Related papers: Unital ${A}_\infty$-categories

We prove formulas for the unitary SK_1 of a semiramified graded division algebra (or valued division algebra over a Henselian field) with a unitary involution. These formulas generalize earlier formulas of Yanchevskii, (and Platonov and…

To an exact endofunctor of a triangulated category with a split-generator, the notion of entropy is given by Dimitrov-Haiden-Katzarkov-Kontsevich, which is a (possibly negative infinite) real-valued function of a real variable. In this…

This paper is the first in a series of two papers, $\mathbf{Z}$-Categories I and $\mathbf{Z}$-Categories II, which develop the notion of $\mathbf{Z}$-category, the natural bi-infinite analog to strict $\omega$-categories, and show that the…

Linear categories naturally have several identification relations : isomorphisms, categorical equivalences and Morita equivalences. In this thesis, we construct the classifying stacks for these three relations ($\ukcatiso$, $\ukcateq$,…

We present here definitions and constructions basic for the theory of monoidal and tensor categories. We provide references to the original sources, whenever possible. Group-theoretical categories are used as examples

In this paper we prove an $\infty$-categorical version of the reflection theorem of Ad\'amek-Rosick\'y. Namely, that a full subcategory of a presentable $\infty$-category which is closed under limits and $\kappa$-filtered colimits is a…

We relate a construction of Kadeishvili's establishing an A-infinity-structure on the homology of a differential graded algebra or more generally of an A-infinity algebra with certain constructions of Chen and Gugenheim. Thereafter we…

Simple argument in favour of unitarity, to all orders, of space-like noncommutative theory is given.

We introduce the concept of degree to classify the periods in the sense of Kontsevich. Using this notion we give some new understanding of some problems in transcendental number theory.

The Univalence Principle is the statement that equivalent mathematical structures are indistinguishable. We prove a general version of this principle that applies to all set-based, categorical, and higher-categorical structures defined in a…

This paper presents and philosophically assesses three types of results on the observational equivalence of continuous-time measure-theoretic deterministic and indeterministic descriptions. The first results establish observational…

We present an $\ell$-adic trace formula for saturated and admissible dg-categories over a base monoidal dg-category. Moreover, we prove K\"unneth formulas for dg-category of singularities, and for inertia-invariant vanishing cycles. As an…

A categorical generalization of the notion of movability from the inverse systems and shape theory was given by the first author who defined the notion of movable category and interpreted by this the movability of topological spaces. In…

In this work, we prove a generalization of Quillen's Theorem A to 2-categories equipped with a special set of morphisms which we think of as weak equivalences, providing sufficient conditions for a 2-functor to induce an equivalence on…

We define the zeta function of a finite category. And we propose a conjecture which states the relationship between the Euler characteristic of finite categories and the zeta function of finite categories. This conjecture is verified when…

In his $1994$ survey, Kleinert defined formally and formulated the problem to obtain unit theorems for unit groups of orders in a semisimple algebra $A$. If $A$ is a group algebra $FG$, it boils down to classifying all finite groups $G$…

We construct unital extensions of the higher order peak algebras defined by Krob and the third author in [Ann. Comb. 9 (2005), 411--430.], and show that they can be obtained as homomorphic images of certain subalgebras of the…

We present some results on (co)limits of diagrams in $\infty$-categories, as well as those in $(n, 1)$-categories. In particular, we deduce a way to reshape colimit diagrams into simplicial ones, and a characterisations of $n$-cofinality…

This paper is an expository account of the theory of stable infinity categories. We prove that the homotopy category of a stable infinity category is triangulated, and that the collection of stable infinity categories is closed under a…

This paper contains a complete description of classes of the unitary equivalence of the admissible representations of infinite-dimensional classic matrix groups paper.