group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
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 $E^\bullet$ is representable by a spectrum $E$ as
(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 $E_\bullet$ is representable by a spectrum $E$ via
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
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.
Let $Ho(Top_*^c)$ 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 $Ho(Top_*^c)$ has coproducts (given by wedge sums) and also weak pushouts (namely, homotopy pushouts).
(Brown-Adams)
A functor $F:Ho(Top_*^c)^{op} \to Set_*$ is representable precisely if
it takes coproducts to products,
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:
(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 $G$ 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 $Ho(Top)$ on connected homotopy 1-types, we have an endomorphism $B g:B G \to B G$ of the classifying space of $G$ which does not split in $Ho(Top)$.
Thus, if $F:Ho(Top)^{op} \to Set$ splits the idempotent $[-,B g]$ of $[-,B G]$, 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 $Ho(Top_*)$.
There is also another example due to Heller, which fais to be representable for cardinality reasons.
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 $L_{whe} Top\simeq$ ∞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).
Let $\mathcal{C}$ be a locally presentable (∞,1)-category. A functor
(from the opposite of the homotopy category of $\mathcal{C}$ to Set)
is called a Brown functor if
it sends small coproducts to products;
it sends (∞,1)-pushouts in $\mathcal{C}\to Ho(\mathcal{C})$ to weak pullbacks in Set.
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
in $\mathcal{C}$, then the induced universal morphism
into the actual pullback is an epimorphism.
Say that a locally presentable (∞,1)-category $\mathcal{C}$ is compactly generated by cogroup objects closed under suspensions if
$\mathcal{C}$ is generated by a set
of compact objects (i.e. every object of $\mathcal{C}$ is an (∞,1)-colimit of the objects $S_i$.)
each $S_i$ admits the structure of a cogroup object in the homotopy category $Ho(\mathcal{C})$;
the set $\{S_i\}$ is closed under forming reduced suspensions.
(suspensions are H-cogroup objects)
Let $\mathcal{C}$ be an (∞,1)-category with finite (∞,1)-colimits and with a zero object. Write $\Sigma \colon X \mapsto 0 \underset{X}{\coprod} 0$ for the reduced suspension functor.
Then the fold map
exhibits cogroup structure on the image of any suspension object $\Sigma X$ in the homotopy category.
This is equivalently the group-structure of the first (fundamental) homotopy group of the values of functor co-represented by $\Sigma X$:
In bare pointed homotopy types $\mathcal{C} =$∞Grpd${}^{\ast/}$, the (homotopy types of) n-spheres $S^n$ are cogroup objects for $n \geq 1$, but not for $n = 0$, by example 1. And of course they are compact objects.
So while $\{S^n\}_{n \in \mathbb{N}}$ generates all of $\infty Grpd^{\ast/}$, the latter is not an example of def. 2 due to the failure of $S^0$ to have cogroup structure.
Removing that generator, the $(\infty,1)$-category generated by $\{S^n\}_{{n \in \mathbb{N}} \atop {n \geq 1}}$ is $\infty Grpd^{\ast/}_{\geq 1}$, 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.
In an $(\infty,1)$-category compactly generated by cogroup objects $\{S_i\}_{i \in I}$ closed under forming suspensions, according to def. 2, a morphism $f\colon X \longrightarrow Y$ is an equivalence precisely if for each $i \in I$ the induced function of maps in the homotopy category
is an isomorphism (a bijection).
By the (∞,1)-Yoneda lemma, the morphism $f$ is an equivalence precisely if for all objects $A \in \mathcal{C}$ the induced morphism
is an equivalence in ∞Grpd. By assumption of compact generation and since the hom-functor $\mathcal{C}(-,-)$ sends $\infty$-colimits in the first argument to $\infty$-limits, this is the case precisely already if it is the case for $A \in \{S_i\}_{i \in I}$.
Now by the standard Whitehead theorem in ∞Grpd (being a hypercomplete (∞,1)-topos), the morphisms
in ∞Grpd are equivalences precisely if they are weak homotopy equivalences, hence precisely if they induce isomorphisms on all homotopy groups $\pi_n$ for all basepoints.
It is this last condition of testing on all basepoints that the assumed cogroup structure on the $S_i$ allows to do away with: this cogroup structure implies that $\mathcal{C}(S_i,-)$ has the structure of an $H$-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 $\pi_n$ based on the connected component of the zero morphism
Now in this pointed situation we may use that
to find that $f$ is an equivalence in $\mathcal{C}$ precisely if the induced morphisms
are isomorphisms for all $i \in I$ and $n \in \mathbb{N}$.
Finally by the assumption that each suspension $\Sigma^n S_i$ of a generator is itself among the set of generators, the claim follows.
(Brown representability)
Let $\mathcal{C}$ be an $(\infty,1)$-category compactly generated by cogroup objects closed under forming suspensions, according to def. 2. Then a functor
(from the opposite of the homotopy category of $\mathcal{C}$ to Set) is representable precisely if it is a Brown functor, def. 1.
Due to the version of the Whitehead theorem of prop. 1 we are essentially reduced to showing that Brown functors $F$ are representable on the $S_i$. To that end consider the following lemma. (In the following we notationally identify, via the Yoneda lemma, objects of $\mathcal{C}$, hence of $Ho(\mathcal{C})$, with the functors they represent.)
Lemma ($\star$): Given $X \in \mathcal{C}$ and $\eta \in F(X)$, hence $\eta \colon X \to F$, then there exists a morphism $f \colon X \to X'$ and an extension $\eta' \colon X' \to F$ of $\eta$ which induces for each $S_i$ a bijection $\eta'\circ (-) \colon Ho(\mathcal{C})(S_i,X') \stackrel{\simeq}{\longrightarrow} PSh(Ho(\mathcal{C}))(S_i,F) \simeq F(S_i)$.
To see this, first notice that we may directly find an extension $\eta_0$ along a map $X\to X_o$ such as to make a surjection: simply take $X_0$ to be the coproduct of all possible elements in the codomain and take
to be the canonical map. (Using that $F$, by assumption, turns coproducts into products, we may indeed treat the coproduct in $\mathcal{C}$ 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 $X_0$. To that end proceed by induction and suppose that $\eta_n \colon X_n \to F$ has been constructed. Then for $i \in I$ let
be the kernel of $\eta_n$ evaluated on $S_i$. These $K_i$ are the pieces that need to go away in order to make a bijection. Hence define $X_{n+1}$ to be their joint homotopy cofiber
Then by the assumption that $F$ takes this homotopy cokernel to a weak fiber (as in remark 2), there exists an extension $\eta_{n+1}$ of $\eta_n$ along $X_n \to X_{n+1}$:
It is now clear that we want to take
and extend all the $\eta_n$ to that colimit. Since we have no condition for evaluating $F$ on colimits other than pushouts, observe that this sequential colimit is equivalent to the following pushout:
where the components of the top and left map alternate between the identity on $X_n$ and the above successor maps $X_n \to X_{n+1}$. Now the excision property of $F$ applies to this pushout, and we conclude the desired extension $\eta' \colon X' \to F$:
It remains to confirm that this indeed gives the desired bijection. Surjectivity is clear. For injectivity use that all the $S_i$ are, by assumption, compact, hence they may be taken inside the sequential colimit:
With this, injectivity follows because by construction we quotiented out the kernel at each stage. Because suppose that $\gamma$ is taken to zero in $F(S_i)$, then by the definition of $X_{n+1}$ above there is a factorization of $\gamma$ through the point:
This concludes the proof of Lemma ($\star$).
Now apply the construction given by this lemma to the case $X_0 \coloneqq 0$ and the unique $\eta_0 \colon 0 \stackrel{\exists !}{\to} F$. Lemma $(\star)$ then produces an object $X'$ which represents $F$ on all the $S_i$, and we want to show that this $X'$ actually represents $F$ generally, hence that for every $Y \in \mathcal{C}$ the function
is a bijection.
First, to see that $\theta$ is surjective, we need to find a preimage of any $\rho \colon Y \to F$. Applying Lemma $(\star)$ to $(\eta',\rho)\colon X'\sqcup Y \longrightarrow F$ we get an extension $\kappa$ of this through some $X' \sqcup Y \longrightarrow Z$ and the morphism on the right of the following commuting diagram:
Moreover, Lemma $(\star)$ gives that evaluated on all $S_i$, the two diagonal morphisms here become isomorphisms. But then prop. 1 implies that $X' \longrightarrow Z$ is in fact an equivalence. Hence the component map $Y \to Z \simeq Z$ is a lift of $\kappa$ through $\theta$.
Second, to see that $\theta$ is injective, suppose $f,g \colon Y \to X'$ have the same image under $\theta$. Then consider their homotopy pushout
along the codiagonal of $Y$. Using that $F$ sends this to a weak pullback by assumption, we obtain an extension $\bar \eta$ of $\eta'$ along $X' \to Z$. Applying Lemma $(\star)$ to this gives a further extension $\bar \eta' \colon Z' \to Z$ which now makes the following diagram
such that the diagonal maps become isomorphisms when evaluated on the $S_i$. As before, it follows via prop. 1 that the morphism $h \colon X' \longrightarrow Z'$ is an equivalence.
Since by this construction $h\circ f$ and $h\circ g$ are homotopic
it follows with $h$ being an equivalence that already $f$ and $g$ were homotopic, hence that they represented the same element.
Let $\mathcal{C}$ be an (∞,1)-category with (∞,1)-pushouts, and with a zero object $0 \in \mathcal{C}$. Write $\Sigma \colon \mathcal{C} \to \mathcal{C}\colon X\mapsto 0 \underset{X}{\sqcup} 0$ for the corresponding suspension (∞,1)-functor.
A reduced additive generalized (Eilenberg-Steenrod) cohomology theory on $\mathcal{C}$ is
a functor
(from the opposite of the homotopy category of $\mathcal{C}$ into $\mathbb{Z}$-graded abelian groups);
a natural isomorphisms (“suspension isomorphisms”) of degree +1
such that $H^\bullet$
(exactness) takes homotopy cofiber sequences to exact sequences.
(additivity) takes small coproducts to products;
Given a generalized cohomology theory $(H^\bullet,\delta)$ on some $\mathcal{C}$ as in def. 3, and given a homotopy cofiber sequence in $\mathcal{C}$
then the corresponding connecting homomorphism is the composite
The connecting homomorphisms of def. 4 are parts of long exact sequences
By the defining exactness of $H^\bullet$, def. 3, and the way this appears in def. 4, using that $\delta$ is by definition an isomorphism.
Given a reduced addive generalized cohomology functor $H^\bullet \colon Ho(\mathcal{C})^{op}\to Ab^{\mathbb{Z}}$, def. 3, its underlying Set-valued functors $H^n \colon Ho(\mathcal{C})^{op}\to Ab\to Set$ are Brown functors, def. 1.
The first condition on a Brown functor holds by additivity. For the second condition, given a homotopy pushout square
in $\mathcal{C}$, consider the induced morphism of the long exact sequences given by prop. 2
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.
Let $\mathcal{C}$be an (∞,1)-category which satisfies the conditions of theorem 2, and let $(H^\bullet, \delta)$ be a generalized cohomology functor on $\mathcal{C}$, def. 3. Then there exists a spectrum object $E \in Stab(\mathcal{C})$ such that
$H\bullet$ is degreewise represented by $E$:
the suspension isomorphism $\delta$ is given by the structure morphisms $\tilde \sigma_n \colon E_n \to \Omega E_{n+1}$ of the spectrum, in that
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).
For triangulated categories a discussion is in (Neeman 96).
For model categories a discussion is in (Jardine 09):
Let $C$ be a 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 morphisms.
(Jardine)
Let $F : C^{op} \to Set_{*}$ be a pointed homotopical functor from $C^{op}$ to the category of pointed sets such that
$F$ preserves small coproducts of cofibrant objects (including preserving the zero object).
(Mayer-Vietoris property) $F$ takes any pushout diagram
with all objects cofibrant and $i$ a cofibration, to a weak pullback.
Then $F$ is representable.
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).
The original theorem was proven in
The categorical generalization was proven in
A simplified version of the proof was spelled out in
and a strengthening in
Textbook accounts include
Frank Adams, part III, section 6 of Stable homotopy and generalised homology, 1974
Robert Switzer, theorem 8.42, theorem 9.27 in Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.
Stanley Kochmann, section 3.4 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Marcelo Aguilar, Samuel Gitler, Carlos Prieto, section 12 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)
Dai Tamaki, Akira Kono, chapter 2.5 Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (pdf)
Quick reviews include
Fernando Muro, Representability of cohomology theories, 2010 (pdf)
Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 17 Phantom maps (pdf)
Generalization to triangulated categories is discussed in
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
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
The counterexamples to nonconnected and unpointed Brown representability are from
The relation to Grothendieck contexts (six operations) is highlighted in