Lax orthogonal factorisation systems

This paper introduces lax orthogonal algebraic weak factorisation systems on 2-categories and describes a method of constructing them. This method rests in the notion of simple 2-monad, that is a generalisation of the simple reflections studied by Cassidy, H\'ebert and Kelly. Each simple 2-monad on a finitely complete 2-category gives rise to a lax orthogonal algebraic weak factorisation system, and an example of a simple 2-monad is given by completion under a class of colimits. The notions of KZ lifting operation, lax natural lifting operation and lax orthogonality between morphisms are studied.Comment: 59 page

A Note on Local Compactness

We propose a categorical definition of locally-compact Hausdorff object which gives the right notion both, for topological spaces and for locales. Stability properties follow from easy categorical arguments. The map version of the notion leads to an investigation of restrictions of perfect maps to open subspaces

Exponentiable functors between quantaloid-enriched categories

Exponentiable functors between quantaloid-enriched categories are characterized in elementary terms. The proof goes as follows: the elementary conditions on a given functor translate into existence statements for certain adjoints that obey some lax commutativity; this, in turn, is precisely what is needed to prove the existence of partial products with that functor; so that the functor's exponentiability follows from the works of Niefield [1980] and Dyckhoff and Tholen [1987].Comment: 10 pages; correction of flaw in proo

From lax monad extensions to Topological theories

We investigate those lax extensions of a Set-monad T = (T,m, e) to the category V-Rel of sets and V-valued relations for a quantale V = (V,⊗, k) that are fully determined by ξ maps : TV ⟶ V. We pay special attention to those maps ξ that make V a T-algebra and, in fact, (V,⊗, k) a monoid in the category Set^T with its cartesian structure. Any such map ξ forms the main ingredient to Hofmann’s notion of topological theory

Topological semi-abelian algebras

Given an algebraic theory whose category of models is semi-abelian, we study the category of topological models of and generalize to it most classical results on topological groups. In particular, is homological, which includes Barr regularity and forces the Mal'cev property. Every open subalgebra is closed and every quotient map is open. We devote special attention to the Hausdorff, compact, locally compact, connected, totally disconnected and profinite -algebras.http://www.sciencedirect.com/science/article/B6W9F-4CB07X6-1/1/61cf6d089f1d054878b360422bce8da

Separated and Connected Maps

Using on the one hand closure operators in the sense of Dikranjan and Giuli and on the other hand left- and right-constant subcategories in the sense of Herrlich, Preuß, Arhangel'skii and Wiegandt, we apply two categorical concepts of connectedness and separation/disconnectedness to comma categories in order to introduce these notions for morphisms of a category and to study their factorization behaviour. While at the object level in categories with enough points the first approach exceeds the second considerably, as far as generality is concerned, the two approaches become quite distinct at the morphism level. In fact, left- and right-constant subcategories lead to a straight generalization of Collins' concordant and dissonant maps in the category $$\mathcal{T}op$$ of topological spaces. By contrast, closure operators are neither able to describe these types of maps in $$\mathcal{T}op$$, nor the more classical monotone and light maps of Eilenberg and Whyburn, although they give all sorts of interesting and closely related types of maps. As a by-product we obtain a negative solution to the ten-year-old problem whether the Giuli–Hušek Diagonal Theorem holds true in every decent category, and exhibit a counter-example in the category of topological spaces over the 1-sphere

Lax comma categories of ordered sets

Let $\mathsf{Ord}$ be the category of (pre)ordered sets. Unlike $\mathsf{Ord}/X$, whose behaviour is well-known, not much can be found in the literature about the lax comma 2-category $\mathsf{Ord}//X$. In this paper, we show that, when $X$ is complete, the forgetful functor $\mathsf{Ord}//X\to \mathsf{Ord}$ is topological. Moreover, $\mathsf{Ord}// X$ is complete and cartesian closed if and only if $X$ is. We end by analysing descent in this category. Namely, when $X$ is complete and cartesian closed, we show that, for a morphism in $\mathsf{Ord}//X$, being pointwise effective for descent in $\mathsf{Ord}$ is sufficient, while being effective for descent in $\mathsf{Ord}$ is necessary, to be effective for descent in $\mathsf{Ord}//X$.Comment: 10 page