Page 97 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 97
NCOMMUTATIVE STRUCTURES WITHIN ORDER STRUCTURES, SEMIGROUPS
AND UNIVERSAL ALGEBRA (MS-67)
Eilenberg-Moore, Kleisli and descent factorizations
Fernando Lucatelli Nunes, f.lucatellinunes@uu.nl
Utrecht University, Netherlands
Every functor that has a left adjoint has two well-known factorizations. The first one is through
the category of Eilenberg-Moore algebras of the induced monad, while the second one is the fac-
torization through the category of free coalgebras (co-Kelsili category) of the induced comonad.
As usual in category theory, we also have the dual cases: a functor that has a right adjoint has a
factorization through the category of Eilenberg-Moore coalgebras of the induced comonad, and
other factorization through the Kleisli category of the induced monad.
More generally, if the functor has a codensity monad (resp. a op-codensity comonad), the
functor has a factorization through the category of Eilenberg-Moore algebras (resp. the free
coalgebras) (see, for instance, [5, Section 3]).
In [5], given a 2-category A satisfying suitable hypothesis, we show that every morphism
inside a 2-category with opcomma objects (and pushouts) has a 2-dimensional cokernel diagram
which, in the presence of the descent objects, induces a factorization of the morphism. We show
that these factorizations generalize the usual Eilenberg-Moore and Kleisli factorizations.
The result is new even in the case A = Cat. In this case, we have that every functor
has a factorization through the category of descent data of its 2-dimensional cokernel diagram.
We show that, if a functor F has a left adjoint, this descent factorization coincides with the
factorization through the category of algebras. Dually, if F has a right adjoint, this descent
factorization coincides with the factorization through the category of coalgebras.
This specializes in a new connection between monadicity and descent theory, which can
be seen as a counterpart account to the celebrated Bénabou-Roubaud Theorem (see [6] or, for
instance, [4, Theorem 1.4] and [3, Section 4]). It also leads in particular to a (formal) monadicity
theorem.
In this talk, we shall give a sketch of the ideas and constructions involved in this particular
case of A = Cat.
References
[1] J. Bénabou and J. Roubaud. Monades et descente. C. R. Acad. Sci. Paris Ser. A-B,
270:A96–A98, 1970.
[2] F. Lucatelli Nunes. Pseudo-Kan extensions and descent theory. Theory Appl. Categ.,
33:No. 15, 390–444, 2018.
[3] F. Lucatelli Nunes. Semantic Factorization and Descent. DMUC preprints, 19-03, 2019.
arXiv: 1902.01225
[4] F. Lucatelli Nunes. Descent Data and Absolute Kan Extensions. DMUC preprints, 19-19,
2019. arXiv: 1606.04999
95
AND UNIVERSAL ALGEBRA (MS-67)
Eilenberg-Moore, Kleisli and descent factorizations
Fernando Lucatelli Nunes, f.lucatellinunes@uu.nl
Utrecht University, Netherlands
Every functor that has a left adjoint has two well-known factorizations. The first one is through
the category of Eilenberg-Moore algebras of the induced monad, while the second one is the fac-
torization through the category of free coalgebras (co-Kelsili category) of the induced comonad.
As usual in category theory, we also have the dual cases: a functor that has a right adjoint has a
factorization through the category of Eilenberg-Moore coalgebras of the induced comonad, and
other factorization through the Kleisli category of the induced monad.
More generally, if the functor has a codensity monad (resp. a op-codensity comonad), the
functor has a factorization through the category of Eilenberg-Moore algebras (resp. the free
coalgebras) (see, for instance, [5, Section 3]).
In [5], given a 2-category A satisfying suitable hypothesis, we show that every morphism
inside a 2-category with opcomma objects (and pushouts) has a 2-dimensional cokernel diagram
which, in the presence of the descent objects, induces a factorization of the morphism. We show
that these factorizations generalize the usual Eilenberg-Moore and Kleisli factorizations.
The result is new even in the case A = Cat. In this case, we have that every functor
has a factorization through the category of descent data of its 2-dimensional cokernel diagram.
We show that, if a functor F has a left adjoint, this descent factorization coincides with the
factorization through the category of algebras. Dually, if F has a right adjoint, this descent
factorization coincides with the factorization through the category of coalgebras.
This specializes in a new connection between monadicity and descent theory, which can
be seen as a counterpart account to the celebrated Bénabou-Roubaud Theorem (see [6] or, for
instance, [4, Theorem 1.4] and [3, Section 4]). It also leads in particular to a (formal) monadicity
theorem.
In this talk, we shall give a sketch of the ideas and constructions involved in this particular
case of A = Cat.
References
[1] J. Bénabou and J. Roubaud. Monades et descente. C. R. Acad. Sci. Paris Ser. A-B,
270:A96–A98, 1970.
[2] F. Lucatelli Nunes. Pseudo-Kan extensions and descent theory. Theory Appl. Categ.,
33:No. 15, 390–444, 2018.
[3] F. Lucatelli Nunes. Semantic Factorization and Descent. DMUC preprints, 19-03, 2019.
arXiv: 1902.01225
[4] F. Lucatelli Nunes. Descent Data and Absolute Kan Extensions. DMUC preprints, 19-19,
2019. arXiv: 1606.04999
95
