A theorem of A.M.Pitts (1986) states that essential surjections of toposes bounded over a base topos $\mathscr{S}$ are of effective lax descent. The symmetric monad $\mathscr{M}$ on the 2-category of toposes bounded over $\mathscr{S}$ is a KZ-monad (Bunge-Carboni 1995) and the $\mathscr{M}$-maps are precisely the $\mathscr{S}$-essential geometric morphisms (Bunge-Funk 2006). These last two results led me to conjecture¹ and then prove² the general lax descent theorem that is the subject matter of this paper. By a ‘Pitts KZ-monad’ on a 2-category $\mathscr{K}$ it is meant here a locally fully faithful equivariant KZ-monad $\mathscr{M}$ on $\mathscr{K}$ that is required to satisfy an analogue of Pitts' theorem on bicomma squares along essential geometric morphisms. The main result of this paper states that, for a Pitts KZ-monad $\mathscr{M}$ on a 2-category $\mathscr{K}$ (‘of spaces’), every surjective $\mathscr{M}$-map is of effective lax descent. There is a dual version of this theorem for a Pitts co-KZ-monad $\mathscr{N}$. These theorems have (known and new) consequences regarding (lax) descent for morphisms of toposes and locales.

In a previous paper (“Hexagonal Logic of the Field $\mathbb F_8$ as a Boolean Logic with Three Involutive Modalities”, in The road to Universal Logic), we proved that elements of $\mathbb P(8)$, i.e. functions of all finite arities on the Galois field $\mathbb F_8$, are compositions of logical functions of a given Boolean structure, plus three geometrical cross product operations. Here we prove that $\mathbb P(8)$ admits a purely logical presentation, as a Boolean manifold, generated by a diagram of $4$ Boolean systems of logical operations on $\mathbb F_8$. In order to obtain this result we provide various systems of parameters of the set of unordered bases on $\mathbb F_2^3$, and consequently parametrical polynomial expressions for the corresponding conjunctions, which in fact are enough to characterize these unordered bases (and the corresponding Boolean structures).

We introduce a weakened notion of double functor, which we call wobbly, and which arises naturally in the study of double adjoints. We then show how horizontal invariance can be used to lift results about wobbly double functors to genuine ones.

The bicategory of fractions of the 2-category of internal groupoids and internal functors in groups with respect to weak equivalences (i.e., functors which are internally full, faithful and essentially surjective) has an easy description: one has just to replace internal functors by monoidal functors. In the present paper, we generalize this result from groups to any monadic category over a regular category $\mathcal C,$ assuming that the axiom of choice holds in $\mathcal C.$ For $\mathbb T$ a monad on $\mathcal C,$ the bicategory of fractions of Grpd$({\mathcal C}^{\mathbb T})$ with respect to weak equivalences is now obtained replacing internal functors by what we call $\mathbb T$-monoidal functors. The notion of $\mathbb T$-monoidal functor is related to the notion of pseudo-morphism between strict algebras for a pseudo-monad on a 2-category.

The notion of pseudocategory is extended from the context of a 2-category to the more general one of a sesquicategory, which is considered as a category equipped with a 2-cell structure. Some particular examples of 2-cells arising from internal transformations in internal categories, conjugations in groups, derivations in crossed-modules or homotopies in abelian chain complexes are studied in this context, namely their behaviour as abstract 2-cells in a 2-cell structure. Issues such as naturality of a 2-cell structure are investigated. This article is intended as a preliminary starting work towards the study of the geometric aspects of the 2-cell structures from an algebraic point of view.