Codensity Monads Research Articles

Semantic Factorization and Descent

Let {mathbb {A}} be a 2-category with suitable opcomma objects and pushouts. We give a direct proof that, provided that the codensity monad of a morphism p exists and is preserved by a suitable morphism, the factorization given by the lax descent object of the two-dimensional cokernel diagram of p is up to isomorphism the same as the semantic factorization of p, either one existing if the other does. The result can be seen as a counterpart account to the celebrated Bénabou–Roubaud theorem. This leads in particular to a monadicity theorem, since it characterizes monadicity via descent. It should be noted that all the conditions on the codensity monad of p trivially hold whenever p has a left adjoint and, hence, in this case, we find monadicity to be a two-dimensional exact condition on p, namely, to be an effective faithful morphism of the 2-category {mathbb {A}}.

D-ultrafilters and their monads

For a number of locally finitely presentable categories K we describe the codensity monad of the full embedding of all finitely presentable objects into K. We introduce the concept of D-ultrafilter on an object, where D is a “nice” cogenerator of K. We prove that the codensity monad assigns to every object an object representing all D-ultrafilters on it. Our result covers e.g. categories of sets, vector spaces, posets, semilattices, graphs and M-sets for finite commutative monoids M.

Quantifiers on languages and codensity monads

AbstractThis paper contributes to the techniques of topo-algebraic recognition for languages beyond the regular setting as they relate to logic on words. In particular, we provide a general construction on recognisers corresponding to adding one layer of various kinds of quantifiers and prove a corresponding Reutenauer-type theorem. Our main tools are codensity monads and duality theory. Our construction hinges on a measure-theoretic characterisation of the profinite monad of the free S-semimodule monad for finite and commutative semirings S, which generalises our earlier insight that the Vietoris monad on Boolean spaces is the codensity monad of the finite powerset functor.

Codensity: Isbell duality, pro-objects, compactness and accessibility

We study codensity monads T induced by (mostly small, mostly dense) full subcategories A⊂K. These monads behave quite similarly, we show some connections with the Isbell duality, pro-finite objects and compact spaces. We prove that they are quite unlikely to be accessible. Finally, we introduce the notion of generically idempotent monad and comment its properties.

Codensity, profiniteness and algebras of semiring-valued measures

We show that, if S is a finite semiring, then the free profinite S-semimodule on a Boolean Stone space X is isomorphic to the algebra of all S-valued measures on X, which are finitely additive maps from the Boolean algebra of clopens of X to S. These algebras naturally appear in the logic approach to formal languages as well as in idempotent analysis. Whenever S is a (pro)finite idempotent semiring, the S-valued measures are all given uniquely by continuous density functions. This generalises the classical representation of the Vietoris hyperspace of a Boolean Stone space in terms of continuous functions into the Sierpiński space.We adopt a categorical approach to profinite algebra which is based on profinite monads. The latter were first introduced by Adámek et al. as a special case of the notion of codensity monads.

A Formula for Codensity Monads and Density Comonads

For a functor F whose codomain is a cocomplete, cowellpowered category $$\mathcal {K}$$ with a generator S we prove that a codensity monad exists iff for every object s in S all natural transformations from $$\mathcal {K}(X, F-)$$ to $$\mathcal {K}(s, F-)$$ form a set. Moreover, the codensity monad has an explicit description using the above natural transformations. Concrete examples are presented, e.g., the codensity monad of the power-set functor $$\mathcal {P}$$ assigns to every set X the set of all nonexpanding endofunctions of $$\mathcal {P}X$$ . Dually, a set-valued functor F is proved to have a density comonad iff all natural transformations from $$X^F$$ to $$2^F$$ form a set. Moreover, that comonad assigns to X the set of all those transformations. For preimages-preserving endofunctors F of $${\mathsf {Set}}$$ we prove that F has a density comonad iff F is accessible.

Codensity and stone spaces

AbstractWe present a detailed computation of two codensity monads associated to two canonical functors – the inclusion functor ofFinSetintoTopand the inclusion functor of the category of the powers of the Sierpiński space intoTop. We show that the categories of algebras of the two monads are the categories of Stone spaces and of sober spaces, respectively. A new motivation for defining these classes of spaces is therefore obtained.

Commutants for Enriched Algebraic Theories and Monads

We define and study a notion of commutant for $$\mathscr {V}$$ -enriched $${\mathscr {J}}$$ -algebraic theories for a system of arities $${\mathscr {J}}$$ , recovering the usual notion of commutant or centralizer of a subring as a special case alongside Wraith’s notion of commutant for Lawvere theories as well as a notion of commutant for $$\mathscr {V}$$ -monads on a symmetric monoidal closed category $$\mathscr {V}$$ . This entails a thorough study of commutation and Kronecker products of operations in $${\mathscr {J}}$$ -theories. In view of the equivalence between $${\mathscr {J}}$$ -theories and $${\mathscr {J}}$$ -ary monads we reconcile this notion of commutation with Kock’s notion of commutation of cospans of monads and, in particular, the notion of commutative monad. We obtain notions of $${\mathscr {J}}$$ -ary commutant and absolute commutant for $${\mathscr {J}}$$ -ary monads, and we show that for finitary monads on $$\text {Set}$$ the resulting notions of finitary commutant and absolute commutant coincide. We examine the relation of the notion of commutant to both the notion of codensity monad and the notion of algebraic structure in the sense of Lawvere.

Eilenberg–Moore Monoids and Backtracking Monad Transformers

We develop an algebraic underpinning of backtracking monad transformers in the general setting of monoidal categories. As our main technical device, we introduce Eilenberg--Moore monoids, which combine monoids with algebras for strong monads. We show that Eilenberg--Moore monoids coincide with algebras for the list monad transformer ('done right') known from Haskell libraries. From this, we obtain a number of results, including the facts that the list monad transformer is indeed a monad, a transformer, and an instance of the MonadPlus class. Finally, we construct an Eilenberg--Moore monoid of endomorphisms, which, via the codensity monad construction, yields a continuation-based implementation a la Hinze.

Codensity and the Giry monad

The Giry monad on the category of measurable spaces sends a space to a space of all probability measures on it. There is also a finitely additive Giry monad in which probability measures are replaced by finitely additive probability measures. We give a characterisation of both finitely and countably additive probability measures in terms of integration operators giving a new description of the Giry monads. This is then used to show that the Giry monads arise as the codensity monads of forgetful functors from certain categories of convex sets and affine maps to the category of measurable spaces.