Related papers: Condensed Sets via free resolutions

In recent work, the authors developed a simple method of constructing topological spaces from certain well-behaved partially ordered sets -- those coming from sequences of relations between finite sets. This method associates a given poset…

We introduce analogues of the Hopf algebra of Free quasi-symmetric functions with bases labelled by colored permutations. As applications, we recover in a simple way the descent algebras associated with wreath products $\Gamma\wr\SG_n$ and…

What are the subcomplexes of a free resolution? This question is simple to state, but the naive approach leads to a computational quagmire that is infeasible even in small cases. In this paper, we invoke the Bernstein--Gelfand--Gelfand…

Staton has shown that there is an equivalence between the category of presheaves on (the opposite of) finite sets and partial bijections and the category of nominal restriction sets: see [2, Exercise 9.7]. The aim here is to see that this…

We discuss which groups can be realized as the fundamental groups of compact Hausdorff spaces. In particular, we prove that the claim ``every group can be realized as the fundamental group of a compact Hausdorff space'' is consistent with…

We present the foundational theory of condensed sets and basic condensed algebra after having introduced key concepts from category theory and homological algebra. In the later sections, we indicate the relevance of condensed mathematics to…

Let $\mathbb{D}$ be the category of pro-sets (or abelian pro-groups). It is proved that for any Grothendieck site $X$, there exists a reflector from the category of precosheaves on $X$ with values in $\mathbb{D}$ to the full subcategory of…

A famous result of Hausdorff states that a sphere with countably many points removed can be partitioned into three pieces A,B,C such that A is congruent to B (i.e., there is an isometry of the sphere which sends A to B), B is congruent to…

We consider the intersection map on the family of non-empty $\omega$-Scott-open sets of the lattice of opens of a topological space. We prove that in a certain class of topological spaces the intersection map forms a continuous retraction…

We prove that the Lipschitz-free space over a countable compact metric space is isometric to a dual space and has the metric approximation property.

This paper extends the Kadison duality between compact convex sets and function systems to the setting of partial convexity. A partially convex set is a set that is convex in a designated set of convex variables when the others are held…

We prove a quantitative distortion theorem for iterated function systems that generate sets of continued fractions. As a consequence, we obtain upper and lower bounds on the Hausdorff dimension of any set of real or complex continued…

Free products of semisimple tesnor categories are constructed with the help of polygonal presentation. The semisimplicity criterion is obtained for the Bisch-Jones' planar algebras as a byproduct.

In this paper, we examine the relationship between a CW-basis for a free simplicial group and methods of freely generating the corresponding crossed complex. Attention is concentrated on the case of resolutions, thus comparing free…

In this technical note we describe a pair of results on heterotic compactifications. First, we give an example demonstrating that the usual statement of the anomaly-freedom constraint for perturbative heterotic compactifications (meaning,…

We describe the canonical correspondence between set of all finite metric spaces and set of special symmetric convex polytopes, and formulate the problem about classification of the metric spaces in terms of combinatorial structure of those…

We introduce the group-compact coarse structure on a Hausdorff topological group in the context of coarse structures on an abstract group which are compatible with the group operations. We develop asymptotic dimension theory for the…

We collect some classical results on the versal families of ordinary compact Riemann surfaces needed in `Versal Families of Compact Super Riemann Surfaces'.

We characterize the notion of definable compactness for topological spaces definable in o-minimal structures, answering questions of Peterzil and Steinhorn (1999) and Johnson (2018). Specifically, we prove the equivalence of various…

The KC property, a separation axiom between weakly Hausdorff and Hausdorff, requires compact subsets to be closed. Various assumptions involving local conditions, dimension, connectivity, and homotopy show certain KC-spaces are in fact…