We give an equivalence between the derived category of a locally finitely presented category and the derived category of contravariant functors from its finitely presented subcategory to the category of abelian groups, in the spirit of Krause’s work [H. Krause, Approximations and adjoints in homotopy categories, Math. Ann. 353 (2012), 765–781]. Then we provide a criterion for the existence of r...

The path category P (X) of a simplicial set X defines a functor X 7→ P (X) which is left adjoint to the nerve functor B : cat → sSet. The first section of this paper contains an elementary description of this functor, with some of its basic properties. This construction specializes to a path category functor for cubical sets which appears in the second section. The path category construction is...

In this paper we introduce the category Apro-ANR called the approximate pro-category of ANR’s, whose objects are all systems of ANR’s and whose morphisms are obtained as equivalence classes of system maps for some equivalence relation. We show that any 2-sink X f −→ Z g ←− Y in Apro-ANR admits a weak pull-back and it admits a pull-back if they are systems of compact ANR’s. Moreover, it admits a...

The category of effect algebras is the Eilenberg-Moore category for the monad arising from the free-forgetful adjunction between categories of bounded posets and orthomodular posets. In the category of effect algebras, an observable is a morphism whose domain is a Boolean algebra. The characterization of subsets of ranges of observables is an open problem. For an interval effect algebra E, a wi...

This paper provides an abstract definition of some kinds of logics, called diagrammatic logics, together with a definition of morphisms and of 2-morphisms between diagrammatic logics. The definition of the 2-category of diagrammatic logics rely on category theory, mainly on adjunction, categories of fractions and limit sketches. This framework is applied to the formalization of a parameterizati...

for any smooth projective variety X over a field k. Here CH(X)Q is the Chow group of codimension p algebraic cycles modulo rational equivalence on X with Q-coefficients, and DMM(X) is the bounded derived category of the (conjectural) abelian category of mixed motivic sheaves onX . By the adjoint relation for the structure morphism aX : X → Spec k, the conjecture would be equivalent to the bijec...

We try to combine the ‘syntactic composition of tree transducers’ [FV98, KV01] on the one hand side and ‘short cut fusion’ [GLP93] on the other hand side. Short cut fusion is based on the ‘cata/build-rule’ [Joh01] or ‘acid rain theorem’ [TM95]. Therefore it is necessary to represent the recursive functions as catamorphisms. A catamorphism is a generalization of the well known list-function fold...

(3) Let A, B be transitive non empty category structures with units, F be a precontravariant feasible functor structure from A to B, and G be a precontravariant feasible functor structure from B to A. Suppose F is bijective and G = F−1. Let a1, a2 be objects of A. Suppose 〈a1,a2〉 6= / 0. Let f be a morphism from a1 to a2 and g be a morphism from F(a2) to F(a1). Then F( f ) = g if and only if G(...

A pointed endofunctor (and in particular a reflector) (R, r) in a category X is direct iff for each morphism f : X→ Y the pullback of Rf against rY exists and the unique fill-in morphism u from X to the pullback is such that Ru is an isomorphism. (This is close to the concept of a simple reflector introduced by Cassidy, Hébert and Kelly in 1985.) We give sufficient conditions for directness, an...

The notion of overlap algebra introduced by G. Sambin provides a constructive version of complete Boolean algebra. Here we show that his notion of overlap morphism corresponds classically to that of map preserving arbitrary joins. Moreover we prove that the power collection of a set is the free overlap algebra join-generated from the set. Then, we generalize the concept of overlap algebra and o...