Contents

category theory

Contents

Idea

Every right adjoint functor $F\dashv G:\mathcal{B}\to\mathcal{A}$ yields by a classical result a monad on $\mathcal{A}$ with endofunctor $G\circ F$. The codensity monad $\mathbb{T}^G$ is a generalization of this monad to functors $G:\mathcal{B}\to\mathcal{A}$ merely admitting a right Kan extension $Ran_G G$ of $G$ along itself, with both monads coinciding in case $G:\mathcal{B}\to\mathcal{A}$ is a right adjoint.

The name ‘codensity monad’ stems from the fact that $\mathbb{T}^G$ reduces to the identity monad iff $G:\mathcal{B}\to\mathcal{A}$ is a codense functor. Thus, in general, the codensity monad “measures the failure of $G$ to be codense”.

Definition

Definition

Let $G:\mathcal{B}\to\mathcal{A}$ be a functor such that the right Kan extension $Ran_G G=(T^G,\;\alpha)$ of $G$ along itself exists with $\alpha :T^G\circ G\Rightarrow G$ the universal 2-cell of the functor $T^G:\mathcal{A}\to\mathcal{A}$. The codensity monad of $G$ is given by the monad

$\mathbb{T}^G:=\langle T^G:\mathcal{A}\to\mathcal{A},\;\eta^G:id_\mathcal{A}\Rightarrow T^G,\;\mu^G:T^G\circ T^G\Rightarrow T^G\rangle$

where the unit $\eta^G:id_\mathcal{A}\Rightarrow T^G$ is the natural transformation given by the universal property of $(T^G,\;\alpha)$ with respect to the pair $(id_\mathcal{A},\;1_G)\;$, whereas the multiplication $\mu^G:T^G\circ T^G\Rightarrow T^G$ results from the universal property of $(T^G,\;\alpha)$ with respect to the pair $(T^G\circ T^G,\;\alpha\circ (1_{T^G}\ast\alpha))$.

That this indeed defines a monad follows from the universal properties of the Kan extension. Concerning existence, $Ran_G G$ exists for $G:\mathcal{B}\to\mathcal{A}$ e.g. when $\mathcal{B}$ is small and $\mathcal{A}$ is complete.

In this circumstance, when $\mathcal{B}$ is small and $\mathcal{A}$ is complete, then the codensity monad is equivalently the one that arises from the adjunction

$\mathcal{A} \underoverset {\underset{}{\longleftarrow}} {\overset{hom(-,G)}{\longrightarrow}} {\bot} [\mathcal{B},Set]^{op}$

where the left adjoint $hom(-,G):\mathcal{A}\to [\mathcal{B},Set]^{op}$ takes an object $a$ to the functor $hom(a,G-):\mathcal{B}\to Set$. The right adjoint $[\mathcal{B},Set]^{op}\to \mathcal{A}$ is the canonical functor from the free completion of $\mathcal{B}$ to the category $\mathcal{A}$ which has limits. $G$ is codense if and only if the left adjoint is full and faithful.

Examples

• The Giry monad (as well as a finitely additive version) arise as codensity monads of forgetful functors from subcategories of the category of convex sets to the category of measurable spaces (Avery 14).

• The codensity monad of the inclusion FinSet $\hookrightarrow$Set is the ultrafilter monad.

Properties

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A very nice overview is provided by

Codensity monads arising from subcategory inclusions are studied in

• Ivan di Liberti, Codensity: Isbell duality, pro-objects, compactness and accessibility, arXiv:1910.01014 (2019). (abstract)

The role in shape theory is discussed in

• Armin Frei, On categorical shape theory , Cah. Top. Géom. Diff. XVII no.3 (1976) pp.261-294. (numdam)

• D. Bourn, J.-M. Cordier, Distributeurs et théorie de la forme, Cah. Top. Géom. Diff. Cat. 21 no.2 (1980) pp.161-189. (pdf)

• J.-M. Cordier, T. Porter, Shape Theory: Categorical Methods of Approximation , (1989), Mathematics and its Applications, Ellis Horwood. Reprinted Dover (2008).

For a description of the Giry monad as a codensity monad, see

Other references include

• C. Casacuberta, A. Frei, Localizations as idempotent approximations to completions , JPAA 142 (1999) no. 1 pp.25–33. (draft)

• Yves Diers, Complétion monadique , Cah. Top. Géom. Diff. Cat. XVII no.4 (1976) pp.362-379. (numdam)

• S. Katsumata, T. Sato, T. Uustala?, Codensity lifting of monads and its dual , arXiv:1810.07972 (2012). (abstract)

• J. Lambek, B. A. Rattray, Localization and Codensity Triples , Comm. Algebra 1 (1974) pp.145-164.