### Context

#### Higher algebra

higher algebra

universal algebra

## Theorems

#### 2-Category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

## Definition

One point in favour of the terminology “nuclear” over “bimonadic” is that the term bimonadic functor is sometimes used in the literature for a functor that is both monadic and comonadic, which is a different concept that is related to bimonads.

## Properties

• Consider an adjunction $F \dashv G$, generating a monad $T$ and a comonad $D$. Under minimal assumptions on the categories, there is an adjunction between the category of $T$-algebras, and the category of $D$-coalgebras, and this adjunction is nuclear. See Pavlovic–Hughes and below.

• Consider an adjunction $F \dashv G : B \to A$, generating a monad $T$ and a comonad $D$. Assuming that $B$ is Cauchy complete, $F$ is comonadic if and only if the free–forgetful adjunction associated to the Eilenberg–Moore category of $T$ is nuclear. See Mesablishvili.

Let $\ell \dashv r$ be an adjunction, inducing a monad $T_1$. One can form the Eilenberg–Moore category of $T$, which induces an adjunction $f_{T_1} \dashv u_{T_1}$. This in turn induces a comonad $D_2$ on the Eilenberg–Moore category. One can form the Eilenberg–Moore category of $D_2$, which induces an adjunction $u_{D_2} \dashv c_{D_2}$. This in turn induces a monad $T_3$ on the Eilenberg–Moore of $D_2$. We can then continue to iterate this construction, giving monads and comonads $T_1, D_2, T_3, D_4, T_5, \dots$.

An obvious question is: does this process ever terminate? In unpublished work of Lack, it is claimed that this process terminates at $T_3$. In other words, $f_{T_3} \dashv u_{T_3}$ is a nuclear adjunction. Thus $T_3 \simeq T_5 \simeq \dots$ and $D_4 \simeq D_6 \simeq \dots$. Furthermore, if the domain of $\ell$ is Cauchy complete, then it terminates at $D_2$.

Under the simplifying assumptions of Cauchy completeness, this is proven in Pavlovic–Hughes. They observe that this construction forms a pseudo-idempotent 2-monad? on a 2-category of monads on $Cat$.

## References

• Michael Barr, Coalgebras in a category of algebras, in: Category Theory, Homology Theory and their Applications I, Lecture Notes in Mathematics 86, Springer (1969) 1-12 [doi:10.1007/BFb0079381, pdf, pdf]

• Armin Frei and John L. MacDonald, Algebras, coalgebras and cotripleability, Archiv der Mathematik 22 (1971): 1-6.

• Stefano Kasangian, Stephen Lack, and Enrico Vitale. Coalgebras, braidings, and distributive laws, Theory and Applications of Categories 13.8 (2004): 129-146. (html)

• Bart Jacobs. Coalgebras and approximation. Logical Foundations of Computer Science: Third International Symposium, LFCS’94 St. Petersburg, Russia, July 11–14, 1994 Proceedings. Berlin, Heidelberg: Springer Berlin Heidelberg, 2005.

• Bachuki Mesablishvili. Monads of effective descent type and comonadicity. Theory and Applications of Categories 16.1 (2006): 1-45. (pdf)

• Matias Menni. Bimonadicity and the explicit basis property. Theory and Applications of Categories 26.22 (2012): 554-581. (pdf)

• Bart Jacobs. Bases as coalgebras. Logical Methods in Computer Science 9 (2013).

• Dusko Pavlovic, Dominic J. D. Hughes, The nucleus of an adjunction and the Street monad on monads (arXiv:2004.07353)

Last revised on February 11, 2024 at 18:47:45. See the history of this page for a list of all contributions to it.