒞A \in \mathcal{C} the induced morphism

𝒞(A,f):𝒞(A,X)𝒞(A,Y) \mathcal{C}(A,f) \;\colon\; \mathcal{C}(A,X) \longrightarrow \mathcal{C}(A,Y)

is an equivalence in ∞Grpd. By assumption of compact generation and since the hom-functor 𝒞(,)\mathcal{C}(-,-) sends \infty-colimits in the first argument to \infty-limits, this is the case precisely already if it is the case for A{S i} iIA \in \{S_i\}_{i \in I}.

Now by the standard Whitehead theorem in ∞Grpd (being a hypercomplete (∞,1)-topos), the morphisms

𝒞(S i,f):𝒞(S i,X)𝒞(S i,Y) \mathcal{C}(S_i,f) \;\colon\; \mathcal{C}(S_i,X) \longrightarrow \mathcal{C}(S_i,Y)

in ∞Grpd are equivalences precisely if they are weak homotopy equivalences, hence precisely if they induce isomorphisms on all homotopy groups π n\pi_n for all basepoints.

It is this last condition of testing on all basepoints that the assumed cogroup structure on the S iS_i allows to do away with: this cogroup structure implies that 𝒞(S i,)\mathcal{C}(S_i,-) has the structure of an HH-group, and this implies (by group multiplication), that all connected components have the same homotopy groups, hence that all homotopy groups are independent of the choice of basepoint, up to isomorphism.

Therefore the above morphisms are equivalences precisely if they are so under applying π n\pi_n based on the connected component of the zero morphism

π n𝒞(S i,f):π n𝒞(S i,X)π n𝒞(S i,Y). \pi_n\mathcal{C}(S_i,f) \;\colon\; \pi_n \mathcal{C}(S_i,X) \longrightarrow \pi_n\mathcal{C}(S_i,Y) \,.

Now in this pointed situation we may use that

π n𝒞(,) π 0𝒞(,Ω n()) π 0𝒞(Σ n(),) Ho(𝒞)(Σ n(),) \begin{aligned} \pi_n \mathcal{C}(-,-) & \simeq \pi_0 \mathcal{C}(-,\Omega^n(-)) \\ & \simeq \pi_0\mathcal{C}(\Sigma^n(-),-) \\ & \simeq Ho(\mathcal{C})(\Sigma^n(-),-) \end{aligned}

to find that ff is an equivalence in 𝒞\mathcal{C} precisely if the induced morphisms

Ho(𝒞)(Σ nS i,f):Ho(𝒞)(Σ nS i,X)Ho(𝒞)(Σ nS i,Y) Ho(\mathcal{C})(\Sigma^n S_i, f) \;\colon\; Ho(\mathcal{C})(\Sigma^n S_i,X) \longrightarrow Ho(\mathcal{C})(\Sigma^n S_i,Y)

are isomorphisms for all iIi \in I and nn \in \mathbb{N}.

Finally by the assumption that each suspension Σ nS i\Sigma^n S_i of a generator is itself among the set of generators, the claim follows.

Theorem

(Brown representability)

Let 𝒞\mathcal{C} be an (,1)(\infty,1)-category compactly generated by cogroup objects closed under forming suspensions, according to def. 2. Then a functor

F:Ho(𝒞) opSet F \;\colon\; Ho(\mathcal{C})^{op} \longrightarrow Set

(from the opposite of the homotopy category of 𝒞\mathcal{C} to Set) is representable precisely if it is a Brown functor, def. 1.

(Lurie, theorem 1.4.1.2)

Proof

Due to the version of the Whitehead theorem of prop. 1 we are essentially reduced to showing that Brown functors FF are representable on the S iS_i. To that end consider the following lemma. (In the following we notationally identify, via the Yoneda lemma, objects of 𝒞\mathcal{C}, hence of Ho(𝒞)Ho(\mathcal{C}), with the functors they represent.)

Lemma (\star): Given X𝒞X \in \mathcal{C} and ηF(X)\eta \in F(X), hence η:XF\eta \colon X \to F, then there exists a morphism f:XXf \colon X \to X' and an extension η:XF\eta' \colon X' \to F of η\eta which induces for each S iS_i a bijection η():Ho(𝒞)(S i,X)PSh(Ho(𝒞))(S i,F)F(S i)\eta'\circ (-) \colon Ho(\mathcal{C})(S_i,X') \stackrel{\simeq}{\longrightarrow} PSh(Ho(\mathcal{C}))(S_i,F) \simeq F(S_i).

To see this, first notice that we may directly find an extension η 0\eta_0 along a map XX oX\to X_o such as to make a surjection: simply take X 0X_0 to be the coproduct of all possible elements in the codomain and take

η 0:X(iI,γ:S iFS i)F \eta_0 \;\colon\; X \sqcup \left( \underset{{i \in I,} \atop {\gamma \colon S_i \stackrel{}{\to} F}}{\coprod} S_i \right) \longrightarrow F

to be the canonical map. (Using that FF, by assumption, turns coproducts into products, we may indeed treat the coproduct in 𝒞\mathcal{C} on the left as the coproduct of the corresponding functors.)

To turn the surjection thus constructed into a bijection, we now successively form quotients of X 0X_0. To that end proceed by induction and suppose that η n:X nF\eta_n \colon X_n \to F has been constructed. Then for iIi \in I let

K iker(Ho(𝒞)(S i,X n)η n()F(S i)) K_i \coloneqq ker \left( Ho(\mathcal{C})(S_i, X_n) \stackrel{\eta_n \circ (-)}{\longrightarrow} F(S_i) \right)

be the kernel of η n\eta_n evaluated on S iS_i. These K iK_i are the pieces that need to go away in order to make a bijection. Hence define X n+1X_{n+1} to be their joint homotopy cofiber

X n+1coker((iI,γK iS i)(γ) iIγK iX n). X_{n+1} \coloneqq coker\left( \left( \underset{{i \in I,} \atop {\gamma \in K_i}}{\sqcup} S_i \right) \overset{(\gamma)_{{i \in I} \atop {\gamma\in K_i}}}{\longrightarrow} X_n \right) \,.

Then by the assumption that FF takes this homotopy cokernel to a weak fiber (as in remark 2), there exists an extension η n+1\eta_{n+1} of η n\eta_n along X nX n+1X_n \to X_{n+1}:

(iIγK iS i) (γ) iIγK i X n η n F (po h) η n+1 * X n+1 F(X n+1) * η n+1 epi * η n ker((γ *) iIγK i) * η n (pb) F(X n) (γ *) iIγK i iIγK iF(S i). \array{ \left( \underset{{i \in I}\atop {\gamma \in K_i}}{\sqcup} S_i \right) &\overset{(\gamma)_{{i \in I}\atop \gamma \in K_i}}{\longrightarrow}& X_n &\overset{\eta_n}{\longrightarrow}& F \\ \downarrow &(po^{h})& \downarrow & \nearrow_{\mathrlap{\exists \eta_{n+1}}} \\ \ast &\longrightarrow& X_{n+1} } \;\;\;\;\;\;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\;\;\;\;\;\; \array{ && F(X_{n+1}) &\longrightarrow& \ast \\ &{}^{\mathllap{\exists \eta_{n+1}}}\nearrow& \downarrow^{\mathrlap{epi}} && \downarrow \\ \ast &\overset{\eta_n}{\longrightarrow}& ker\left((\gamma^\ast\right)_{{i \in I} \atop {\gamma \in K_i}}) &\longrightarrow& \ast \\ &{}_{\mathllap{\eta_n}}\searrow& \downarrow &(pb)& \downarrow \\ && F(X_n) &\underset{(\gamma^\ast)_{{i \in I} \atop {\gamma \in K_i}} }{\longrightarrow}& \underset{{i \in I}\atop {\gamma\in K_i}}{\prod}F(S_i) } \,.

It is now clear that we want to take

Xlim nX n X' \coloneqq \underset{\rightarrow}{\lim}_n X_n

and extend all the η n\eta_n to that colimit. Since we have no condition for evaluating FF on colimits other than pushouts, observe that this sequential colimit is equivalent to the following pushout:

nX n nX 2n (po h) nX 2n+1 X, \array{ \underset{n}{\sqcup} X_n &\longrightarrow& \underset{n}{\sqcup} X_{2n} \\ \downarrow &(po^h)& \downarrow \\ \underset{n}{\sqcup} X_{2n+1} &\longrightarrow& X' } \,,

where the components of the top and left map alternate between the identity on X nX_n and the above successor maps X nX n+1X_n \to X_{n+1}. Now the excision property of FF applies to this pushout, and we conclude the desired extension η:XF\eta' \colon X' \to F:

nX n nX 2n+1 X nX 2n (η 2n+1) n η (η 2n) n F F(X) η epi *(η n) n lim nF(X n) nF(X 2n+1) n(X 2n) nF(X n), \array{ && \underset{n}{\sqcup} X_n \\ & \swarrow && \searrow \\ \underset{n}{\sqcup} X_{2n+1} &\longrightarrow& X' &\longleftarrow& \underset{n}{\sqcup} X_{2n} \\ & {}_{\mathllap{(\eta_{2n+1})_{n}}}\searrow& \downarrow^{\mathrlap{\exists \eta'}} & \swarrow_{\mathrlap{(\eta_{2n})_n}} \\ && F } \;\;\;\;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\;\;\;\; \array{ && F(X') \\ &{}^{\mathllap{\exists \eta'}}\nearrow& \downarrow^{\mathrlap{epi}} \\ &\ast \overset{(\eta_n)_n}{\longrightarrow}& \underset{\longleftarrow}{\lim}_n F(X_n) \\ & \swarrow && \searrow \\ \underset{n}{\prod}F(X_{2n+1}) && && \underset{n}{\prod}(X_{2n}) \\ & \searrow && \swarrow \\ && \underset{n}{\prod}F(X_n) } \,,

It remains to confirm that this indeed gives the desired bijection. Surjectivity is clear. For injectivity use that all the S iS_i are, by assumption, compact, hence they may be taken inside the sequential colimit:

X n(γ) γ^ S i γ X=lim nX n. \array{ && X_{n(\gamma)} \\ &{}^{\mathllap{ \exists \hat \gamma}}\nearrow& \downarrow \\ S_i &\overset{\gamma}{\longrightarrow}& X' = \underset{\longrightarrow}{\lim}_n X_n } \,.

With this, injectivity follows because by construction we quotiented out the kernel at each stage. Because suppose that γ\gamma is taken to zero in F(S i)F(S_i), then by the definition of X n+1X_{n+1} above there is a factorization of γ\gamma through the point:

0: S i γ^ X n(γ) η n F * X n(γ)+1 X \array{ 0 \colon & S_i &\overset{\hat \gamma}{\longrightarrow}& X_{n(\gamma)} &\overset{\eta_n}{\longrightarrow}& F \\ & \downarrow && \downarrow & \\ & \ast &\longrightarrow& X_{n(\gamma)+1} \\ & && \downarrow \\ & && X' }

This concludes the proof of Lemma (\star).

Now apply the construction given by this lemma to the case X 00X_0 \coloneqq 0 and the unique η 0:0!F\eta_0 \colon 0 \stackrel{\exists !}{\to} F. Lemma ()(\star) then produces an object XX' which represents FF on all the S iS_i, and we want to show that this XX' actually represents FF generally, hence that for every Y𝒞Y \in \mathcal{C} the function

θη():Ho(𝒞)(Y,X)F(Y) \theta \coloneqq \eta'\circ (-) \;\colon\; Ho(\mathcal{C})(Y,X') \stackrel{}{\longrightarrow} F(Y)

is a bijection.

First, to see that θ\theta is surjective, we need to find a preimage of any ρ:YF\rho \colon Y \to F. Applying Lemma ()(\star) to (η,ρ):XYF(\eta',\rho)\colon X'\sqcup Y \longrightarrow F we get an extension κ\kappa of this through some XYZX' \sqcup Y \longrightarrow Z and the morphism on the right of the following commuting diagram:

Ho(𝒞)(,X) Ho(𝒞)(,Z) η() κ() F(). \array{ Ho(\mathcal{C})(-,X') && \longrightarrow && Ho(\mathcal{C})(-, Z) \\ & {}_{\mathllap{\eta'\circ(-)}}\searrow && \swarrow_{\mathrlap{\kappa \circ (-)}} \\ && F(-) } \,.

Moreover, Lemma ()(\star) gives that evaluated on all S iS_i, the two diagonal morphisms here become isomorphisms. But then prop. 1 implies that XZX' \longrightarrow Z is in fact an equivalence. Hence the component map YZZY \to Z \simeq Z is a lift of κ\kappa through θ\theta.

Second, to see that θ\theta is injective, suppose f,g:YXf,g \colon Y \to X' have the same image under θ\theta. Then consider their homotopy pushout

YY (f,g) X Y Z \array{ Y \sqcup Y &\stackrel{(f,g)}{\longrightarrow}& X' \\ \downarrow && \downarrow \\ Y &\longrightarrow& Z }

along the codiagonal of YY. Using that FF sends this to a weak pullback by assumption, we obtain an extension η¯\bar \eta of η\eta' along XZX' \to Z. Applying Lemma ()(\star) to this gives a further extension η¯:ZZ\bar \eta' \colon Z' \to Z which now makes the following diagram

Ho(𝒞)(,X) Ho(𝒞)(,Z) η() η¯() F() \array{ Ho(\mathcal{C})(-,X') && \longrightarrow && Ho(\mathcal{C})(-, Z) \\ & {}_{\mathllap{\eta'\circ(-)}}\searrow && \swarrow_{\mathrlap{\bar \eta' \circ (-)}} \\ && F(-) }

such that the diagonal maps become isomorphisms when evaluated on the S iS_i. As before, it follows via prop. 1 that the morphism h:XZh \colon X' \longrightarrow Z' is an equivalence.

Since by this construction hfh\circ f and hgh\circ g are homotopic

YY (f,g) X h Y Z Z \array{ Y \sqcup Y &\stackrel{(f,g)}{\longrightarrow}& X' \\ \downarrow && \downarrow & \searrow^{\mathrlap{\stackrel{h}{\simeq}}} \\ Y &\longrightarrow& Z &\longrightarrow& Z' }

it follows with hh being an equivalence that already ff and gg were homotopic, hence that they represented the same element.

Definition

Let 𝒞\mathcal{C} be an (∞,1)-category with (∞,1)-pushouts, and with a zero object 0𝒞0 \in \mathcal{C}. Write Σ:𝒞𝒞:X0X0\Sigma \colon \mathcal{C} \to \mathcal{C}\colon X\mapsto 0 \underset{X}{\sqcup} 0 for the corresponding suspension (∞,1)-functor.

A reduced additive generalized (Eilenberg-Steenrod) cohomology theory on 𝒞\mathcal{C} is

  1. a functor

    H :Ho(𝒞) opAb H^\bullet \;\colon \; Ho(\mathcal{C})^{op} \longrightarrow Ab^{\mathbb{Z}}

    (from the opposite of the homotopy category of 𝒞\mathcal{C} into \mathbb{Z}-graded abelian groups);

  2. a natural isomorphisms (“suspension isomorphisms”) of degree +1

    δ:H H +1Σ \delta \; \colon \; H^\bullet \longrightarrow H^{\bullet+1} \circ \Sigma

such that H H^\bullet

  1. (exactness) takes homotopy cofiber sequences to exact sequences.

  2. (additivity) takes small coproducts to products;

Definition

Given a generalized cohomology theory (H ,δ)(H^\bullet,\delta) on some 𝒞\mathcal{C} as in def. 3, and given a homotopy cofiber sequence in 𝒞\mathcal{C}

XfYgZcoker(g)ΣX, X \stackrel{f}{\longrightarrow} Y \stackrel{g}{\longrightarrow} Z \stackrel{coker(g)}{\longrightarrow} \Sigma X \,,

then the corresponding connecting homomorphism is the composite

:H (X)δH +1(ΣX)coker(g) *H +1(Z). \partial \;\colon\; H^\bullet(X) \stackrel{\delta}{\longrightarrow} H^{\bullet+1}(\Sigma X) \stackrel{coker(g)^\ast}{\longrightarrow} H^{\bullet+1}(Z) \,.
Proposition

The connecting homomorphisms of def. 4 are parts of long exact sequences

H (Z)H (Y)H (X)H +1(Z). \cdots \stackrel{\partial}{\longrightarrow} H^{\bullet}(Z) \longrightarrow H^\bullet(Y) \longrightarrow H^\bullet(X) \stackrel{\partial}{\longrightarrow} H^{\bullet+1}(Z) \to \cdots \,.
Proof

By the defining exactness of H H^\bullet, def. 3, and the way this appears in def. 4, using that δ\delta is by definition an isomorphism.

Proposition

Given a reduced addive generalized cohomology functor H :Ho(𝒞) opAb H^\bullet \colon Ho(\mathcal{C})^{op}\to Ab^{\mathbb{Z}}, def. 3, its underlying Set-valued functors H n:Ho(𝒞) opAbSetH^n \colon Ho(\mathcal{C})^{op}\to Ab\to Set are Brown functors, def. 1.

Proof

The first condition on a Brown functor holds by additivity. For the second condition, given a homotopy pushout square

X 1 f 1 Y 1 X 2 f 2 Y 2 \array{ X_1 &\stackrel{f_1}{\longrightarrow}& Y_1 \\ \downarrow^{} && \downarrow \\ X_2 &\stackrel{f_2}{\longrightarrow}& Y_2 }

in 𝒞\mathcal{C}, consider the induced morphism of the long exact sequences given by prop. 2

H (coker(f 2)) H (Y 2) f 2 * H (X 2) H +1(Σcoker(f 2)) H (coker(f 1)) H (Y 1) f 1 * H (X 1) H +1(Σcoker(f 1)) \array{ H^\bullet(coker(f_2)) &\longrightarrow& H^\bullet(Y_2) &\stackrel{f^\ast_2}{\longrightarrow}& H^\bullet(X_2) &\stackrel{}{\longrightarrow}& H^{\bullet+1}(\Sigma coker(f_2)) \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow && \downarrow && \downarrow^{\mathrlap{\simeq}} \\ H^\bullet(coker(f_1)) &\longrightarrow& H^\bullet(Y_1) &\stackrel{f^\ast_1}{\longrightarrow}& H^\bullet(X_1) &\stackrel{}{\longrightarrow}& H^{\bullet+1}(\Sigma coker(f_1)) }

Here the outer vertical morphisms are isomorphisms, as shown, due to the pasting law (see also at fiberwise recognition of stable homotopy pushouts). This means that the four lemma applies to this diagram. Inspection shows that this implies the claim.

Corollary

Let 𝒞\mathcal{C}be an (∞,1)-category which satisfies the conditions of theorem 2, and let (H ,δ)(H^\bullet, \delta) be a generalized cohomology functor on 𝒞\mathcal{C}, def. 3. Then there exists a spectrum object EStab(𝒞)E \in Stab(\mathcal{C}) such that

  1. HH\bullet is degreewise represented by EE:

    H Ho(𝒞)(,E ), H^\bullet \simeq Ho(\mathcal{C})(-,E_\bullet) \,,
  2. the suspension isomorphism δ\delta is given by the structure morphisms σ˜ n:E nΩE n+1\tilde \sigma_n \colon E_n \to \Omega E_{n+1} of the spectrum, in that

    δ:H n()Ho(𝒞)(,E n)Ho(𝒞)(,σ˜ n)Ho(𝒞)(,ΩE n+1)Ho(𝒞)(Σ(),E n+1)H n+1(Σ()). \delta \colon H^n(-) \simeq Ho(\mathcal{C})(-,E_n) \stackrel{Ho(\mathcal{C})(-,\tilde\sigma_n) }{\longrightarrow} Ho(\mathcal{C})(-,\Omega E_{n+1}) \simeq Ho(\mathcal{C})(\Sigma (-), E_{n+1}) \simeq H^{n+1}(\Sigma(-)) \,.
Proof

Via prop. 3, theorem 2 gives the first clause. With this, the second clause follows by the (∞,1)-Yoneda lemma (in fact just with the Yoneda lemma).

Further variants

For triangulated categories and model categories

For triangulated categories a discussion is in (Neeman 96).

For model categories a discussion is in (Jardine 09):

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 morphisms.

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.

(Jardine 09)

This follows essentially immediately from Theorem \ref{Categorical} applied to Ho(C)Ho(C). Jardine also proves a more refined version.

For RO(G)RO(G)-graded equivariant cohomology

The generalization of the Brown representability theorem to equivariant cohomology says that RO(G)-graded equivariant cohomology is represented by genuine G-spectra (e.g. May 96, chapter XIII.3).

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)

A simplified version of the proof was spelled out in

  • Edwin Spanier, section 7.7 of Algebraic topology, McGraw-Hill, 1966

and a strengthening in

  • John Frank Adams, A variant of E. H. Brown’s representability theorem, Topology, 10:185-198, 1971

Textbook accounts include

Quick reviews include

Generalization to triangulated categories is discussed in

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

A model category 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

  • Rick Jardine, Representability theorems for simplicial presheaves, 2009 (pdf)

A version in (infinity,1)-category theory is in

Exposition of this is in

The case of RO(G)-graded equivariant cohomology theory is discussed for instance in

  • Peter May, chapter XIII.3 of Equivariant homotopy and cohomology theory, CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. (pdf)

The counterexamples to nonconnected and unpointed Brown representability are from

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

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 6, 2016 09:21:53 by Urs Schreiber (131.220.184.222)