nLab
reflected limit

Reflection 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 reflects limits of JJ if whenever we have a cone η:const x IJ\eta\colon const^I_x \to J over JJ in CC such that F(η)F(\eta) is a limit of FJF\circ J in DD, then η\eta was already a limit of JJ in CC.

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

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

Remarks

Reflection of limits is distinct from preservation of limits, although there are relationships. For instance, a conservative functor reflects any limits which exist in its domain and that it preserves. For if JJ above has some limit θ\theta which is preserved by FF, then there is a unique induced map ηθ\eta\to\theta by the universal property of a limit, which becomes an isomorphism in DD since F(η)F(\eta) and F(θ)F(\theta) are both limits of FJF\circ J; hence if FF is conservative then it must already have been an isomorphism in CC, and so η\eta was already also a limit of JJ.

A functor which both reflects and preserves limits, and such that limits exist in its domain whenever they do in its codomain, is said to create them.

Revised on March 6, 2012 21:06:55 by David Corfield (81.155.107.228)