nLab
created limit

Creation of limits

Definition

Let F:CDF\colon C\to D be a functor and J:ICJ\colon I\to C a diagram. We say that FF creates limits for JJ if JJ has a limit whenever the composite FJF\circ J has a limit, and FF both preserves and reflects limits of JJ. This means that, in addition to JJ having a limit whenever FJF \circ J does, a cone over JJ in CC is a limiting cone if and only if its image in DD is a limiting cone over FJF\circ J.

Of course, a functor FF creates a colimit if F opF^{op} creates the corresponding limit.

If FF creates all limits or colimits of a given type (i.e. over a given category II), we simply say that FF creates that sort of limit (e.g. FF creates products, FF creates equalizers, etc.).

Remarks

A monadic functor creates all limits that exist in its codomain, and all colimits that exist in its codomain and are preserved by the corresponding monad (or, equivalently, by the monadic functor itself). Creation of a particular sort of split coequalizer figures prominently in Beck’s monadicity theorem.

One should beware that in Categories Work, a more restrictive notion of “creation” is used which requires the every limit in DD to lift to one in CC uniquely on the nose, rather than merely up to isomorphism. This corresponds to a version of the monadicity theorem which asserts an isomorphism of categories, rather than merely an equivalence.

Revised on November 2, 2012 23:43:38 by Sridhar Ramesh (98.248.229.177)