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In mathematics, especially in
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, the codensity monad is a fundamental construction associating a
monad Monad may refer to: Philosophy * Monad (philosophy), a term meaning "unit" **Monism, the concept of "one essence" in the metaphysical and theological theory ** Monad (Gnosticism), the most primal aspect of God in Gnosticism * ''Great Monad'', an ...
to a wide class of
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
s.


Definition

The codensity monad of a functor G: D \to C is defined to be the right Kan extension of G along itself, provided that this Kan extension exists. Thus, by definition it is in particular a functor T^G : C \to C. The monad structure on T^G stems from the universal property of the right Kan extension. The codensity monad exists whenever D is a small category (has only a set, as opposed to a
proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
, of morphisms) and C possesses all (small, i.e., set-indexed) limits. It also exists whenever G has a left adjoint. By the general formula computing right Kan extensions in terms of
ends End, END, Ending, or variation, may refer to: End *In mathematics: **End (category theory) ** End (topology) **End (graph theory) ** End (group theory) (a subcase of the previous) **End (endomorphism) *In sports and games ** End (gridiron footbal ...
, the codensity monad is given by the following formula: T^G(c) = \int_ G(d)^, where C(c, G(d)) denotes the set of
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s in C between the indicated objects and the integral denotes the end. The codensity monad therefore amounts to considering maps from c to an object in the image of G, and maps from the set of such morphisms to G(d), compatible for all the possible d. Thus, as is noted by Avery, codensity monads share some kinship with the concept of
integration Integration may refer to: Biology *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technology, ...
and double dualization.


Examples


Codensity monads of right adjoints

If the functor G admits a left adjoint F, the codensity monad is given by the composite G \circ F, together with the standard unit and multiplication maps.


Concrete examples for functors not admitting a left adjoint

In several interesting cases, the functor G is an inclusion of a
full subcategory In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, ...
not admitting a left adjoint. For example, the codensity monad of the inclusion of
FinSet In the mathematical field of category theory, FinSet is the category whose objects are all finite sets and whose morphisms are all functions between them. FinOrd is the category whose objects are all finite ordinal numbers and whose morphisms are a ...
into
Set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
is the ultrafilter monad associating to any set M the set of
ultrafilter In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter on ...
s on M. This was proven by Kennison and Gildenhuys, though without using the term "codensity". In this formulation, the statement is reviewed by Leinster. A related example is discussed by Leinster: the codensity monad of the inclusion of finite-dimensional vector spaces (over a fixed field k) into all vector spaces is the double dualization monad given by sending a vector space V to its
double dual In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
V^ = \operatorname(\operatorname(V, k), k). Thus, in this example, the end formula mentioned above simplifies to considering (in the notation above) only one object d, namely a one-dimensional vector space, as opposed to considering all objects in D. Adámek show that, in a number of situations, the codensity monad of the inclusion D := C^ \subseteq C of finitely presented objects (also known as
compact object In astronomy, the term compact star (or compact object) refers collectively to white dwarfs, neutron stars, and black holes. It would grow to include exotic stars if such hypothetical, dense bodies are confirmed to exist. All compact objects ha ...
s) 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
algebras In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition a ...
over the codensity monad of the inclusion of finite sets (regarded as
discrete topological space In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
s) into topological spaces are equivalent to
Stone space In topology and related areas of mathematics, a Stone space, also known as a profinite space or profinite set, is a compact totally disconnected Hausdorff space. Stone spaces are named after Marshall Harvey Stone who introduced and studied them in t ...
s. Avery shows that the Giry monad arises as the codensity monad of natural forgetful functors between certain categories of convex vector spaces to
measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. Definition Consider a set X and a σ-algebra \mathcal A on X. Then the ...
s.


Relation to Isbell duality

Di Liberti shows that the codensity monad is closely related to
Isbell duality Isbell conjugacy (a.k.a. Isbell duality or Isbell adjunction) (named after John R. Isbell John Rolfe Isbell (October 27, 1930 – August 6, 2005) was an American mathematician, for many years a professor of mathematics at the University at Buffalo ...
: for a given small category C, Isbell duality refers to the adjunction \mathcal O : Set^ \rightleftarrows (Set^C)^ : Spec between the category of
presheaves In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
on C (that is, functors from the opposite category of C to sets) and the opposite category of copresheaves on C. The monad Spec \circ \mathcal O induced by this adjunction is shown to be the codensity monad of the
Yoneda embedding In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewi ...
y: C \to Set^. Conversely, the codensity monad of a full small dense subcategory K in a cocomplete category C is shown to be induced by Isbell duality.


See also

*


References

* * Footnotes {{reflist, refs= {{Cite journal, last1=Avery , first1=Tom , date=2016 , title=Codensity and the Giry monad , journal=Journal of Pure and Applied Algebra , volume=220 , issue=3 , pp=1229–1251 , doi=10.1016/j.jpaa.2015.08.017 , arxiv=1410.4432 {{Cite journal, last1=Kennison , first1=J.F. , last2=Gildenhuys , first2=Dion , date=1971 , title=Equational completion, model induced triples and pro-objects , journal=Journal of Pure and Applied Algebra , volume=1 , issue=4 , pp=317–346 , doi=10.1016/0022-4049(71)90001-6 , doi-access=free {{Cite book, last1=Adámek , first1=Jirí , last2=Sousa , first2=Lurdes , date=2019 , title=D-Ultrafilters and their Monads , arxiv=1909.04950 {{Cite journal, last1=Sipoş , first1=Andrei , date=2018 , title=Codensity and stone spaces , journal=Mathematica Slovaca , volume=68 , pp=57–70 , doi=10.1515/ms-2017-0080 , arxiv=1409.1370 Category theory