Related papers: Rigid Algebras and Cospans

We show that weak monoidal Quillen equivalences induce equivalences of symmetric monoidal $\infty$-categories with respect to the Dwyer-Kan localization of the symmetric monoidal model categories. The result will induce a Dold-Kan…

Quantitative algebras are algebras enriched in the category $\mathsf{Met}$ of metric spaces so that all operations are nonexpanding. Mardare, Plotkin and Panangaden introduced varieties (aka $1$-basic varieties) as classes of quantitative…

To any rigid analytic space (in the sense of Fujiwara-Kato) we assign an $\mathbb{A}^1$-invariant rigid analytic homotopy category with coefficients in any presentable category. We show some functorial properties of this assignment as a…

We establish a correspondence between modules and spans of algebras within a general monoidal 2-category $\mathfrak{C}$. Specifically, for an algebra $A$ in $\mathfrak{C}$, we construct a normalized lax 3-functor from the 2-category of…

One goal of applied category theory is to better understand networks appearing throughout science and engineering. Here we introduce "structured cospans" as a way to study networks with inputs and outputs. Given a functor $L \colon…

We give an operadic definition of a genuine symmetric monoidal G-category, and we prove that its classifying space is a genuine E_\infty G-space. We do this by developing some very general categorical coherence theory. We combine results of…

We prove that the category of (strictly unital) A$_\infty$-categories, linear over a commutative ring $R$, with strict A$_\infty$-morphisms has a cofibrantly generated model structure. In this model structure every object is fibrant and the…

The notion of conformal algebras was introduced by Victor G. Kac using the axiomatic description of the operator product expansion of chiral fields in conformal field theory. The structure theory, representations and cohomology of Lie and…

For a commutative ring $A$, we have the category of (bounded-below) chain complexes of $A$-modules $Ch_{+}(A\mymod)$, a closed symmetric monoidal category with a compatible stable Quillen model structure. The associated homotopy category is…

This article is a survey of algebra in the $\infty$-categorical context, as developed by Lurie in "Higher Algebra", and is a chapter in the "Handbook of Homotopy Theory". We begin by introducing symmetric monoidal stable…

We demonstrate that any full and faithful $*$-functor between approximable categories of locally finite coarse spaces induces a coarse embedding between the underlying spaces. Furthermore, we establish a general characterisation of such…

We define closed model category structures on different categories connected to the world of operad algebras over the category C(k) of (unbounded) complexes of k-modules: on the category of operads, on the category of algebras over a fixed…

Fusion categories are fundamental objects in quantum algebra, but their definition is narrow in some respects. By definition a fusion category must be k-linear for some field k, and every simple object V is strongly simple, meaning that (V)…

We show that for a monoidal model category $\M=(\ul{M}, \otimes, I)$, certain co-Segal $\M$-categories are equivalent to strict ones.

A PROP is a symmetric monoidal category, whose set of objects is the set of natural numbers and on objects the monoidal structure is given by the addition. An algebra over a PROP is a symmetric strict monoidal functor to the tensor category…

We study monoidal categorifications of certain monoidal subcategories $\mathcal{C}_J$ of finite-dimensional modules over quantum affine algebras, whose cluster algebra structures coincide and arise from the category of finite-dimensional…

In this paper we study compact closed categories within the context of homotopical algebra. We construct two new model category structures by localizing two (Quillen equivalent) model categories of symmetric monoidal categories with the…

By arithmeticity and superrigidity, a commensurability class of lattices in a higher rank Lie group is defined by a unique algebraic group over a unique number subfield of $\mathbb{R}$ or $\mathbb{C}$. We prove an adelic version of…

In this paper we explore a generic notion of superrigidity for von Neumann algebras $L(G)$ and reduced $C^*$-algebras $C^*_r(G)$ associated with countable discrete groups $G$. This allows us to classify these algebras for various new…

For a finite group $G$, we construct a simplified model for the $G$-symmetric monoidal $G$-$\infty$-category of rational $G$-spectra. Using this model, we classify $\mathcal{I}$-normed algebras in rational $G$-spectra for a given indexing…