A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often model-theoretic constructions that add sets to models are cited as evidence in favor of the latter. This paper informs this debate by developing a way for a Universist to interpret talk that seems to necessitate the addition of sets to V. We argue that, despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We present a method of interpreting extension-talk (V-logic), and show how it captures satisfaction in ‘ideal’ outer models and relates to impredicative class theories. We provide some reasons to regard the technique as philosophically virtuous, and argue that it opens new doors to philosophical and mathematical discussions for the Universist.

We give a complete description of a basis of the extension spaces between indecomposable string and quasi-simple band modules in the module category of a gentle algebra.

Assume that $A$ is a finite-dimensional algebra over some field, and assume that $A$ is weakly symmetric and indecomposable, with radical cube zero and radical square nonzero. We show that such an algebra of wild representation type does not have a nonprojective module $M$ whose ext-algebra is finite dimensional. This gives a complete classification of weakly symmetric indecomposable algebras which have a nonprojective module whose ext-algebra is finite dimensional. This shows in particular that existence of ext-finite nonprojective modules is not equivalent with the failure of the finite generation condition (Fg), which ensures that modules have support varieties.

Soit $G$ un groupe réductif connexe déployé sur une extension finie $F$ de $\mathbb{Q}_{p}$. Nous déterminons les extensions entre séries principales continues unitaires $p$-adiques et lisses modulo $p$ de $G(F)$ dans le cas générique. Pour cela, nous calculons le delta-foncteur $\text{H}^{\bullet }\text{Ord}_{B(F)}$ des parties ordinaires dérivées d’Emerton relatif à un sous-groupe de Borel sur certaines représentations induites de $G(F)$ en utilisant une filtration de Bruhat. Ces extensions interviennent dans le programme de Langlands $p$-adique et modulo $p$.

Let $\mathfrak{A}$ be a ${{C}^{*}}$ -algebra with real rank zero that has the stable weak cancellation property. Let $\Im $ be an ideal of $\mathfrak{A}$ such that $\Im $ is stable and satisfies the corona factorization property. We prove that

$$0\,\to \,\Im \,\to \mathfrak{A}\,\to \,\mathfrak{A}/\Im \,\to \,0$$

is a full extension if and only if the extension is stenotic and $K$ -lexicographic. As an immediate application, we extend the classification result for graph ${{C}^{*}}$ -algebras obtained by Tomforde and the first named author to the general non-unital case. In combination with recent results by Katsura, Tomforde, West, and the first named author, our result may also be used to give a purely $K$ -theoretical description of when an essential extension of two simple and stable graph ${{C}^{*}}$ -algebras is again a graph ${{C}^{*}}$ -algebra.

We characterize the abelian groups $G$ for which the functors $\mathrm{Ext} (G, - )$ or $\mathrm{Ext} (- , G)$ commute with or invert certain direct sums or direct products.

Let C*(E) be the graph C*-algebra associated to a graph E and let J be a gauge-invariant ideal in C*(E). We compute the cyclic six-term exact sequence in K-theory associated to the extension

in terms of the adjacency matrix associated to E. The ordered six-term exact sequence is a complete stable isomorphism invariant for several classes of graph C*-algebras, for instance those containing a unique proper nontrivial ideal. Further, in many other cases, finite collections of such sequences constitute complete invariants.

Our results allow for explicit computation of the invariant, giving an exact sequence in terms of kernels and cokernels of matrices determined by the vertex matrix of E.

This paper gives a complete classification of essentially commutative C*-algebras whose essential spectrum is homeomorphic to S2n−1 by their characteristic numbers. Let 1, 2 be such two C*-algebras; then they are C*-isomorphic if and only if they have the same n-th characteristic number. Furthermore, let γn() = m then is C*-isomorphic to C*(Mzl, …, Mzn) if m = 0, is C*-isomorphic C*(Tz1, …, Tzn−1, Tznm) if m ≠ 0. Some examples are given to show applications of the classfication theorem. We finally remark that the proof of the theorem depends on a construction of a complete system of representatives of Ext(S2n−1).