Related papers: What do homotopy algebras form?

We prove an equivalence of categories from formal complex structures with formal holomorphic maps to homotopy algebras over a simple operad with its associated homotopy morphisms. We extend this equivalence to complex manifolds. A complex…

We give an alternative to Postnikov's homotopy classification of maps from 3-dimensional CW-complexes to homogeneous spaces G/H of Lie groups. It describes homotopy classes in terms of lifts to the group G and is suitable for extending the…

A kind of unstable homotopy theory on the category of associative rings (without unit) is developed. There are the notions of fibrations, homotopy (in the sense of Karoubi), path spaces, Puppe sequences, etc. One introduces the notion of a…

Both simplicial sets and simplicial spaces are used pervasively in homotopy theory as presentations of spaces, where in both cases we extract the "underlying space" by taking geometric realization. We have a good handle on the category of…

We define a strong homotopy derivation of (cohomological) degree k of a strong homotopy algebra over an operad P. This involves resolving the operad obtained from P by adding a generator with "derivation relations". For a wide class of…

We compute the homotopy groups at each unital abelian C*-algebra $C(T)$ in the Morita $3$-category of abelian C*-algebras, C*-algebras with central maps, C*-correspondences, and adjointable bimodule maps. We describe these groups in terms…

We introduce simple models for associative algebras and bimodules in the context of non-symmetric $\infty$-operads, and use these to construct an $(\infty,2)$-category of associative algebras, bimodules, and bimodule homomorphisms in a…

In this paper we address the classification problem for purely infinite simple Leavitt path algebras of finite graphs over a field $\ell$. Each graph $E$ has associated a Leavitt path $\ell$-algebra $L(E)$. There is an open question which…

We study quantization of a class of inhomogeneous Lie bialgebras which are crossproducts in dual sectors with Abelian invariant parts. This class forms a category stable under dualization and the double operations. The quantization turns…

We construct a left semi-model structure on the category of intensional type theories (precisely, on $\mathrm{CxlCat_{Id,1,\Sigma(,\Pi_{ext})}}$). This presents an $\infty$-category of such type theories; we show moreover that there is an…

We extend the homotopy theories based on point reduction for finite spaces and simplicial complexes to finite acyclic categories and $\Delta$-complexes, respectively. The functors of classifying spaces and face posets are compatible with…

We extend some classical results of Bousfield on homology localizations and nilpotent completions to a presentably symmetric monoidal stable $\infty$-category $\mathscr{M}$ admitting a multiplicative left-complete $t$-structure. If $E$ is a…

First we survey and explain the strategy of some recent results that construct holomorphic $\text{sl}(2, \mathbb C)$-differential systems over some Riemann surfaces $\Sigma_g$ of genus $g\geq 2$, satisfying the condition that the image of…

We extend the category of (super)manifolds and their smooth mappings by introducing a notion of microformal or "thick" morphisms. They are formal canonical relations of a special form, constructed with the help of formal power expansions in…

We define $A_{\infty}$-structures -- algebras, coalgebras, modules, and comodules -- in an arbitrary monoidal DG category or bicategory by rewriting their definitions in terms of unbounded twisted complexes. We develop new notions of strong…

Given a good homology theory E and a topological space X, the E-homology of X is not just an E_{*}-module but also a comodule over the Hopf algebroid (E_{*}, E_{*}E). We establish a framework for studying the homological algebra of…

We prove, under mild assumptions, that a Quillen equivalence between symmetric monoidal model categories gives rise to a Quillen equivalence between their model categories of (non-symmetric) operads, and also between model categories of…

We prove that a weak equivalence between two cofibrant (colored) props in chain complexes induces a Dwyer-Kan equivalence between the simplicial localizations of the associated categories of algebras. This homotopy invariance under base…

Over suitable monoidal model categories, we construct a Dwyer-Kan model category structure on the category of algebras over an augmented operadic collection. As examples we obtain Dwyer-Kan model category structure on the categories of…

This is a survey article with the goal to advertise spectrum valued versions of $K$- and $KK$- theory for $C^{*}$-algebras via a (stable and symmetric monoidal) $\infty$-categorical enhancement of Kasparov's classical $KK$-theory. The main…