Related papers: Batanin higher groupoids and homotopy types

We continue to develop the theory of separable higher categories, including center functors, higher centralizers, modular extensions and group theoretical higher fusion categories. Moreover, we outline a theory of orthogonal higher…

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…

In our recent papers [Sh1,2], we introduced a {\it twisted tensor product} of dg categories, and provided, in terms of it, {\it a contractible 2-operad $\mathcal{O}$}, acting on the category of small dg categories, in which the "natural…

We characterize all the strongly monotypic polytopes. Hadwiger's conjecture for this class of polytopes is deduced from the characterization.

It has been proved by Bergh and Thaule that the higher mapping cone axiom is equivalent to the higher octahedral axiom for n-angulated categories. In this note, we use homotopy cartesian diagrams to give several new equivalent statements of…

In this paper we introduce the notion of weak operator and the theory of Yetter-Drinfeld modules over a weak braided Hopf algebra with invertible antipode in a strict monoidal category. We prove that the class of such objects constitutes a…

Using a previous classification result on symmetric additive 2-cocycles, we collect a variety of facts about the Lubin-Tate cohomology of formal groups to compute the 2-primary component of the scheme of symmetric multiplicative 2-cocycles.…

We study here the higher derived limits of mod $p$ cohomology on the centric orbit category of a saturated fusion system on a finite $p$-group. It is an open problem whether all such higher limits vanish. This is known in many cases,…

We give an explicit handy (and cocycle-free) description of the groupoid of weak maps between two crossed-modules in terms of certain digrams of groups which we we call a {\em butterflies}. We define composition of butterflies and this way…

Clemens Berger showed that Weak Omega Categories of Michael Batanin can be defined as model of a certain kind of theories that he called "homogeneous theories". By using the work of Mark Weber on the Abstract Nerves for the specific case of…

We provide the expected constructions of weakly $\omega$-categorified models (in the sense of Bressie) of the theory of groups and quandles which arise by replacing the homotopies used to give equivalence relations in the theory of…

Let k be a field. We attach a CW-complex to any Schurian k-category and we prove that the fundamental group of this CW-complex is isomorphic to the intrinsic fundamental group of the k-category. This extends previous results by J.C.…

We construct a `weak' version EM^w(K) of Lack & Street's 2-category of monads in a 2-category K, by replacing their compatibility constraint of 1-cells with the units of monads by an additional condition on the 2-cells. A relation between…

With an explicit, algebraic indexing $(2,1)$-category, we develop an efficient homotopy theory of cyclonic objects: circle-equivariant objects relative to the family of finite subgroups. We construct an $\infty$-category of cyclotomic…

In order to diagnose the cause of some defects in the category of canonical hypergroups, we investigate several categories of hyperstructures that generalize hypergroups. By allowing hyperoperations with possibly empty products, one obtains…

We introduce higher analogs for cleavages in the context of (Kan) simplicial fibrations. We apply them to obtain geometric models for representations up to homotopy of (higher) Lie groupoids. Concretely, we set an equivalence between…

Let~$G$ be a group and let~$\mathcal{F}$ be a family of subgroups of~$G$. The generalised Lusternik--Schnirelmann category~$\operatorname{cat}_\mathcal{F}(G)$ is the minimal cardinality of covers of~$BG$ by open subsets with fundamental…

We show that if a complex has free finitely generated reduced homology groups for two consecutive dimensions and trivial homology for all other dimensions, then it must have the homotopy type of a wedge of spheres of two consecutive…

In a previous article, we introduced notions of finiteness obstruction, Euler characteristic, and L^2-Euler characteristic for wide classes of categories. In this sequel, we prove the compatibility of those notions with homotopy colimits of…

For a locally finitely presented Grothendieck category $\mathcal{A}$, we consider a certain subcategory of the homotopy category of FP-injective objects in $\mathcal{A}$ which we show is compactly generated. In the case where $\mathcal{A}$…