Related papers: Rational relative sectional category

In this paper, we study the Lusternik-Schnirelmann category of a simplicial map between simplicial complexes, generalizing the simplicial category of a complex to that of a map. Several properties of this new invariant are shown, including…

The first goal of this paper is to provide an abstract framework in which to formulate and study local duality in various algebraic and topological contexts. For any stable $\infty$-category $\mathcal{C}$ together with a collection of…

We prove that holomorphic maps from an open subset of a complex smooth projective curve to a complex smooth projective rationally simply connected variety can be approximated by algebraic maps for the compact-open topology. This theorem can…

We show that the classification diagram of a relative $\infty$-category arising from a relative simplicial category is equivalent to the levelwise nerve. Applications include the comparison of the diagonal of the levelwise nerve and the…

Let X, Y be nonsingular real algebraic sets. A map fi:X-->Y is said to be k-regulous, where k is a nonnegative integer, if it is of class C^k and the restriction of fi to some Zariski open dense subset of X is a regular map. Assuming that Y…

We classify the homotopy classes of proper Fredholm maps from an infinite dimensional Hilbert manifold into its model space in terms of a suitable version of framed cobordism. Our construction is an alternative approach to the…

The completely positive rank is an analogue of topological covering dimension, defined for nuclear C*-algebras via completely positive approximations. These may be thought of as simplicial approximations of the algebra, which leads to the…

These notes are based on a series of three lectures given (online) by the first named author at the workshop "Higher Structures and Operadic Calculus" at CRM Barcelona in June 2021. The aim is to give a concise introduction to rational…

We develop a new, intrinsic, computationally friendly approach to Lie coalgebras through graph coalgebras, which are new and likely to be of independent interest. Our graph coalgebraic approach has advantages both in finding relations…

We present an intrinsic and concrete development of the subdivision of small categories, give some simple examples and derive its fundamental properties. As an application, we deduce an alternative way to compare the homotopy categories of…

This paper lays the foundations of an approach to applying Gromov's ideas on quantitative topology to topological data analysis. We introduce the "contiguity complex", a simplicial complex of maps between simplicial complexes defined in…

We provide a foundation for working with homological and homotopical methods in categorical algebra. This involves two mutually complementary components, namely (a) the strategic selection of suitable axiomatic frameworks, some well known…

This paper establishes the separation of complexity classes $\mathbf{P}$ and $\mathbf{NP}$ through a novel homological algebraic approach grounded in category theory. We construct the computational category $\mathbf{Comp}$, embedding…

Given a simple undirected graph, one can construct from it a $c$-step nilpotent Lie algebra for every $c \geq 2$ and over any field $K$, in particular also over the real and complex numbers. These Lie algebras form an important class of…

We study the Section Conjecture in \'etale homotopy theory for varieties over $\mathbb{R}$. We prove its pro-$2$ variant for equivariantly triangulable varieties. Examples include all smooth varieties as well as all (possibly singular)…

In this paper, Lusternik-Schinrelmann and geometric category of finite spaces are considered. We define new numerical invariants of these spaces derived from the geometric category and present an algorithmic approach for its effective…

We give an overview of relative tensor products (RTPs) for von Neumann algebra modules. For background, we start with the categorical definition and go on to examine its algebraic formulation, which is applied to Morita equivalence and…

Categorification is the process of finding category-theoretic analogs of set-theoretic concepts by replacing sets with categories, functions with functors, and equations between functions by natural isomorphisms between functors, which in…

This is an introduction to the study of abstract homotopy theory by means of model categories and $(\infty,1)$-categories. The only prerequisites are very basic general topology and abstract algebra. None categorical background is needed.…

In the rational cohomology of a 1-connected space a structure of $C_{\infty}$-algebra is constructed and it is shown that this object determines the rational homotopy type