Editing Limit (category theory) (section)

===Lifting of limits===
A functor ''G'' : ''C'' → ''D'' is said to '''lift limits''' for a diagram ''F'' : ''J'' → ''C'' if whenever (''L'', ''φ'') is a limit of ''GF'' there exists a limit (''L''&prime;, ''φ''&prime;) of ''F'' such that ''G''(''L''&prime;, ''φ''&prime;) = (''L'', ''φ''). A functor ''G'' '''lifts limits of shape ''J''''' if it lifts limits for all diagrams of shape ''J''. One can therefore talk about lifting products, equalizers, pullbacks, etc. Finally, one says that ''G'' '''lifts limits''' if it lifts all limits. There are dual definitions for the lifting of colimits.

A functor ''G'' '''lifts limits uniquely''' for a diagram ''F'' if there is a unique preimage cone (''L''&prime;, ''φ''&prime;) such that (''L''&prime;, ''φ''&prime;) is a limit of ''F'' and ''G''(''L''&prime;, ''φ''&prime;) = (''L'', ''φ''). One can show that ''G'' lifts limits uniquely if and only if it lifts limits and is [[amnestic functor|amnestic]].

Lifting of limits is clearly related to preservation of limits. If ''G'' lifts limits for a diagram ''F'' and ''GF'' has a limit, then ''F'' also has a limit and ''G'' preserves the limits of ''F''. It follows that:
*If ''G'' lifts limits of all shape ''J'' and ''D'' has all limits of shape ''J'', then ''C'' also has all limits of shape ''J'' and ''G'' preserves these limits.
*If ''G'' lifts all small limits and ''D'' is complete, then ''C'' is also complete and ''G'' is continuous.
The dual statements for colimits are equally valid.