nLab nuclear adjunction

Definition

A nuclear adjunction or bimonadic adjunction is an adjunction that is both monadic and comonadic.

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.

Construction of nuclear adjunction

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 In unpublished work of Lack, it is claimed that this process terminates at $T_3$.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 in $Cat$.

Note that this construction is distinct from monadic decomposition, which instead constructs new monads from comparison functors, rather than (co)free–forgetful adjunctions.

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.

• Paul Taylor’s email to the categories mailing list “monadic completion of adjunctions” (link)

• 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)