Codensity monad

The codensity monad of a functor ${\displaystyle G:D\to C}$ is defined to be the right Kan extension of ${\displaystyle G}$ along itself, provided that this Kan extension exists. Thus, by definition it is in particular a functor

$${\displaystyle T^{G}:C\to C.}$$

The monad structure on ${\displaystyle T^{G}}$ stems from the universal property of the right Kan extension.

The codensity monad exists whenever ${\displaystyle D}$ is a small category (has only a set, as opposed to a proper class, of morphisms) and ${\displaystyle C}$ possesses all (small, i.e., set-indexed) limits. It also exists whenever ${\displaystyle G}$ has a left adjoint.

By the general formula computing right Kan extensions in terms of ends, the codensity monad is given by the following formula:

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

where ${\displaystyle C(c,G(d))}$ denotes the set of morphisms in ${\displaystyle C}$ between the indicated objects and the integral denotes the end. The codensity monad therefore amounts to considering maps from ${\displaystyle c}$ to an object in the image of ${\displaystyle G,}$ and maps from the set of such morphisms to ${\displaystyle G(d),}$ compatible for all the possible ${\displaystyle d.}$ Thus, as is noted by Avery,[1] codensity monads share some kinship with the concept of integration and double dualization.

Di Liberti[7] shows that the codensity monad is closely related to Isbell duality: for a given small category ${\displaystyle C,}$ Isbell duality refers to the adjunction

$${\displaystyle {\mathcal {O}}:Set^{C^{op}}\rightleftarrows (Set^{C})^{op}:Spec}$$

between the category of presheaves on ${\displaystyle C}$ (that is, functors from the opposite category of ${\displaystyle C}$ to sets) and the opposite category of copresheaves on ${\displaystyle C.}$ The monad

$${\displaystyle Spec\circ {\mathcal {O}}}$$

induced by this adjunction is shown to be the codensity monad of the Yoneda embedding

$${\displaystyle y:C\to Set^{C^{op}}.}$$

Conversely, the codensity monad of a full small dense subcategory ${\displaystyle K}$ in a cocomplete category ${\displaystyle C}$ is shown to be induced by Isbell duality.[8]

• Monadic functor – Operation in algebra and mathematicsPages displaying short descriptions of redirect targets

• 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. arXiv:1209.3606. Bibcode:2012arXiv1209.3606L.

Footnotes