Codensity monad

In mathematics, especially in category theory, the codensity monad is a fundamental construction associating a monad to a wide class of functors.

Definition

The codensity monad of a functor $G:D\to C$ is defined to be the right Kan extension of G along itself, provided that this Kan extension exists. Thus, by definition it is a functor, denoted $T^{G}:C\to C$. The monad structure on $T^{G}$ stems from the universal property of the right Kan extension. By the general formula computing right Kan extensions in terms of ends, the codensity monad is given by the following formula:

$T^{G}(c)=\int _{d\in D}G(d)^{C(c,G(d))},$

where $C(c,G(d))$ denotes the set of morphisms in C between the indicated objects and the integral denotes the end. The codensity monad therefore amounts to considering maps from c to an object in the image of G, and maps from set of such morphisms to G(d), compatible for all the possible d. Thus, as is noted by Avery (2016), codensity monads share some kinship with the concept of integration and double dualization.

Examples

Codensity monads of right adjoints

If the functor G admits a left adjoint F, the codensity monad is given by the composite $G\circ F$, together with the standard unit and multiplication maps.

Concrete examples for functors not admitting a left adjoint

In several interesting cases, the functor G is an inclusion of a full subcategory not admitting a left adjoint. For example, 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 k) into all vector spaces is the double dualization monad given by sending a vector space V to its double dual

$V^{**}=\operatorname {Hom} (\operatorname {Hom} (V,k),k).$

Thus, in this example, the end formula mentioned above simplifies to considering (in the notation above only one object d, namely a one-dimensional vector space), as opposed to considering all objects in D.

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

• Monadic functor

References

• Avery, Tom (2016), "Codensity and the Giry monad", Journal of Pure and Applied Algebra, 220 (3): 1229–1251, arXiv:1410.4432, 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