A functor is said to reflect limits (colimits) of a given shape if a cone (cocone) is (co-)limiting whenever its image under is.
Let be a functor and a diagram. We say that reflects limits of if whenever we have a cone over in such that is a limit of in , then was already a limit of in .
Of course, a functor reflects a colimit if reflects the corresponding limit.
If reflects all limits or colimits of a given type (i.e. over a given category ), we simply say that reflects that sort of limit (e.g. reflects products, reflects equalizers, etc.).
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.
Reflection of limits is distinct from preservation of limits, although there are relationships, e.g prop. .
A faithful functor reflects epimorphisms and monomorphisms.
(The simple proof is spelled out here at epimorphism.)
A fully faithful functor (hence a full subcategory inclusion) reflects all limits and colimits.
This is evident from inspection of the defining universal property.
A conservative functor reflects any limits which exist in its domain and that it preserves.
If in def. has some limit which is preserved by , then there is a unique induced map by the universal property of a limit, which becomes an isomorphism in since and are both limits of ; hence if is conservative then it must already have been an isomorphism in , and so was already also a limit of .
reflected limit
Last revised on August 11, 2022 at 19:48:31. See the history of this page for a list of all contributions to it.