Relative plus constructions

Let $h$ be a connective homology theory. We construct a functorial relative plus construction as a Bousfield localization functor in the category of maps of spaces. It allows us to associate to a pair $(X,H)$, consisting of a connected space $X$ and an $h$-perfect normal subgroup $H$ of the fundamental group ${\pi }_1\left(X\right)$, an $h$-acyclic map $X\to X_H^{+h}$ inducing the quotient by $H$ on the fundamental group. We show that this map is terminal among the $h$-acyclic maps that kill a subgroup of $H$. When $h$ is an ordinary homology theory with coefficients in a commutative ring with unit $R$, this provides a functorial and well-defined counterpart to a construction by cell attachment introduced by Broto, Levi, and Oliver in the spirit of Quillen’s plus construction. We also clarify the necessity to use a strongly $R$-perfect group $H$ in characteristic zero.     

On the Baer–Lovász–Tutte construction of groups from graphs: Isomorphism types and homomorphism notions

Let $p$ be an odd prime. From a simple undirected graph $G$, through the classical procedures of Baer (1938), Tutte (1947) and Lovász (1989), there is a $p$-group $P_G$ of class 2 and exponent $p$ that is naturally associated with $G$. Our first result is to show that this construction of groups from graphs respects isomorphism types. That is, given two graphs $G$ and $H$, $G$ and $H$ are isomorphic as graphs if and only if $P_G$ and $P_H$ are isomorphic as groups. Our second contribution is a new homomorphism notion for graphs. Based on this notion, a category of graphs can be defined, and the Baer–Lovász–Tutte construction naturally leads to a functor from this category of graphs to the category of groups.     

Representation and factorization theorems for almost-Lp-spaces

We extend the notions of $p$-convexity and $p$-concavity for Banach ideals of measurable functions following an asymptotic procedure. We prove a representation theorem for the spaces satisfying both properties as the one that works for the classical case: each almost $p$-convex and almost $p$-concave space is order isomorphic to an almost-$L^p$-space. The class of almost-$L^p$-spaces contains, in particular, direct sums of (infinitely many) $L^p$-spaces with different norms, that are not in general $p$-convex – nor $p$-concave –. We also analyze in this context the extension of the Maurey–Rosenthal factorization theorem that works for $p$-concave operators acting in $p$-convex spaces. In this way we provide factorization results that allow to deal with more general factorization spaces than $L^p$-spaces.     

C-system of a module over a Jf-relative monad

Let $\mathbb{F}$ be the category with the set of objects $\mathbb{N}$ and morphisms given by the functions between the standard finite sets of the corresponding cardinalities. Let $Jf:\mathbb{F}\to Sets(U)$ be the obvious functor from this category to the category of sets in a given Grothendieck universe U. In this paper we construct, for any Jf-relative monad RR and any left RR-module LM, a C-system $C(\mathbf{RR},\mathbf{LM})$ and explicitly compute the action of the four B-system operations on its B-sets.In the introduction we explain in detail the relevance of this result to the construction of the term C-systems of type theories.     

The space of bounded analytic functions and weighted Bergman spaces

In this paper, we construct a class of weighted Bergman spaces such that they can generate $H^{\infty }$ in the sense of Möbius invariance. $H^{\infty }$ is also equal to the intersection of these spaces. Applying the relation between $H^{\infty }$ and these spaces, we show that the characterization via higher order derivatives does not hold for some of these weighted Bergman spaces.     

A short note on band projections in partially ordered vector spaces

Consider an Archimedean partially ordered vector space $X$ with generating cone (or, more generally, a pre-Riesz space $X$). Let $P$ be a linear projection on $X$ such that both $P$ and its complementary projection $I?P$ are positive; we prove that the range of $P$ is a band. This shows that the well-known concept of band projections on vector lattices can, to a certain extent, be transferred to the framework of ordered vector spaces.     

Mixed quantum skew Howe duality and link invariants of type A

We define a ribbon category $\mathsf{Sp}(\beta )$, depending on a parameter β, which encompasses Cautis, Kamnitzer and Morrison's spider category, and describes for $\beta =m?n$ the monoidal category of representations of $U_q({\mathfrak{gl}}_{m|n})$ generated by exterior powers of the vector representation and their duals. We identify this category $\mathsf{Sp}(\beta )$ with a direct limit of quotients of a dual idempotented quantum group ${\dot{\mathbf{U}}}_q({\mathfrak{gl}}_{r+s})$, proving a mixed version of skew Howe duality in which exterior powers and their duals appear at the same time. We show that the category $\mathsf{Sp}(\beta )$ gives a unified natural setting for defining the colored ${\mathfrak{gl}}_{m|n}$ link invariant (for $\beta =m?n$) and the colored HOMFLY-PT polynomial (for β generic).