Related papers: On the linear independency of monoidal natural tra…

Gaussian elimination answers any question about a finitely presented vector space. However, a "uniform family" of such presentations--given as generic relations among an unspecified number of generators--is susceptible to elimination only…

In this paper, we show that there are infinitely many semisimple tensor (or monoidal) categories of rank two over an algebraically closed field $\mathbb F$.

If $\mathbb{F}$ is an ordered field and $M$ is a finite-rank torsion-free monoid, then one can embed $M$ into a finite-dimensional vector space over $\mathbb{F}$ via the inclusion $M \hookrightarrow \text{gp}(M) \hookrightarrow \mathbb{F}…

The purpose of this paper is to show that various convolution products are fully homotopical, meaning that they preserve weak equivalences in both variables without any cofibrancy hypothesis. We establish this property for diagrams of…

Let ${\mathcal M}$ be a moduli space of stable vector bundles of rank $r$ and determinant $\xi$ on a compact Riemann surface $X$. Fix a semistable holomorphic vector bundle $F$ on $X$ such that $\chi(E\otimes F)= 0$ for $E \in \mathcal M$.…

We suggest two approaches to a definition of unitarity for pseudonatural transformations between unitary pseudofunctors on pivotal dagger 2-categories. The first is to require that the 2-morphism components of the transformation be unitary.…

We construct a modular functor which takes its values in the monoidal bicategory of finite categories, left exact functors and natural transformations. The modular functor is defined on bordisms that are 2-framed. Accordingly we do not need…

We provide conditions on a monoidal model category $\mathcal{M}$ so that the category of commutative monoids in $\mathcal{M}$ inherits a model structure from $\mathcal{M}$ in which a map is a weak equivalence or fibration if and only if it…

Using a generalization of forward elimination, it is proved that functions $f_1,...,f_n:X\to\mathbb{A}$, where $\mathbb{A}$ is a field, are linearly independent if and only if there exists a nonsingular matrix $[f_i(x_j)]$ of size $n$,…

We construct an equivalence between the 2-categories VMonCat of rigid V-monoidal categories for a braided monoidal category V and VModTens of oplax braided functors from V into the Drinfeld centers of ordinary rigid monoidal categories. The…

Let $G=\{e^{tA}:t\in\mathbb{R}\}$ be a closed one-parameter subgroup of the general linear group of matrices of order $n$ acting on $\mathbb{R}^{n}$ by matrix-vector multiplications. We assume that all eigenvalues of $A$ are rationally…

Given a functor $F: \mathcal{C} \to \mathcal{D}$ and a model-theoretic independence relation on $\mathcal{D}$, we can lift that independence relation along $F$ to $\mathcal{C}$ by declaring a commuting square in $\mathcal{C}$ to be…

Let $K$ be a complete non-archimedean field over $\mathbb{Q}_p$, $G$ be a rigid group over $K$, and $X$ be a perfectoid space over $K$. We consider the natural morphism of sites $\nu: X_v \to X_{\mathrm{\acute{e}t}}$. It is known from work…

For two not necessarily commutative topological groups G and T, let H(G,T) denote the space of all continuous homomorphisms from G to T with the compact-open topology. We prove that if G is metrizable and T is compact then H(G,T) is a…

A graded tensor category over a group $G$ will be called a crossed product tensor category if every homogeneous component has at least one multiplicatively invertible object. Our main result is a description of the crossed product tensor…

The category of double categories and double functors is equipped with a symmetric closed monoidal structure. For any double category $\mathbb A$, the corresponding internal hom functor $|[ \mathbb A,-]|$ sends a double category $\mathbb B$…

Let $G$ be a compact $p$-adic analytic group and $k$ a field positive characteristic. We prove that for every smooth representation of $G$ on a $k$-vector space $V$, every 1-cocycle $G\to V$ is continuous. We deduce that the first derived…

We interpret ontological models for finite-dimensional quantum theory as functors from the category of finite-dimensional Hilbert spaces and bounded linear maps to the category of measurable spaces and Markov kernels. This uniformises…

We study positive subunital maps on ordered effect spaces and introduce the defect $d(T) = u - T(u)$, which satisfies a cocycle identity under composition. Using only this identity and elementary order-theoretic arguments -- requiring no…

We investigate tensor products of matrix factorisations. This is most naturally done by formulating matrix factorisations in terms of bimodules instead of modules. If the underlying ring is C[x_1,...,x_N] we show that bimodule matrix…