Codensity monad

In mathematics, especially in category theory, the codensity monad is a fundamental construction associating a monad to a functor. If the functor G in question admits a left adjoint F, the codensity monad is given by the usual unit map $id\to G\circ F$, but the codensity monad exists for functors not admitting a left ajoint.

Examples

In several interesting cases, the functor G is an inclusion of a full subcategory. Such examples include:

• The codensity monad of the inclusion of FinSet into Set is the ultrafilter monad associating to any set M the set of ultrafilters on M.

• The codensity monad of the inclusion of finite-dimensional vector spaces (over a fixed field) into all vector spaces is the double dualization monad given by sending a vector space V to its double dual V^**.

See also

• Monadic functor

References

• Leinster, Tom (2013), "Codensity and the ultrafilter monad", Theory and Applications of Categories, 28: 332–370, arXiv:1209.3606, Bibcode:2012arXiv1209.3606L

• Sipoş, Andrei (2018), "Codensity and stone spaces", Mathematica Slovaca, 68: 57–70, arXiv:1409.1370, doi:10.1515/ms-2017-0080

• Avery, Tom (2016), "Codensity and the Giry monad", Journal of Pure and Applied Algebra, 220 (3): 1229–1251, doi:10.1016/j.jpaa.2015.08.017