Generalized Long-Moody functors

In this paper, we generalize the Long-Moody construction for representations of braid groups to other groups, such as mapping class groups of surfaces. Moreover, we introduce Long-Moody endofunctors over a functor category that encodes representations of a family of groups. In this context, notions of polynomial functor are defined; these play an important role in the study of homological stability. We prove that, under some additional assumptions, a Long-Moody functor increases the (very) strong (respectively weak) polynomial degree of functors by one

Weights for Objects of Monoids

The main objective of the paper is to define the construction of the object of monoids, over a monoidal category object in any 2-category with finite products, as a weighted limit. To simplify the definition of the weight, we use matrices of symmetric (possibly colored) operads that define some auxiliary categories and 2-categories. Systematic use of these matrices of operads allows us to define several similar objects as weighted limits. We show, among others, that the constructions of the object of bi-monoids over a symmetric monoidal category object or the object of actions of monoids along an action of a monoidal category object can be also described as weighted limits.Comment: 19 page

Localisations of cobordism categories and invertible TFTs in dimension two

Cobordism categories have played an important role in classical geometry and more recently in mathematical treatments of quantum field theory. Here we will compute localisations of two-dimensional discrete cobordism categories. This allows us, up to equivalence, to determine the category of invertible two-dimensional topological field theories in the sense of Atiyah. We are able to treat the orientable, non-orientable, closed and open cases.Comment: 30 pages, accepted for publication by Homology, Homotopy and Application

Operads within monoidal pseudo algebras

A general notion of operad is given, which includes as instances, the operads originally conceived to study loop spaces, as well as the higher operads that arise in the globular approach to higher dimensional algebra. In the framework of this paper, one can also describe symmetric and braided analogues of higher operads, likely to be important to the study of weakly symmetric, higher dimensional monoidal structures