We introduce the notion of a matrad $M=\{M_{n,m}\}$ whose submodules $M_{*,1}$ and $M_{1,*}$ are non-$\Sigma$ operads. We define the free matrad $\mathcal{H}_\infty$ generated by a singleton $\theta^n_m$ in each bidegree $(m,n)$ and realize $\mathcal{H}_\infty$ as the cellular chains on a new family of polytopes $\{KK_{n,m}=KK_{m,n}\}$, called biassociahedra, of which $KK_{n,1}$ is the associahedron $K_n$. We construct the universal enveloping functor from matrads to PROPs and define an $A_\infty$-bialgebra as an algebra over $\mathcal{H}_\infty$.

From the viewpoint of semi-abelian homology, some recent results on homology of Leibniz $n$-algebras can be explained categorically. In parallel with these results, we develop an analogous theory for Lie $n$-algebras. We also consider the relative case: homology of Leibniz $n$-algebras relative to the subvariety of Lie $n$-algebras.

Let $R$ be a ring and let $\mathcal{A}$ be a hereditary torsion class of $R$-modules. The inclusion of the localizing subcategory generated by $\mathcal{A}$ into the derived category of $R$ has a right adjoint, denoted CellA. Recently, Benson has shown how to compute $\rm{Cell}_{\mathcal{A}}R$ when $R$ is a group ring of a finite group over a prime field and $\mathcal{A}$ is the hereditary torsion class generated by a simple module. We generalize Benson’s construction to the case where $\mathcal{A}$ is any hereditary torsion class on $R$. It is shown that for every $R$-module $M$ there exists an injective $R$-module $E$ such that:

$$H^n(\rm{Cell}_{\mathcal{A}}M)\cong \rm{Ext}^{n-1}_{\rm{End}_R(E)} (\rm{Hom}_R (M,E),E)\hbox{ for }n\ge 2. $$

We develop a localisation theory for certain categories, yielding a 3-arrow calculus: Every morphism in the localisation is represented by a diagram of length 3, and two such diagrams represent the same morphism if and only if they can be embedded in a 3-by-3 diagram in an appropriate way. Applications include the localisation of an arbitrary Quillen model category with respect to its weak equivalences as well as the localisation of its full subcategories of cofibrant, fibrant and bifibrant objects, giving the homotopy category in all four cases. In contrast to the approach of Dwyer, Hirschhorn, Kan and Smith, the Quillen model category under consideration does not need to admit functorial factorisations.

We prove that the folk model structure on strict $\infty$-categories transfers to the category of strict $\infty$-groupoids (and more generally to the category of strict $(\infty; n)$-categories), and that the resulting model structure on strict $\infty$-groupoids coincides with the one defined by Brown and Golasiński via crossed complexes.

We establish a relation between smooth 2-functors defined on the path 2-groupoid of a smooth manifold and differential forms on this manifold. This relation can be understood as a part of a dictionary between fundamental notions from category theory and differential geometry. We show that smooth 2-functors appear in several fields, namely as connections on (non-abelian) gerbes, as derivatives of smooth functors and as critical points in BF theory. We demonstrate further that our dictionary provides a powerful tool to discuss the transgression of geometric objects to loop spaces.

We show that every monomial ring can be realized topologically by a certain topological space. This space is called a generalized Davis-Januszkiewicz space and can be thought of as a colimit over a multicomplex, a combinatorial object generalizing a simplicial complex. Furthermore, we show that such a space is obtained as the homotopy fiber of a certain map with total space the classical Davis-Januszkiewicz space.

We study cohomology for $p$-local finite groups with nonconstant coefficient systems. In particular we show that under certain restrictions there exists a cohomology transfer map in this context, and deduce the standard consequences.

It is proved that Kaygun’s Hopf-Hochschild cochain complex for a module-algebra is a brace algebra with multiplication. As a result, an analogue of Deligne’s Conjecture holds for module-algebras, and the Hopf-Hochschild cohomology of a module-algebra has a Gerstenhaber algebra structure.

In this paper we use Peter May’s algebraic approach to Steenrod operations to construct Steenrod’s reduced power operations in simplicial Bredon-Illman cohomology with local coefficients of a one vertex $G$-Kan complex, $G$ being a discrete group.

This note discusses the cyclic cohomology of a left Hopf algebroid ($\times_A$-Hopf algebra) with coefficients in a right module-left comodule, defined using a straightforward generalisation of the original operators given by Connes and Moscovici for Hopf algebras. Lie-Rinehart homology is a special case of this theory. A generalisation of cyclic duality that makes sense for arbitrary para-cyclic objects yields a dual homology theory. The twisted cyclic homology of an associative algebra provides an example of this dual theory that uses coefficients that are not necessarily stable anti Yetter-Drinfel’d modules

In this paper, we show that the étale index of a torsion cohomological Brauer class is divisible by the period of the class. The tool used to make this computation is the Čech approximation of the title. To create the approximation, we use the folklore theorem that the homotopy limit and Postnikov spectral sequences for a cosimplicial space agree beginning with the $E_2$-page. As far we know, this folklore theorem has no proof in the literature, so we include a proof.

Let $M$ be a homogeneous space admitting a left translation by a connected Lie group $G$. The adjoint to the action gives rise to a map from $G$ to the monoid of self-homotopy equivalences of $M$. The purpose of this paper is to investigate the injectivity of the homomorphism which is induced by the adjoint map on the rational homotopy group. In particular, the visibility degrees are determined explicitly for all the cases of simple Lie groups and their associated homogeneous spaces of rank one which are classified by Oniscik.

Let $G$ be a finite group and $k$ be a field of characteristic $p > 0$. A cohomology class $\zeta\in H^n (G,k)$ is called productive if it annihilates $\rm{Ext}^*_{kG}(L_\zeta,L_\zeta)$. We consider the chain complex $\mathbf{P}(\zeta)$ of projective $kG$-modules which has the homology of an $(n - 1)$-sphere and whose $k$-invariant is $\zeta$ under a certain polarization. We show that $\zeta$ is productive if and only if there is a chain map $\Delta : \mathbf{P}(\zeta)\to \mathbf{P}(\zeta)\otimes \mathbf{P}(\zeta)$ such that $(\rm{id} \otimes \epsilon) \Delta \simeq \rm{id}$ and $(\epsilon \otimes \rm{id}) \Delta \simeq \rm{id}$. Using the Postnikov decomposition of $\mathbf{P}(\zeta) \otimes \mathbf{P}(\zeta)$, we prove that there is a unique obstruction for constructing a chain map $\Delta$ satisfying these properties. Studying this obstruction more closely, we obtain theorems of Carlson and Langer on productive elements.

In [1] (L. Fajstrup, Dicovering spaces, Homology Homotopy Appl. 5 (2003), no. 2, 1–17), we study coverings in the setting of directed topology. Unfortunately, there is a condition missing in the definition of a directed covering. Some of the results in [1] require this extra condition and in fact it was claimed to follow from the original definition. It is the purpose of this note to give the right definition and point out how this affects the statements in that paper. Moreover, we give an example of a dicovering in the sense of [1], which does not satisfy the extra condition. Fortunately, with the extra condition, the subsequent results are now correct.