Editing Brown's representability theorem

{{short description|On representability of a contravariant functor on the category of connected CW complexes}}
In mathematics, '''Brown's representability theorem''' in [[homotopy theory]]<ref>{{Citation | last1=Switzer | first1=Robert M. | title=Algebraic topology---homotopy and homology | publisher=Springer-Verlag | location=Berlin, New York | series=Classics in Mathematics | isbn=978-3-540-42750-6 | mr=1886843  | year=2002 | pages = 152–157}}</ref> gives [[necessary and sufficient condition]]s for a [[contravariant functor]] ''F'' on the [[homotopy category]] ''Hotc'' of pointed connected [[CW complex]]es, to the [[category of sets]] '''Set''', to be a [[representable functor]].

More specifically, we are given

:''F'': ''Hotc''<sup>op</sup> → '''Set''',

and there are certain obviously necessary conditions for ''F'' to be of type ''Hom''(&mdash;, ''C''), with ''C'' a pointed connected CW-complex that can be deduced from [[category theory]] alone.  The statement of the substantive part of the theorem is that these necessary conditions are then sufficient.  For technical reasons, the theorem is often stated for functors to the category of [[pointed set]]s; in other words the sets are also given a base point.

==Brown representability theorem for CW complexes==

The representability theorem for CW complexes, due to [[Edgar H. Brown]],<ref>{{Citation | last1=Brown | jstor=1970209| first1=Edgar H. | title=Cohomology theories | mr=0138104  | year=1962 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=75 | issue=3| pages=467–484 | doi=10.2307/1970209 }}</ref> is the following.  Suppose that:
# The functor ''F'' maps [[coproducts]] (i.e. [[wedge sum]]s) in ''Hotc'' to products in ''Set'': <math>F(\vee_\alpha X_\alpha) \cong \prod_\alpha F(X_\alpha),</math>
# The functor ''F'' maps [[Mapping cylinder|homotopy pushouts]] in ''Hotc'' to [[weak pullbacks]].  This is often stated as a [[Mayer–Vietoris sequence|Mayer–Vietoris]] axiom: for any CW complex ''W'' covered by two subcomplexes ''U'' and ''V'', and any elements ''u'' ∈ ''F''(''U''), ''v'' ∈ ''F''(''V'') such that ''u'' and ''v'' restrict to the same element of ''F''(''U'' ∩ ''V''), there is an element ''w'' ∈ ''F''(''W'') restricting to ''u'' and ''v'', respectively.
Then ''F'' is representable by some CW complex ''C'', that is to say there is an isomorphism 
:''F''(''Z'') ≅ ''Hom''<sub>''Hotc''</sub>(''Z'', ''C'')
for any CW complex ''Z'', which is [[Natural transformation|natural]] in ''Z'' in that for any morphism from ''Z'' to another CW complex ''Y'' the induced maps ''F''(''Y'') → ''F''(''Z'') and ''Hom''<sub>''Hot''</sub>(''Y'', ''C'') → ''Hom''<sub>''Hot''</sub>(''Z'', ''C'') are compatible with these isomorphisms.

The converse statement also holds: any functor represented by a CW complex satisfies the above two properties. This direction is an immediate consequence of basic category theory, so the deeper and more interesting part of the equivalence is the other implication.

The representing object ''C'' above can be shown to depend functorially on ''F'': any [[natural transformation]] from ''F'' to another functor satisfying the conditions of the theorem necessarily induces a map of the representing objects. This is a consequence of [[Yoneda's lemma]].

Taking ''F''(''X'') to be the [[singular cohomology]] group ''H''<sup>''i''</sup>(''X'',''A'') with coefficients in a given abelian group ''A'', for fixed ''i'' > 0; then the representing space for ''F'' is the [[Eilenberg–MacLane space]] ''K''(''A'', ''i''). This gives a means of showing the existence of Eilenberg-MacLane spaces.

==Variants==

Since the homotopy category of CW-complexes is equivalent to the localization of the category of all topological spaces at the [[weak homotopy equivalence]]s, the theorem can equivalently be stated for functors on a category defined in this way.

However, the theorem is false without the restriction to ''connected'' pointed spaces, and an analogous statement for unpointed spaces is also false.<ref>{{Citation
 | last1 = Freyd  | first1 = Peter
 | last2 = Heller | first2 = Alex
 | title = Splitting homotopy idempotents. II.
 | journal = [[Journal of Pure and Applied Algebra]]
 | volume = 89 | issue = 1–2 | pages = 93–106
 | year = 1993
 | doi = 10.1016/0022-4049(93)90088-b
| doi-access = free
 }}</ref>

A similar statement does, however, hold for [[spectrum (homotopy theory)|spectra]] instead of CW complexes.  Brown also proved a general categorical version of the representability theorem,<ref>{{Citation | last1=Brown | first1=Edgar H. | title=Abstract homotopy theory | year=1965 | journal=[[Transactions of the American Mathematical Society]] | volume=119 | issue=1| pages=79–85 | url=https://www.ams.org/journals/tran/1965-119-01/S0002-9947-1965-0182970-6/ | doi=10.2307/1994231| doi-access=free | jstor=1994231 }}</ref> which includes both the version for pointed connected CW complexes and the version for spectra.

A version of the representability theorem in the case of [[triangulated category|triangulated categories]] is due to Amnon Neeman.<ref>{{Citation | last1=Neeman | first1=Amnon | title=The Grothendieck duality theorem via Bousfield's techniques and Brown representability | url=https://www.ams.org/jams/1996-9-01/S0894-0347-96-00174-9/home.html | mr=1308405  | year=1996 | journal=[[Journal of the American Mathematical Society]] | issn=0894-0347 | volume=9 | issue=1 | pages=205–236 | doi=10.1090/S0894-0347-96-00174-9| doi-access=free }}</ref> Together with the preceding remark, it gives a criterion for a (covariant) functor ''F'': ''C'' → ''D'' between triangulated categories satisfying certain technical conditions to have a right [[adjoint functor]]. Namely, if ''C'' and ''D'' are triangulated categories with ''C'' compactly generated and ''F'' a triangulated functor commuting with arbitrary direct sums, then ''F'' is a left adjoint. Neeman has applied this to proving the [[Coherent duality|Grothendieck duality theorem]] in algebraic geometry.

[[Jacob Lurie]] has proved a version of the Brown representability theorem<ref>{{Citation | last1=Lurie | first1=Jacob | title=Higher Algebra | year=2011 | url=http://math.harvard.edu/~lurie/papers/higheralgebra.pdf | url-status=dead | archiveurl=https://web.archive.org/web/20110609013026/http://www.math.harvard.edu/~lurie/papers/higheralgebra.pdf | archivedate=2011-06-09 }}</ref> for the homotopy category of a pointed [[quasicategory]] with a compact set of generators which are cogroup objects in the homotopy category. For instance, this applies to the homotopy category of pointed connected CW complexes, as well as to the unbounded [[derived category]] of a [[Grothendieck category|Grothendieck abelian category]] (in view of Lurie's higher-categorical refinement of the derived category).

==References==
{{Reflist|colwidth=30em}}

[[Category:Category theory]]
[[Category:Representable functors]]
[[Category:Theorems in homotopy theory]]