Trees in Partial Higher Dimensional Automata

We prove that the category of (strictly unital) -categories, linear over a commutative ring R, with strict -morphisms has a cofibrantly generated model structure. In this model structure every object is fibrant and the cofibrant objects have cofibrant morphisms. As a consequence we prove that the semi-free -categories (resp. resolutions) are cofibrant objects (resp. resolution) in this model structure.

Given a complete hereditary cotorsion pair$(\mathcal{X}, \mathcal{Y})$, we introduce the concept of$(\mathcal{X}, \mathcal{X} \cap \mathcal{Y})$-Gorenstein projective modules and study its stability properties. As applications, we first get two model structures related to Gorenstein flat modules over a right coherent ring. Secondly, for any non-negative integern, we construct a cofibrantly generated model structure on Mod(R) in which the class of fibrant objects are the modules of Gorenstein injective dimension ≤nover a left Noetherian ringR. Similarly, ifRis a left coherent ring in which all flat leftR-modules have finite projective dimension, then there is a cofibrantly generated model structure on Mod(R) such that the cofibrant objects are the modules of Gorenstein projective dimension ≤n. These structures have their analogous in the category of chain complexes.

The flat stable module category of a coherent ring

A-cellular homotopy theories

In a directed graph with the unique path property, there is a unique directed path of a given length between any ordered pair of its vertices. Such graphs, their paths, cycles, factors, and other properties, were extensively studied during the years, mainly in the context of de Bruijn graphs, forming the most important family of such graphs. Such a graph is equivalent to a binary matrix for which , where is the all‐ones matrix. In this paper, some new results on the structural properties of such graphs, which are not necessarily de Bruijn graphs, are proved. We mainly concentrate on such graphs for which each vertex has in‐degree two and out‐degree two. The minimum number and the maximum number of vertices which are reached from a given vertex using all paths of length smaller than are considered. We provide an exact formula for the number of factors in these graphs and show that this number depends only on the number of alternating cycles in the graph, where in an alternating cycle every two consecutive edges have opposite directions in the cycle. Constructions of such graphs using several methods are presented. In particular, we present a new construction which provides a large set of such graphs with long alternating cycles. We define the integral graph and the derivative graph and implement these definitions on this family of graphs. Various properties from the point of view of the de Bruijn graph and the theory of nonsingular shift‐registers sequences are also considered. Finally, generalizations for graphs in which the in‐degree and the out‐degree of each vertex are greater than two are also discussed.

We prove the existence of Morita model structures on the categories of small simplicial categories, simplicial sets, simplicial operads and dendroidal sets, modelling the Morita homotopy theory of ( ∞ , 1 ) {(\infty,1)} -categories and ∞ {\infty} -operads. We give a characterization of the weak equivalences in terms of simplicial presheaves, simplicial algebras and slice categories. In the case of the Morita model structure for simplicial categories and simplicial operads, we also show that each of these model structures can be obtained as an explicit left Bousfield localization of the Bergner model structure on simplicial categories and the Cisinski–Moerdijk model structure on simplicial operads, respectively.

We take the point of view that, if transition systems are coalgebras for a functor T, then an adequate logic for these transition systems should arise from the ‘Stone dual’ L of T. We show that such a functor always gives rise to an ‘abstract’ adequate logic for T-coalgebras and investigate under which circumstances it gives rise to a ‘concrete’ such logic, that is, a logic with an inductively defined syntax and proof system. We obtain a result that allows us to prove adequateness of logics uniformly for a large number of different types of transition systems and give some examples of its usefulness.KeywordsModal LogicTransition SystemPropositional LogicFree AlgebraModal AlgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

José Maria Fernando Medina, diretor da Diretoria de Planejamento Social e Urbano da Fundação João Pinheiro (DPSU/FJP).

We study the issue of sound multi-agent design trying to keep a system engineering perspective; that is, both when analyzing a problem and when synthesizing a solution the focus is kept on the whole, large-scale structure of the software artifact that is to be realized. The formal tool we evaluate to provide effective assistance to the sound design of multi-agent systems (MASs) applies (labeled) transition systems to the description of software interactive behavior [3], as elaborated and promoted in the field of concurrency theory. When specifying a transition system semantics for an agent, a number of rules are given that describe in a quite declarative way the dynamics of its inner machinery, also providing insights on its internal architecture – expressed at a given level of abstraction. We believe that this formalism is fit for design because of its calculus-like nature, which combines the description of a system with the prescription of its possible evolutions; this property buys designers some generativity, while remaining in a well grounded mathematical landscape. Moreover, transition systems allow us to address both composability and extensibility. An agent model is divided into several LTS specifications that we call tiers, capturing different aspects of an agent design. A partial specification of an agent can be further detailed by adding new tiers, with a technique known as grey-boxing [4, 5]. As a use case for our approach we consider ParADE [2], a development environment exploiting the formal semantics of a FIPA-like Agent Communication Language (ACL) to gain agents semantic interoperability, and providing interaction laws as a design mechanism to specify the agent social role.

This monograph introduces a framework for genuine proper equivariant stable homotopy theory for Lie groups. The adjective ‘proper’ alludes to the feature that equivalences are tested on compact subgroups, and that the objects are built from equivariant cells with compact isotropy groups; the adjective ‘genuine’ indicates that the theory comes with appropriate transfers and Wirthmüller isomorphisms, and the resulting equivariant cohomology theories support the analog of an R O ( G ) R O(G) -grading. Our model for genuine proper G G -equivariant stable homotopy theory is the category of orthogonal G G -spectra; the equivalences are those morphisms that induce isomorphisms of equivariant stable homotopy groups for all compact subgroups of G G . This class of π ∗ \pi _* -isomorphisms is part of a symmetric monoidal stable model structure, and the associated tensor triangulated homotopy category is compactly generated. Consequently, every orthogonal G G -spectrum represents an equivariant cohomology theory on the category of G G -spaces. These represented cohomology theories are designed to only depend on the ‘proper G G -homotopy type’, tested by fixed points under all compact subgroups. An important special case of our theory are infinite discrete groups. For these, our genuine equivariant theory is related to finiteness properties in the sense of geometric group theory; for example, the G G -sphere spectrum is a compact object in our triangulated equivariant homotopy category if the universal space for proper G G -actions has a finite G G -CW-model. For discrete groups, the represented equivariant cohomology theories on finite proper G G -CW-complexes admit a more explicit description in terms of parameterized equivariant homotopy theory, suitably stabilized by G G -vector bundles. Via this description, we can identify the previously defined G G -cohomology theories of equivariant stable cohomotopy and equivariant K-theory as cohomology theories represented by specific orthogonal G G -spectra.

Relative categories: Another model for the homotopy theory of homotopy theories

Pre-synthesis of Petri nets based on prime cycles and distance paths

On the parameterized complexity of the synthesis of Boolean nets with restricted place environments

In previous work, we showed that there are appropriate model category structures on the category of simplicial categories and on the category of Segal precategories, and that they are Quillen equivalent to one another and to Rezk’s complete Segal space model structure on the category of simplicial spaces. Here, we show that these results still hold if we instead use groupoid or \invertible cases. Namely, we show that model structures on the categories of simplicial groupoids, Segal pregroupoids, and invertible simplicial spaces are all Quillen equivalent to one another and to the standard model structure on the category of spaces. We prove this result using two difierent approaches to invertible complete Segal spaces and Segal groupoids.

The codiagonal functor $$\overline{W}$$ transfers a Quillen closed model structure on the bisimplicial set category from the ordinary model category of simplicial sets. This bisimplicial model structure is different from the so called Moerdijk model structure, which is similarly transferred from simplicial sets but through the diagonal functor. We show the mutual relationship of these two closed model structures on the category of bisimplicial sets.