Let be a locally presentable (∞,1)-category. A functor
(from the opposite of the homotopy category of to Set)
is called a Brown functor if
it sends small coproducts to products;
it sends (∞,1)-pushouts in 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. says that for a (∞,1)-pushout square
in , then the induced universal morphism
into the actual pullback is an epimorphism.
(Brown representability theorem)
Let be a locally presentable (∞,1)-category.
If
is generated by a set
of compact objects (i.e. every object of is an (∞,1)-colimit of the objects .)
each admits the structure of a cogroup object in the homotopy category ,
then a functor
(from the opposite of the homotopy category of to Set)
is representable precisely if it is a Brown functor, def. .
A right Brown functor between categories of fibrant objects is a functor that preserves weak equivalences between fibrant objects.
Dually for a left Brown functor (for instance between Waldhausen categories).
Last revised on March 11, 2016 at 15:04:45. See the history of this page for a list of all contributions to it.