continuous algebra

Continuous algebras

Continuous algebras


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

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

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


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



Consider the following conditions on a TT-algebra AA:

  1. AA is a continuous algebra.
  2. AA is a coreflective sub-TT-algebra of a free TT-algebra.
  3. AA is a retract (up to isomorphism) of a free TT-algebra.

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


This is B1.1.15 in the Elephant. If the algebra structure a:TAAa : T A \to A has a left adjoint \ell, then a\ell \dashv a is an adjunction in the 2-category of TT-algebras (since any left adjoint between TT-algebras is a TT-morphism). And since the counit of aη Aa\dashv \eta_A is an isomorphism, the unit of a\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.


Let GG be the pseudo 2-comonad? on the 2-category TAlgT Alg of TT-algebras induced by the free-forgetful adjunction. Then GG is lax-idempotent, and a TT-algebra AA is continuous if and only if it is a GG-coalgebra.


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


  • 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.

Last revised on November 22, 2013 at 09:34:29. See the history of this page for a list of all contributions to it.