Editing Brown's representability theorem (section)

==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).