We give a definition of atlases for ineffective orbifolds, and prove that this definition leads to the same notion of orbifold as that defined via topological groupoids.
In this paper, we introduce the notion of a topological groupoid extension and relate it to the already existing notion of a gerbe over a topological stack. We further study the properties of a gerbe over a Serre, Hurewicz stack.
We extend the notion of Hitchin component from surface groups to orbifold groups and prove that this gives new examples of higher Teichm{u}ller spaces. We show that the Hitchin component of an orbifold group is homeomorphic to an open ball and we com
pute its dimension explicitly. We then give applications to the study of the pressure metric, cyclic Higgs bundles, and the deformation theory of real projective structures on $3$-manifolds.
The aim of this sequel to arXiv:1812.02935 is to set up the cornerstones of Koszul duality and Koszulity in the context of a large class of operadic categories. In particular, we will prove that operads, in the generalized sense of Batanin-Markl, gov
erning important operad- and/or PROP-like structures such as the classical operads, their variants such as cyclic, modular or wheeled operads, and also diver
In this work we provide a definition of a coloured operad as a monoid in some monoidal category, and develop the machinery of Grobner bases for coloured operads. Among the examples for which we show the existance of a quadratic Grobner basis we con
sider the seminal Lie-Rinehart operad whose algebras include pairs (functions, vector fields).
We introduce acyclic polygraphs, a notion of complete categorical cellular model for (small) categories, containing generators, relations and higher-dimensional globular syzygies. We give a rewriting method to construct explicit acyclic polygraphs fr
om convergent presentations. For that, we introduce higher-dimensional normalisation strategies, defined as homotopically coherent ways to relate each cell of a polygraph to its normal form, then we prove that acyclicity is equivalent to the existence of a normalisation strategy. Using acyclic polygraphs, we define a higher-dimensional homotopical finiteness condition for higher categories which extends Squiers finite derivation type for monoids. We relate this homotopical property to a new homological finiteness condition that we introduce here.