nLab continuous algebra

Definition

Let $T$ be a lax-idempotent 2-monad or pseudo 2-monad on a 2-category $K$, and $A$ a pseudo $T$-algebra witnessed by a left adjoint $a : T A \to A$ to the unit $\eta_A : A \to T A$. We say that $A$ is a continuous $T$-algebra if $a : T A \to A$ has a further left adjoint, forming an adjoint triple.

Note that every free $T$-algebra $T B$ is continuous: it is a property of lax-idempotent 2-monads that the multiplication $\mu_B:T T B \to T B$ has left adjoint $T\eta_B$.

For the rest of this page we will say “$T$-algebra” to mean “pseudo $T$-algebra”.

Examples

• If $T A = Ind(A)$ is the ind-completion monad on Cat, whose algebras are categories with filtered colimits, then a continuous $T$-algebra is precisely a continuous category. This is the origin of the name.

• Specializing the previous example, if $T A = Idl(A)$ is the ideal completion monad on Poset, whose algebras are posets with directed joins, then a continuous $T$-algebra is a continuous poset.

Note also that if an adjoint functor theorem applies (such as if $A$ and $T A$ are complete lattices, or locally presentable categories), then $A$ is continuous if and only if $T A \to A$ is a continuous functor. This provides another justification for the name.

• For instance, if $T$ is the downset monad on Poset, whose algebras are suplattices, then continuous algebras are called constructive completely distributive lattices. Assuming the axiom of choice, these are equivalent to completely distributive lattices. See (WoodFawcett).

• If $T$ is the free small-cocompletion monad on $Cat$, whose algebras are cocomplete categories, then continuous algebras are in particular completely distributive categories. The converse only holds if an adjoint functor theorem applies.

• If $T$ is the presheaf category “monad”, whose “algebras” are total categories, then the “continuous algebras” are totally distributive categories. The scare quotes are because this example is not really a monad for size reasons.

• If $T$ is the free cocompletion monad under coproducts, then continuous algebras are “locally connected” in a sense: the extra left adjoint $A \to T A$ decomposes every object as a coproduct of connected ones.

• If $T$ is a “filter monad” on topological spaces, which can be regarded as a “presheaf category” type construction with topological spaces thought of as generalized multicategories (specifically, as relational beta-modules), then continuous $T$-algebras are called “distributive spaces” in (Hofmann).

Characterizations

Theorem

Consider the following conditions on a $T$-algebra $A$:

1. $A$ is a continuous algebra.
2. $A$ is a coreflective sub-$T$-algebra of a free $T$-algebra.
3. $A$ is a retract (up to isomorphism) of a free $T$-algebra.

Then (1) $\Rightarrow$ (2) $\Rightarrow$ (3), and both converses hold if idempotent 2-cells split in the underlying 2-category.

Proof

This is B1.1.15 in the Elephant. If the algebra structure $a : T A \to A$ has a left adjoint $\ell$, then $\ell \dashv a$ is an adjunction in the 2-category of $T$-algebras (since any left adjoint between $T$-algebras is a $T$-morphism). And since the counit of $a\dashv \eta_A$ is an isomorphism, the unit of $\ell \dashv a$ is an isomorphism, by the theorem on fully faithful adjoint triples. This shows (1) $\Rightarrow$ (2), and (2) $\Rightarrow$ (3) is obvious. For the converse, see the Elephant.

Theorem

Let $G$ be the pseudo 2-comonad on the 2-category $T Alg$ of $T$-algebras induced by the free-forgetful adjunction. Then $G$ is lax-idempotent, and a $T$-algebra $A$ is continuous if and only if it is a $G$-coalgebra.

Proof

Lax-idempotence of $G$ follows from the fact that the adjunction $K \rightleftarrows T Alg$ is a lax-idempotent 2-adjunction. Therefore, the $G$-coalgebras are those $T$-algebras for which the counit $G A \to A$ has a left adjoint in $T Alg$. However, this counit is just the structure map $T A \to A$ of $A$, and since $T$ is lax-idempotent, any left adjoint between $T$-algebras is automatically a (pseudo) $T$-morphism.

This theorem is proven in (Kock) in the special case when the base 2-category is a (1,2)-category (the result is due to Bart Jacobs).

Related pages

• lax-idempotent 2-monad
• adjoint triple

References

• Richard Wood and Barry Fawcett, “Constructive complete distributivity. I”. Math. Proc. Camb. Phil. Soc. (1990), 107, 81

• Anders Kock, “Monads for which structures are adjoint to units”, JPAA 104 (1992).

• Dirk Hofmann, “Duality for distributive spaces”. Theory Appl. Categ. 28 (3) (2013), 66–122, web site.