nLab prorepresentable functor

A functor CSetC\to Set 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 CC’, and may use various descriptions via the (system of) representing objects in CC, but one does not usually say ‘prorepresentable functor in CC’.



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

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