Related papers: Adams' cobar construction as a monoidal $E_{\infty…

In this paper we show that if a cosimplicial space or spectrum $X^\bullet$ has a certain kind of combinatorial structure (we call it a $\Xi^n$-structure) then the total space of $X^\b$ has an action of a certain operad which is weakly…

Given an inclusion $A\hookrightarrow L$ of Lie algebroids sharing the same base manifold $M$, i.e. a Lie pair, we prove that the space $\Gamma(\Lambda^\bullet A^\vee)\otimes_{R} \frac{U(L)}{U(L)\cdot\Gamma(A)}$, where $R=C^\infty(M)$,…

We prove that for a topological operad $P$ the operad of oriented cubical chains, $C^{ord}_\ast(P)$, and the operad of singular chains, $S_\ast(P)$, are weakly equivalent. As a consequence, $C^{ord}_\ast(P;\mathbb{Q})$ is formal if and only…

We give a simple counterexample to the plausible conjecture that Adams-type maps of ring spectra are stable under composition. We then show that over a field, this failure is quite extreme, as any map of $\mathbb{E}_{\infty}$-algebras is a…

We study a special type of $E_\infty$-operads that govern strictly unital $E_\infty$-coalgebras (and algebras) over the ring of integers. Morphisms of coalgebras over such an operad are defined by using universal $E_\infty$-bimodules. Thus…

Given a closed manifold $M$. We give an algebraic model for the Chas-Sullivan product and the Goresky-Hingston coproduct. In the simply-connected case, this admits a particularly nice description in terms of a Poincar\'e duality model of…

This paper studies averaging algebras, say, associative algebras endowed with averaging operators. We develop a cohomology theory for averaging algebras and justify it by interpreting lower degree cohomology groups as formal deformations…

Curved algebras are a generalization of differential graded algebras which have found numerous applications recently. The goal of this foundational article is to introduce the notion of a curved operad, and to develop the operadic calculus…

We propose an alternative representation for linear quantum gravity. It is based on the use of a structure that bears some resemblance to the Abelian loop representation used in electromagnetism but with the difference that space of…

Roe algebras are C*-algebras built using large-scale (or 'coarse') aspects of a metric space (X,d). In the special case that X=G is a finitely generated group and d is a word metric, the simplest Roe algebra associated to (G,d) is…

A characterization of simplicial objects in categories with finite products obtained by the reduced bar construction is given. The condition that characterizes such simplicial objects is a strictification of Segal's condition guaranteeing…

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…

Let $k$ be a field, $k^*=k\setminus\{0\}$ and $C_2$ the cyclic group of order 2. In this note we compute all the braided monoidal structures on the category of $k$-vector spaces graded by the Klein group $C_2\times C_2$. Actually, for the…

In this paper we give an example of duoidal $\infty$-categories. We introduce map $\mathcal{O}$-monoidales in an $\mathcal{O}$-monoidal $(\infty,2)$-category for an $\infty$-operad $\mathcal{O}^{\otimes}$. We show that the endomorphism…

The notion of a complete Boolean algebra, although completely legitimate in constructive mathematics, fails to capture some natural structures such as the lattice of subsets of a given set. Sambin's notion of an overlap algebra, although…

For a fixed closed manifold $P$, we construct a cobordism category of embedded manifolds with a single Baas-Sullivan singularity of type $P$. Our main theorem identifies the homotopy type of the classifying space of this cobordism category…

Consider the semialgebraic structure over the real field. More generally, let an ominimal structure be over a real closed field. We show that a definable metric space X with a definable metric d is embedded into a Euclidean space so that…

The internal bialgebroid -- in a symmetric monoidal category with coequalizers -- is defined. The axioms are formulated in terms of internal entwining structures and alternatively, in terms of internal corings. The Galois property of the…

Our objective in this article is to show a possibly interesting structure of homotopic nature appearing in persistent (co)homology. Assuming that the filtration of the (say) simplicial set embedded in a finite dimensional vector space…

Given a map $f: X\rightarrow Y$ of simply connected spaces of finite type such. The space of based loops at $f$ of the space of maps between $X$ and $Y$ is denoted by $\Omega_{f} Map(X,Y)$. For $n> 0$, we give a model categorical…