nLab codensity monad

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

• Every monad that is induced by an adjunction $L \dashv R$ is the codensity monad of $R$. In particular, every enriched monad is a codensity monad (via its Kleisli category).

• 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. Its algebras are compact Hausdorff spaces.

• The codensity monad of the inclusion $FinGrp \hookrightarrow$ Grp, is the profinite completion monad, whose algebras may be identified with profinite groups – that is, topological groups whose underlying topological space is profinite (Avery 17, Proposition 2.7.10).

• The codensity monad of the inclusion $FinSet \to Top$ computes the Stone spectrum? of the Boolean algebra of clopen subsets of a topological space. Its algebras are precisely the Stone spaces. (Sipoș, Theorem 2).

• The codensity monad of the inclusion $N \to Top$, where $N$ denotes the full subcategory of Top consisting of arbitrary small products of the Sierpiński space, is the localic spectrum? of the frame of opens of a topological space. Its algebras are precisely the sober spaces. (Sipoș, Theorem 6)

• The codensity monad of the inclusion of countable sets in all sets, $Ctbl \hookrightarrow Set$, assigns to each set $X$ the set of ultrafilters on $X$ closed under countable intersections. This still holds for the inclusion of the full subcategory of $Ctbl$ on the single set $\mathbb{N}$.

• More generally, the codensity monad of the inclusion of sets of cardinality less than that of fixed $Y$, $Set_{\lt Y} \hookrightarrow Set$, assigns to each set $X$ the set of $Y$-complete ultrafilters on $X$.

• For the codensity monad induced by the inclusion of homotopy types with finite homotopy groups into all homotopy types see there.

• The codensity monad induced by the Yoneda embedding is isomorphic to the monad induced by the Isbell adjunction.

Properties

….

Related entries

• dense

• dense functor

• shape theory

• Kan extension

• algebraic theory

• ultrafilter

• idempotent monad

Link

• nCafé blog 2012: Where do Monads come from?

References

One of the first references is

• Anders Kock, Continuous Yoneda Representations of a Small Category, Preprint Aarhus University (1966). (pdf)

A very nice overview is provided by

• Tom Leinster, Codensity and the Ultrafilter Monad , TAC 12 no.13 (2013) pp.332-370. (abstract)

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 and other probability monads as codensity monads, see

• Tom Avery, Codensity and the Giry monad, Journal of Pure and Applied Algebra 220 3 (2016) 1229-1251 [arXiv:1410.4432, doi:10.1016/j.jpaa.2015.08.017]

• Ruben Van Belle, Probability monads as codensity monads. Theory and Applications of Categories 38 (2022), 811–842, (tac)

Other references include

• Tom Avery, Structure and Semantics, (arXiv:1708.01050)

• 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. Uustalu, 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.

• Jiří Adámek, Lurdes Sousa?, D-Ultrafilters and their Monads, (arXiv:1909.04950)

• Andrei Sipos?, Codensity and Stone spaces, (arXiv:1409.1370)