We prove that the existence of arbitrarily large supercompact cardinals implies that every absolute epireflective class of objects in a balanced accessible category is a small-orthogonality class. In other words, if L is a localization functor on a balanced accessible category such that the unit morphism X → LX is an epimorphism for all X and the class of L-local objects is defined by an absolu...

A probabilistic structure on sequential dynamical systems is introduced here, the new model will be called Probabilistic Sequential Network, PSN. The morphisms of Probabilistic Sequential Networks are defined using two algebraic conditions. It is proved here that two homomorphic Probabilistic Sequential Networks have the same equilibrium or steady state probabilities if the morphism is either a...

Given a proper morphism X → S, we show that a large class of objects in the derived category of X naturally form an Artin stack locally of finite presentation over S. This class includes S-flat coherent sheaves and, more generally, contains the collection of all S-flat objects which can appear in the heart of a reasonable sheaf of t-structures on X. In this sense, this is the Mother of all Modu...

Let C be a category with finite colimits, and let (E ,M) be a factorisation system on C with M stable under pushouts. Writing C;M for the symmetric monoidal category with morphisms cospans of the form c → m ←, where c ∈ C and m ∈ M, we give method for constructing a category from a symmetric lax monoidal functor F : (C;M,+) → (Set,×). A morphism in this category, termed a decorated corelation, ...

For a symmetric monoidal-closed category X and any object K, the category of K-Chu spaces is small-topological over X and small cotopological over X . Its full subcategory of M-extensive K-Chu spaces is topological over X when X is Mcomplete, for any morphism class M. Often this subcategory may be presented as a full coreflective subcategory of Diers’ category of affine K-spaces. Hence, in addi...

An unpublished result by the first author states that there exists a Hopf algebra H such that for any Möbius category C (in the sense of Leroux) there exists a canonical algebra morphism from the dual H∗ of H to the incidence algebra of C. Moreover, the Möbius inversion principle in incidence algebras follows from a ‘master’ inversion result in H∗. The underlying module of H was originally defi...

For a small category B and a double category D, let LaxN (B,D) denote the category whose objects are vertical normal lax functors B //D and morphisms are horizontal lax transformations. It is well known that LaxN (B,Cat) ≃ Cat/B, where Cat is the double category of small categories, functors, and profunctors. In [19], we generalized this equivalence to certain double categories, in the case whe...

This paper is a sequel to [12]. We are here concerned with properties of theories in full first-order intuitionistic logic; the latter correspond under the identification of theories with categories provided by categorical logic (cf. [8] or [ 1 l]), to Heyting pretoposes, i.e. pretoposes with universal quantification of subobjects along morphisms. Using the lattice-theoretic machinery developed...

In [2] Baez and Dolan established their stabilization hypothesis as one of a list of the key properties that a good theory of higher categories should have. It is the analogue for n-categories of the well-known stabilization theorems in homotopy theory. To explain the statement, recall that Baez-Dolan introduce the notion of k-uply monoidal n-category which is an n+ k-category having only one i...