A functor is prorepresentable (or pro-representable) if it is a small filtered colimit of representables. In other words, it corresponds to a pro-object.
As these functors are ‘exactly’ the left exact functors, (at least with a caveat on size), that latter term can also be used, but there are some standard situations and conventions when this ‘pro-’ terminology is used.
One tends to say ‘pro-object in ’, and may use various descriptions via the (system of) representing objects in , but one does not usually say ‘prorepresentable functor in ’.
A. Grothendieck, M. Raynaud et al. Revêtements étales et groupe fondamental (SGA I), Lecture Notes in Mathematics 224, Springer 1971 (retyped as math.AG/0206203; published version Documents Mathématiques 3, Société Mathématique de France, Paris 2003)
M. Artin and B. Mazur, Étale homotopy theory, 1969, No. 100 in Lecture Notes in Maths., Springer-Verlag, Berlin.
For some applications see
Alexander I. Efimov, Valery Lunts?, Dmitri O. Orlov, Deformation theory of objects in homotopy and derived categories II: Pro-representability of the deformation functor, doi
Revised on October 26, 2015 15:52:52
by Tim Porter