Limit (category theory)

The dual notions of limits and cones are colimits and co-cones. Although it is straightforward to obtain the definitions of these by inverting all morphisms in the above definitions, we will explicitly state them here:

Examples of colimits are given by the dual versions of the examples above:

Limits and colimits are important special cases of universal constructions.

Therefore, the definitions of limits and colimits can then be restated in the form:

An important property of adjoint functors is that every right adjoint functor is continuous and every left adjoint functor is cocontinuous. Since adjoint functors exist in abundance, this gives numerous examples of continuous and cocontinuous functors.

There are examples of functors which lift limits uniquely but neither create nor reflect them.

Older terminology referred to limits as "inverse limits" or "projective limits," and to colimits as "direct limits" or "inductive limits." This has been the source of a lot of confusion.

There are several ways to remember the modern terminology. First of all,