We study a new type of higher categorical structure, called weakly globular n-fold category, previously introduced by the author. We show that this structure is a model of weak ncategories by proving that it is suitably equivalent to the TamsamaniSimpson model. We also introduce groupoidal weakly globular nfold categories and show that they are algebraic models of n-types.

This review focuses on the ways in which the objectification of individuals and groups of people, as well as the self-objectification that typically develops from such treatment, is implicated in positive and negative societal change. Four areas are reviewed: (a) objectification (including dehumanization, infra-humanization, dehumanized perception, sexualization, and colonialism), (b) self-obje...

A popular slogan is that (∞, 1)-categories (also called quasi-categories or ∞categories) sit somewhere between categories and spaces, combining some of the features of both. The analogy with spaces is fairly clear, at least to someone who is happy to regard spaces as Kan complexes, which are simplicial sets in which every horn can be filled. The analogy with categories is somewhat more subtle. ...

Continuing our first paper in this series, we study multiple limits in infinite-dimensional multiple categories. The general setting is chiral multiple categories a weak, partially lax form with directed interchanges. After defining multiple limits we prove that all of them can be constructed from (multiple) products, equalisers and tabulators all of them assumed to be respected by faces and de...

This is an extended account of a short presentation with this title given at the Minneapolis IMA Workshop on ‘n-categories: foundations and applications’, June 7-18, 2004, organised by John Baez and Peter May.

Adhesive high-level replacement (HLR) system have been recently introduced as a new categorical framework for graph transformation in the double pushout approach [1, 2]. They combine the well-known framework of HLR systems with the framework of adhesive categories introduced by Lack and Sobociński [3, 4]. The main concept behind adhesive categories are the so-called van Kampen squares, which en...

We present a weak form of a recognition principle for Quillen model categories due to J.H. Smith. We use it to put a model category structure on the category of small categories enriched over a suitable monoidal simplicial model category. The proof uses a part of the model structure on small simplicial categories due to J. Bergner. We give an application of the weak form of Smith’s result to le...

We introduce weak morphisms of higher dimensional automata and use them to define preorder relations for HDAs, among which homeomor-phic abstraction and trace equivalent abstraction. It is shown that homeomor-phic abstraction is essentially always stronger than trace equivalent abstraction. We also define the trace language of an HDA and show that, for a large class of HDAs, it is invariant und...

1 .l Summary IN [l] QUILLEN introduced the notion of a model category (a category together with three classes of maps: weak equivalences, fibrations and cofibrations, satisfying certain axioms (1.4 (iv))) as a general framework for “doing homotopy theory”. To each model category M there is associated a homotopy category. If W C M denotes the subcategory of the weak equivalences, then this homot...