Related papers: Being Van Kampen is a universal property
We introduce the notion of a $c$-category, which is a kind of category whose behaviour is controlled by connective ring spectra. More precisely, any $c$-category admits a finite step resolution by categories of compact modules over…
We consider a condition for non-degenerate commuting squares of matrix algebras (finite dimensional von Neumann algebras) called the \emph{span condition}, which in the case of the $n$-dimensional standard spin models is shown to be…
Like categories, small 2-categories have well-understood classifying spaces. In this paper, we deal with homotopy types represented by 2-diagrams of 2-categories. Our results extend to homotopy colimits of 2-functors lower categorical…
This paper is the second in a series of two papers about generalizing Quillen's Theorem A to strict $\infty$-categories. In the first one, we presented a proof of this Theorem A of a simplicial nature, direct but somewhat ad hoc. In the…
We consider categories of posets with $\mathfrak{C}$-valued structure sheaves for any category $\mathfrak{C}$ and see how they possess poset-indexed lax colimits that are both easy to describe and "weakly equivalent" to their ordinary…
By means of several examples, we motivate that universal properties are the simplest way to solve a given mathematical problem, explaining in this way why they appear everywhere in mathematics. In particular, we present the co-universal…
A Universal Mapping Property is generally described as a characterization of an object up to a unique isomorphism by considering its relation to every other object; however, the term "by considering its relation to every other object" is…
Based on the intuitive notion of convexity, we formulate a universal property defining interval objects in a category with finite products. Interval objects are structures corresponding to closed intervals of the real line, but their…
We prove that if $D\subset C^n$ is a bounded domain with real analytic boundary and D is pseudoconvex then the compact open topology in the group of holomorphic automorphisms of D is the topology of uniform convergence on D.
The compactness phenomenon is one of the featured aspects of structuralism in mathematics. In simple and broad words, a compactness property holds in a structure if a related property is satisfied by sufficiently many substructures of that…
We define and give explicit construction of the universal tree-graded space with a given collection of pieces. We apply that to proving uniqueness of asymptotic cones of relatively hyperbolic groups whose peripheral subgroups have unique…
A celebrated theorem of Pimsner states that a covariant representation $T$ of a $C^*$-correspondence $E$ extends to a $C^*$-representation of the Toeplitz algebra of $E$ if and only if $T$ is isometric. This paper is mainly concerned with…
We prove a number of results linking properties of actions by compact groups (both quantum and classical) on Banach spaces, such as uniform continuity, spectrum finiteness and extensibility of the actions across several constructions.…
One of the aims of this article is to provide a class of polynomial mappings for which the Jacobian conjecture is true. Also, we state and prove several global univalence theorems and present a couple of applications of them.
The graph complex acts on the spaces of Poisson bi-vectors $P$ by infinitesimal symmetries. We prove that whenever a Poisson structure is homogeneous, i.e. $P = L_{\vec{V}}(P)$ w.r.t. the Lie derivative along some vector field $\vec{V}$,…
A category is adhesive if it has all pullbacks, all pushouts along monomorphisms, and all exactness conditions between pullbacks and pushouts along monomorphisms which hold in a topos. This condition can be modified by considering only…
In the sixties, Grothendieck developed the theory of pro-objects over a category. The fundamental property of the category $Pro(C)$ is that there is an embedding $C \stackrel{c}{\rightarrow} Pro(C)$, $Pro(C)$ is closed under small…
We categorify cocompleteness results of monad theory, in the context of pseudomonads. We first prove a general result establishing that, in any 2-category, weighted bicolimits can be constructed from oplax bicolimits and bicoequalizers of…
Necessary and sufficient conditions for the exactness (in the algebraic sense) of certain sequences of continuous group homomorphisms are established.
Proofs of coherence in category theory, starting from Mac Lane's original proof of coherence for monoidal categories, are sometimes based on confluence techniques analogous to what one finds in the lambda calculus, or in term-rewriting…