Higher dimensional central extensions of groups were introduced by G. Janelidze as particular instances of the abstract notion of covering morphism from categorical Galois theory. More recently, the notion has been extended to and studied in arbitrary semi-abelian categories. Here, we further extend the scope to exact Mal’tsev categories and beyond. For this, we consider conditions on a Galois ...

Introduction The following text arose from the desire to establish a rm relationship between higher order categories and topological spaces. Our approach combines the algebraic features of Michael Batanin's !-operads 1] with the geometric features of Andr e Joyal's cellular sets 15] and tries to mimick as far as possible the classical construction of the simplicial nerve of a category. Higher o...

Reynolds’ theory of relational parametricity formalizes parametric polymorphism for System F, thus capturing the idea that polymorphically typed System F programs always map related inputs to related results. This paper shows that Reynolds’ theory can be seen as the instantiation at dimension 1 of a theory of relational parametricity for System F that holds at all higher dimensions, including i...

We study maximal m-rigid objects in the m-cluster category C H associated with a finite dimensional hereditary algebra H with n nonisomorphic simple modules. We show that all maximal m-rigid objects in these categories have exactly n nonisomorphic indecomposable summands, and that any almost complete m-rigid object in C H has exactly m + 1 nonisomorphic complements. We also show that the maxima...

There are a dozen definitions of weak higher categories, all of which loosen the notion of composition of arrows. A new approach is presented here, where instead the notion of identity arrow is weakened — these are tentatively called fair categories. The approach is simplicial in spirit, but the usual simplicial category ∆ is replaced by a certain ‘fat’ delta of ‘coloured ordinals’, where the d...

Let $f:G\to \mathrm{Pic}(R)$ be a map of $E_\infty$-groups, where $\mathrm{Pic}(R)$ denotes the Picard space an $E_\infty$-ring spectrum $R$. We determine tensor $X\otimes_R Mf$ Thom $E_\infty$-$R$-algebra $Mf$ with $X$; when $X$ is circle, topological Hochschild homology over use theory localizations $\infty$-categories as technical tool: we contribute to this $\infty$-categorical analogue Day...

A calculus for a fragment of category theory is presented. The types in the language denote categories and the expressions functors. The judgements of the calculus systematise categorical arguments such as: an expression is functorial in its free variables; two expressions are naturally isomorphic in their free variables. There are special binders for limits and more general ends. The rules for...