Codensity monad
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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 is defined to be the right Kan extension of along itself, provided that this Kan extension exists. Thus, by definition it is in particular a functor The monad structure on stems from the universal property of the right Kan extension.
The codensity monad exists whenever is a small category (has only a set, as opposed to a
By the general formula computing right Kan extensions in terms of ends, the codensity monad is given by the following formula:
where denotes the set of morphisms in between the indicated objects and the integral denotes the end. The codensity monad therefore amounts to considering maps from to an object in the image of and maps from the set of such morphisms to compatible for all the possible Thus, as is noted by Avery,
Examples
Codensity monads of right adjoints
If the functor admits a left adjoint the codensity monad is given by the composite together with the standard unit and multiplication maps.
Concrete examples for functors not admitting a left adjoint
In several interesting cases, the functor is an inclusion of a
A related example is discussed by Leinster:[4] the codensity monad of the inclusion of finite-dimensional vector spaces (over a fixed field ) into all vector spaces is the
Thus, in this example, the end formula mentioned above simplifies to considering (in the notation above) only one object namely a one-dimensional vector space, as opposed to considering all objects in Adámek and Sousa[5] show that, in a number of situations, the codensity monad of the inclusion of finitely presented objects (also known as compact objects) is a double dualization monad with respect to a sufficiently nice cogenerating object. This recovers both the inclusion of finite sets in sets (where a cogenerator is the set of two elements), and also the inclusion of finite-dimensional vector spaces in vector spaces (where the cogenerator is the ground field).
Sipoş showed that the
Relation to Isbell duality
Di Liberti[7] shows that the codensity monad is closely related to Isbell duality: for a given small category Isbell duality refers to the adjunction between the category of
See also
- Monadic functor– Operation in algebra and mathematics
References
- Di Liberti, Ivan (2019), Codensity: Isbell duality, pro-objects, compactness and accessibility, arXiv:1910.01014
- Leinster, Tom (2013). "Codensity and the ultrafilter monad" (PDF). Theory and Applications of Categories. 28: 332–370. Bibcode:2012arXiv1209.3606L.
Footnotes
- ^ .
- .
- ^ Leinster 2013, §3.
- ^ Leinster 2013, §7.
- arXiv:1909.04950.
- .
- ^ Di Liberti 2019.
- ^ Di Liberti 2019, §2.