Codensity monad
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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 composite , together with the standard unit and multiplication maps; however the codensity monad exists for functors not admitting a left adjoint.
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. This was proven by Kennison & Gildenhuys (1971), though without using the term "codensity". In this formulation, the statement is reviewed by Leinster (2013, §3).
- A related example is discussed by Leinster (2013, §7): 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**.
- Sipoş (2018) showed that the algebras over the codensity monad of the inclusion of finite sets (regarded as discrete topological spaces) into topological spaces are equivalent to Stone spaces.
Avery (2016) shows that the Giry monad arises as the codensity monad of natural forgetful functors between certain categories of convex vector spaces to measurable spaces.
See also
References
- 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
- Leinster, Tom (2013), "Codensity and the ultrafilter monad", Theory and Applications of Categories, 28: 332–370, arXiv:1209.3606, Bibcode:2012arXiv1209.3606L
- Kennison, J.F.; Gildenhuys, Dion (1971), "Equational completion, model induced triples and pro-objects", Journal of Pure and Applied Algebra, 1 (4): 317–346, doi:10.1016/0022-4049(71)90001-6
- Sipoş, Andrei (2018), "Codensity and stone spaces", Mathematica Slovaca, 68: 57–70, arXiv:1409.1370, doi:10.1515/ms-2017-0080