Special and general types
(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
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 is representable by a spectrum 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 is representable by a spectrum 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 18.104.22.168). 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,
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 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 has coproducts (given by wedge sums) and also weak pushouts (namely, homotopy pushouts).
(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 (with ) and any elements , such that , there exists such that and .
For homotopy functors on presentable -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 ∞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).
(suspensions are H-cogroup objects)
Let be an (∞,1)-category with finite (∞,1)-colimits and with a zero object. Write for the reduced suspension functor.
Then the fold map
exhibits cogroup structure on the image of any suspension object in the homotopy category.
This is equivalently the group-structure of the first (fundamental) homotopy group of the values of functor co-represented by :
In bare pointed homotopy types ∞Grpd, the (homotopy types of) n-spheres are cogroup objects for , but not for , by example 1. And of course they are compact objects.
So while generates all of , the latter is not an example of def. 2 due to the failure of to have cogroup structure.
Removing that generator, the -category generated by is , 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 22.214.171.124)
In -categories compactly generated by cogroup objects closed under forming suspensions, the following strenghtening of the Whitehead theorem holds.
In an -category compactly generated by cogroup objects closed under forming suspensions, according to def. 2, a morphism is an equivalence precisely if for each the induced function of maps in the homotopy category
is an isomorphism (a bijection).
(Lurie, p. 114, Lemma star)
By the (∞,1)-Yoneda lemma, the morphism is an equivalence precisely if for all objects the induced morphism
is an equivalence in ∞Grpd. By assumption of compact generation and since the hom-functor sends -colimits in the first argument to -limits, this is the case precisely already if it is the case for .
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 for all basepoints.
It is this last condition of testing on all basepoints that the assumed cogroup structure on the allows to do away with: this cogroup structure implies that has the structure of an -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 based on the connected component of the zero morphism
Now in this pointed situation we may use that
to find that is an equivalence in precisely if the induced morphisms
are isomorphisms for all and .
Finally by the assumption that each suspension of a generator is itself among the set of generators, the claim follows.
Let be an -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 to Set) is representable precisely if it is a Brown functor, def. 1.
(Lurie, theorem 126.96.36.199)
Due to the version of the Whitehead theorem of prop. 1 we are essentially reduced to showing that Brown functors are representable on the . To that end consider the following lemma. (In the following we notationally identify, via the Yoneda lemma, objects of , hence of , with the functors they represent.)
Lemma (): Given and , hence , then there exists a morphism and an extension of which induces for each a bijection .
To see this, first notice that we may directly find an extension along a map such as to make a surjection: simply take to be the coproduct of all possible elements in the codomain and take
to be the canonical map. (Using that , by assumption, turns coproducts into products, we may indeed treat the coproduct in 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 . To that end proceed by induction and suppose that has been constructed. Then for let
be the kernel of evaluated on . These are the pieces that need to go away in order to make a bijection. Hence define to be their joint homotopy cofiber
Then by the assumption that takes this homotopy cokernel to a weak fiber (as in remark 2), there exists an extension of along :
It is now clear that we want to take
and extend all the to that colimit. Since we have no condition for evaluating 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 and the above successor maps . Now the excision property of applies to this pushout, and we conclude the desired extension :
It remains to confirm that this indeed gives the desired bijection. Surjectivity is clear. For injectivity use that all the 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 is taken to zero in , then by the definition of above there is a factorization of through the point:
This concludes the proof of Lemma ().
Now apply the construction given by this lemma to the case and the unique . Lemma then produces an object which represents on all the , and we want to show that this actually represents generally, hence that for every the function
is a bijection.
First, to see that is surjective, we need to find a preimage of any . Applying Lemma to we get an extension of this through some and the morphism on the right of the following commuting diagram:
Moreover, Lemma gives that evaluated on all , the two diagonal morphisms here become isomorphisms. But then prop. 1 implies that is in fact an equivalence. Hence the component map is a lift of through .
Second, to see that is injective, suppose have the same image under . Then consider their homotopy pushout
along the codiagonal of . Using that sends this to a weak pullback by assumption, we obtain an extension of along . Applying Lemma to this gives a further extension which now makes the following diagram
such that the diagonal maps become isomorphisms when evaluated on the . As before, it follows via prop. 1 that the morphism is an equivalence.
Since by this construction and are homotopic
it follows with being an equivalence that already and were homotopic, hence that they represented the same element.
Given a generalized cohomology theory on some as in def. 3, and given a homotopy cofiber sequence in
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 , def. 3, and the way this appears in def. 4, using that is by definition an isomorphism.
Given a reduced addive generalized cohomology functor , def. 3, its underlying Set-valued functors 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 , 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 be an (∞,1)-category which satisfies the conditions of theorem 2, and let be a generalized cohomology functor on , def. 3. Then there exists a spectrum object such that
is degreewise represented by :
the suspension isomorphism is given by the structure morphisms of the spectrum, in that
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 be a model category with a zero object and such that there is a set of cofibrant compact objects such that the weak equivalences in are precisely the -local morphisms.
(Jardine 09, p. 20)
(Jardine 09, theorem 24)
For -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).
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
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
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)