We give a framework for comparing on the one hand theories of ncategories that are weakly enriched operadically, and on the other hand ncategories given as algebras for a contractible globular operad. Examples of the former are the definition by Trimble and variants (Cheng-Gurski) and examples of the latter are the definition by Batanin and variants (Leinster). We will show how to take a theory...

We introduce a general categorical framework for the definition of weak behavioural equivalences, building on and extending recent results in the field. This framework is based on special order enriched categories, i.e. categories whose homsets are endowed with suitable complete orders. Using this structure we provide an abstract notion of saturation, which allows us to define various (weak) be...

In this note, I provide a correction to the proof of Proposition 6.6 of [Rez10]. It forms a key part of the proof of the main Theorem 1.4 of that paper. The error lies in the use of Proposition 2.19, which is simply false as stated. In fact, it is easy to construct counterexamples to 2.19 when the poset P does not have contractible nerve. The error the original paper was pointed out to me by Bi...

We introduce a dependent type theory whose models are weak ω-categories, generalizing Brunerie’s definition of ω-groupoids. Our type theory is based on the definition of ω-categories given by Maltsiniotis, himself inspired by Grothendieck’s approach to the definition of ω-groupoids. In this setup, ω-categories are defined as presheaves preserving globular colimits over a certain category, calle...

In this note, we will introduce a notion of modularity of exact categories due to Masana Harada [Har05]. The naming is coming from the classical modular lattices theory [Bir48]. We will also state and prove so-called “homotopy Grayson-Staffeldt-Jordan-Hölder theorem” which is implicitly appeared in [Gra87] and [Sta89]. The theorem says contractibility of a simplicial set associated to a certain...

Adhesive high-level replacement (HLR) systems have been recently established as a suitable categorical framework for double pushout transformations based on weak adhesive HLR categories. Among different types of graphs and graph-like structures, various kinds of Petri nets and algebraic high-level (AHL) nets are interesting instantiations of adhesive HLR systems. AHL nets combine algebraic spec...

Adhesive high-level replacement (HLR) systems have been recently established as a suitable categorical framework for double pushout transformations based on weak adhesive HLR categories. Among different types of graphs and graph-like structures, various kinds of Petri nets and algebraic high-level (AHL) nets are interesting instantiations of adhesive HLR systems. AHL nets combine algebraic spec...

We introduce a variant on the graphical calculus of Cockett and Seely[2] for monoidal functors and illustrate it with a discussion of Tannaka reconstruction, some of which is known and some of which is new. The new portion is: given a separable Frobenius functor F : A −→ B from a monoidal category A to a suitably complete or cocomplete braided autonomous category B, the usual formula for Tannak...

1. My approach is "foundational". On the one hand, I am motivated by the problem of the foundations of mathematics (an unsolved problem as far as I am concerned). On the other hand-and this is more relevant here-, I start "from scratch", and thus what I say can be understood with little technical knowledge. I only assume a modest amount of category theory as background. I will talk informally a...