Semiadjoint functors and Kan extensions

Siberian Mathematical Journal -     

Characterizing shape theories by Kan extensions

Every pair (C,K) of categories, where K is a proreflective subcategory of C, generates a shape theory. As a main result in this paper we give a characterization of such pairs, showing that these are exactly those having the property that every functor F :K→A has a Kan extension Ran F :C→A, which is preserved by all functors commuting with inverse limits.     

A class of exactness properties characterized via left Kan extensions

We consider a general class of exactness properties on a finitely complete category, all of which can be expressed as the condition that a certain morphism in a diagram is a strong epimorphism. For each such exactness property, we characterize finitely bicomplete categories having the property by restricting the condition to those diagrams built from only one object in the category via a left Kan extension. In the regular context, this generalizes the theory of approximate co-operations introduced by D. Bourn and Z. Janelidze. As an application, we deduce from this a characterization of (essentially) algebraic categories satisfying such a given exactness property. The pointed version of these exactness properties is also studied.     

Finite-product-preserving functors,Kan extensions,and strongly-finitary 2-monads

We study those 2-monads on the 2-categoryCat of categories which, as endofunctors, are the left Kan extensions of their restrictions to the sub-2-category of finite discrete categories, describing their algebras syntactically. Showing that endofunctors of this kind are closed under composition involves a lemma on left Kan extensions along a coproduct-preserving functor in the context of cartesian closed categories, which is closely related to an earlier result of Borceux and Day.The first author gratefully acknowledges the support of the Australian Research Council.     

Maximal extensions of partial orders preserving antimonotonicity

Let (A, ?? _r ) be a partially ordered set and f : A ?? A be an order reversing (antimonotone) map. We characterize the maximal partial order extensions of r preserving the antimonotonicity of f.     

Linear extensions of partial orders preserving monotonicity

Given a poset (A, r) and an acyclic r-monotone function f: AA, we prove that r can be extended to a linear order R with xRyf(x)Rf(y) for all x, yA.     

Measure preserving extensions and minimal wandering rates

For conservative ergodic (infinite) measure preserving transformations we show that existence and asymptotics of minimal wandering rates are preserved if we pass to extensions or factors. Consequently, they are similarity invariants.     

Positivity preserving finite element approximation

We consider finite element operators defined on ``rough' functions in a bounded polyhedron in . Insisting on preserving positivity in the approximations, we discover an intriguing and basic difference between approximating functions which vanish on the boundary of and approximating general functions which do not. We give impossibility results for approximation of general functions to more than first order accuracy at extreme points of . We also give impossibility results about invariance of positive operators on finite element functions. This is in striking contrast to the well-studied case without positivity.

     

Prolongations of valuations to finite extensions

Let ${K=\mathbb{Q}(\theta)}$ be an algebraic number field with θ in the ring A _K of algebraic integers of K and f(x) be the minimal polynomial of θ over the field ${\mathbb{Q}}$ of rational numbers. For a rational prime p, let ${\bar{f}(x)\,=\,\bar{g}_{1}(x)^{e_{1}}....\bar{g}_{r}(x)^{e_{r}}}$ be the factorization of the polynomial ${\bar{f}(x)}$ obtained by reducing coefficients of f(x) modulo p into a product of powers of distinct irreducible polynomials over ${\mathbb{Z}/p\mathbb{Z}}$ with g _i (x) monic. Dedekind proved that if p does not divide [ ${A_{K}:\mathbb{Z}}$ [θ]], then ${pA_{K}=\wp_{1}^{e_{1}}\ldots\wp_{r}^{e_{r}}}$ , where ${\wp_{1},\ldots,\wp_{r}}$ are distinct prime ideals of A _K , ${\wp_{i}=pA_{K}+g_{i}(\theta)A_{K}}$ having residual degree equal to the degree of ${\bar{g}_{i}(x)}$ . He also proved that p does not divide [ ${A_{K}:\mathbb{Z}}$ [θ]] if and only if for each i, either e _i  = 1 or ${\bar{g}_{i}(x)}$ does not divide ${\bar{M}(x)}$ where ${M(x)=\frac{1}{p}(f(x)-g_{1}(x)^{e_{1}}....g_{r}(x)^{e_{r}})}$ . Our aim is to give a weaker condition than the one given by Dedekind which ensures that if the polynomial ${\bar{f}(x)}$ factors as above over ${\mathbb{Z}/p\mathbb{Z}}$ , then there are exactly r prime ideals of A _K lying over p, with respective residual degrees ${\deg \bar {g}_{1}(x),...,\deg \bar {g}_{r}(x)}$ and ramification indices e ₁, ..., e _r . In this paper, the above problem has been dealt with in a more general situation when the base field is a valued field (K, v) of arbitrary rank and K(θ) is any finite extension of K.     

Whitney preserving maps on finite graphs

For a Whitney preserving map f:X→G we show the following: (a) If X is arcwise connected and G is a graph which is not a simple closed curve, then f is a homeomorphism; (b) If X is locally connected and G is a simple closed curve, then X is homeomorphic to either the unit interval [0,1], or the unit circle S¹. As a consequence of these results, we characterize all Whitney preserving maps between finite graphs. We also show that every hereditarily weakly confluent Whitney preserving map between locally connected continua is a homeomorphism.     

Isometric extensions and multiple recurrence of infinite measure preserving systems

Furstenberg’s multiple recurrence theorem is investigated for infinite measure preserving systems arising from isometric extensions which are 1-factor maps. It will be shown that any isometric extension preservesd-recurrence for alld≥1 and multiple recurrence.