internalization and categorical algebra
algebra object (associative, Lie, …)
internal category ($\to$ more)
Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
A polynomial comonad on a category is a comonad whose underlying endofunctor is a polynomial functor.
A polynomial comonad on Set is equivalent to a small category (Ahman-Uustalu, 2016).
A morphism of polynomial comonads on $\mathrm{Set}$ is equivalent to a retrofunctor (Ahman-Uustalu, 2017).
A two-sided comodule $A \nrightarrow B$ between polynomial comonads on $\mathrm{Set}$, viewed as small categories, is equivalent to a parametric right adjoint $[A, \mathrm{Set}] \rightarrow [B, \mathrm{Set}]$ between functor categories (Garner, 2019).
Polynomial comonads were first studied under the name directed containers in:
The equivalence between polynomial comonads on $\mathrm{Set}$ and small categories was shown in:
The observation that morphisms of polynomial comonads are the same as retrofunctors (under the name cofunctor) appears in:
The notion of (two-sided) comodule between polynomial comonads was given in:
A detailed study of polynomial comonads and their morphisms appears in Chapter 7 of the (unpublished) book:
Last revised on May 11, 2024 at 04:55:00. See the history of this page for a list of all contributions to it.