codensity monad




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

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



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

𝕋 G:=T G:𝒜𝒜,η G:id 𝒜T G,μ G:T GT GT G\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 η G:id 𝒜T G\eta^G:id_\mathcal{A}\Rightarrow T^G is the natural transformation given by the universal property of (T G,α)(T^G,\;\alpha) with respect to the pair (id 𝒜,1 G)(id_\mathcal{A},\;1_G)\;, whereas the multiplication μ G:T GT GT G\mu^G:T^G\circ T^G\Rightarrow T^G results from the universal property of (T G,α)(T^G,\;\alpha) with respect to the pair (T GT G,α(1 T G*α))(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 GGRan_G G exists for G:𝒜G:\mathcal{B}\to\mathcal{A} e.g. when \mathcal{B} is small and 𝒜\mathcal{A} is complete.

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

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

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





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

  • 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. 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.

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

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

Last revised on March 5, 2021 at 18:25:04. See the history of this page for a list of all contributions to it.