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.).

Examples

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.

Terminological remarks

  • The condition that FF preserves limits of JJ is often left out of the definition, instead strengthening the first condition to say that for any limiting cone LL of FJF\circ J there is a limiting cone LL' of JJ satisfying the additional condition that FLLF L' \cong L in the appropriate sense. The latter is certainly implied by preservation of limits, since limits are unique up to canonical isomorphism. But conversely, for the same reason, if FF preserves one limit of JJ then it must preserve all such limits. Thus the two definitions are equivalent if all limits of the relevant sort exist in DD.

  • The above modification is used in Categories Work, together with a more restrictive notion of “creation” that requires FL=LF L' = L on the nose, rather than merely up to isomorphism. (Of course, this violates the principle of equivalence.) This corresponds to a version of the monadicity theorem which asserts an isomorphism of categories, rather than merely an equivalence.

  • Kissinger suggested a concise way to state creation/preservation/etc. of limits. However, there is some dispute about its correctness.

Revised on August 28, 2017 18:58:07 by Mike Shulman (69.75.108.203)