arXiv subjects
Publications and source records attributed to Fernando Lucatelli Nunes.
We give sufficient conditions for effective descent in categories of (generalized) internal multicategories. Two approaches to study effective descent morphisms are pursued. The first one relies on establishing the category of internal multicategories as an equalizer of categories of diagrams. The second approach extends the techniques developed by Ivan Le Creurer in his study of descent for internal essentially algebraic structures.
Let $\mathbb{A}$ be a $2$-category with suitable opcomma objects and pushouts. We give a direct proof that, provided that the codensity monad of a morphism $p$ exists and is preserved by a suitable morphism, the factorization given by the lax descent object of the higher cokernel of $p$ is up to isomorphism the same as the semantic factorization of $p$, either one existing if the other does. The result can be seen as a counterpart account to the celebrated Bénabou-Roubaud theorem. This leads in particular to a monadicity theorem, since it characterizes monadicity via descent. It should be noted that all the conditions on the codensity monad of $p$ trivially hold whenever $p$ has a left adjoint and, hence, in this case, we find monadicity to be a $2$-dimensional exact condition on $p$, namely, to be an effective faithful morphism of the $2$-category $\mathbb{A} $.
We study lax epimorphisms in 2-categories, with special attention to $\mathsf{Cat}$ and $\mathcal{V}$-$\mathsf{Cat}$. We show that any 2-category with convenient colimits has an orthogonal $LaxEpi$-factorization system, and we give a concrete description of this factorization in $\mathsf{Cat}$.
The fundamental construction underlying descent theory, the lax descent category, comes with a functor that forgets the descent data. We prove that, in any $2$-category $\mathfrak{A} $ with lax descent objects, the forgetful morphisms create all Kan extensions that are preserved by certain morphisms. As a consequence, in the case $\mathfrak{A} = \mathsf{Cat} $, we get a monadicity theorem which says that a right adjoint functor is monadic if and only if it is, up to the composition with an equivalence, (naturally isomorphic to) a functor that forgets descent data. In particular, within the classical context of descent theory, we show that, in a fibred category, the forgetful functor between the category of internal actions of a precategory $a$ and the category of internal actions of the underlying discrete precategory is monadic if and only if it has a left adjoint. More particularly, this shows that one of the implications of the celebrated Benabou-Roubaud theorem does not depend on the so called Beck-Chevalley condition. Namely, we prove that, in indexed categories, whenever an effective descent morphism induces a right adjoint functor, the induced functor is monadic.
Given a pseudomonad $\mathcal{T} $, we prove that a lax $\mathcal{T} $-morphism between pseudoalgebras is a $\mathcal{T} $-pseudomorphism if and only if there is a suitable (possibly non-canonical) invertible $\mathcal{T} $-transformation. This result encompasses several results on \textit{non-canonical isomorphisms}, including Lack's result on normal monoidal functors between braided monoidal categories, since it is applicable in any $2$-category of pseudoalgebras, such as the $2$-categories of monoidal categories, cocomplete categories, pseudofunctors and so on.
This is the introductory chapter of my PhD Thesis. This thesis consists of one introductory chapter and four single-authored papers written during my PhD studies at the University of Coimbra under supervision of Maria Manuel Clementino. In this first chapter, we give a glance of the scope of our work and briefly describe elements of the original contributions of each paper, including some connections between them. We also give a brief exposition of our main setting, which is $2$-dimensional category theory. In this direction: (1) we give an exposition on the doctrinal adjunction, focusing on the Beck-Chevalley condition as used in Chapter "Pseudo-Kan Extensions and Descent" (arXiv: 1606.04999), (2) we apply the results of "On lifting of biadjoints and lax algebras" (arXiv: 1607.03087) in a generalized setting of the formal theory of monads and (3) we apply the biadjoint triangle theorem of "On biadjoint triangles" (aXiv: 1606.05009) to study (pseudo)exponentiable pseudocoalgebras.
There are two main constructions in classical descent theory: the category of algebras and the descent category, which are known to be examples of weighted bilimits. We give a formal approach to descent theory, employing formal consequences of commuting properties of bilimits to prove classical and new theorems in the context of Janelidze-Tholen "Facets of Descent II", such as Bénabou-Roubaud Theorems, a Galois Theorem, embedding results and formal ways of getting effective descent morphisms. In order to do this, we develop the formal part of the theory on commuting bilimits via pseudomonad theory, studying idempotent pseudomonads and proving a $2$-dimensional version of the adjoint triangle theorem. Also, we work out the concept of pointwise pseudo-Kan extension, used as a framework to talk about bilimits, commutativity and the descent object. As a subproduct, this formal approach can be an alternative perspective/guiding template for the development of higher descent theory.
By the biadjoint triangle theorem, given a pseudomonad $\mathcal{T} $ on a $2$-category $\mathfrak{B} $, if a right biadjoint $\mathfrak{A}\to\mathfrak{B} $ has a lifting to the pseudoalgebras $\mathfrak{A}\to\mathsf{Ps}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} $ then this lifting is also right biadjoint provided that $\mathfrak{A} $ has codescent objects. In this paper, we give general results on lifting of biadjoints. As a consequence, we get a \textit{biadjoint triangle theorem} which, in particular, allows us to study triangles involving the $2$-category of lax algebras, proving analogues of the result described above. More precisely, we prove that, denoting by $\ell :\mathsf{Lax}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} \to\mathsf{Lax}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg}_\ell $ the inclusion, if $R: \mathfrak{A}\to\mathfrak{B} $ is right biadjoint and has a lifting $J: \mathfrak{A} \to \mathsf{Lax}\textrm{-}\mathcal{T}\textrm{-}\mathsf{Alg} $, then $\ell\circ J$ is right biadjoint as well provided that $\mathfrak{A} $ has some needed weighted bicolimits. In order to prove such theorem, we study the descent objects and the lax descent objects. At the last section, we study direct consequences of our theorems in the context of the $2$-monadic approach to coherence. In particular, we give the construction of the left $2$-adjoint to the inclusion of the strict algebras into the lax algebras.
We prove a biadjoint triangle theorem and its strict version, which are $2$-dimensional analogues of the adjoint triangle theorem of Dubuc. Similarly to the $1$-dimensional case, we demonstrate how we can apply our results to get the pseudomonadicity characterization (due to Le Creurer, Marmolejo and Vitale). Furthermore, we study applications of our main theorems in the context of the $2$-monadic approach to coherence. As a direct consequence of our strict biadjoint triangle theorem, we give the construction (due to Lack) of the left $2$-adjoint to the inclusion of the strict algebras into the pseudoalgebras. In the last section, we give two brief applications on lifting biadjunctions and pseudo-Kan extensions.