Our aim is to compare three nerve functors for strict $n$-categories: the Street nerve, cellular and multi-simplicial nerve. We show that these are equivalent in some appropriate sense. In particular, classes of $n$-categorical weak equivalences they define coincide: Thomason equivalences. give two applications this result: first one states a Dyer-Kan-type equivalence equivalence; second one, f...

This paper describes a formalism and implementation for the interpretation and generation of sentences containing context dependent constructs like determiners, pronouns, focus, and ellipsis. A variant of ‘quasi-logical form’ is used as an underspecified meaning representation, related to ‘resolved logical forms’ via ‘conditional equivalences’. These equivalences define the interpretation of co...

In this paper the notion of action atomicity is relaxed by permitting actions to be observed in the middle of their evolution. Non atomic semantic equivalences, based on the notion of bisimulation, are studied over stable event structures. Splitn bisimulation equivalence (denoted n ) considers each event as composed of n phases. ST bisimulation equivalence (denoted ST ) is a slight refinement o...

We show that if C is a proper model category, then the pro-category pro-C has a strict model structure in which the weak equivalences are the levelwise weak equivalences. This is related to a major result of [10]. The strict model structure is the starting point for many homotopy theories of pro-objects such as those described in [5], [17], and [19].

The Joyal model structure on simplicial sets is extended to a model structure on the simplicial presheaves on a small site, in which the cofibrations are monomorphisms and the weak equivalences are local (or stalkwise) Joyal equivalences. The model structure is shown to be left proper.

Let $X$ be a sufficiently nice scheme. We survey some recent progress on dualizing complexes. It turns out that a complex in $kinj X$ is dualizing if and only if tensor product with it induces an equivalence of categories from Murfet's new category $kmpr X$ to the category $kinj X$. In these terms, it becomes interesting to wonder how to glue such equivalences.

The aim of this paper is to introduce the notion of cat$^{bf {1}}-$groupoids which are the groupoid version of cat$^{bf {1}}-$groups and to prove the categorical equivalence between crossed modules over groupoids and cat$^{bf {1}}-$groupoids. In section 4 we introduce the notions of crossed squares over groupoids and of cat$^{bf {2}}-$groupoids, and then we show their categories are equivalent....

We present a spectrum of trace-based, testing, and bisimulation equivalences for nondeterministic and probabilistic processes whose activities are all observable. For every equivalence under study, we examine the discriminating power of three variants stemming from three approaches that differ for the way probabilities of events are compared when nondeterministic choices are resolved via determ...

Image formation is described by means of modern geometry in terms of topological and geometric curvatures. These curvatures and derived equivalences are a quantisation of the set of rules for constructing the image in particular along discontinuity, singularity and bifurcation sets. A simpliication of the set of construction rules is proposed by applying a dynamic scale-space paradigm. Dynamic ...