We deene families of posets, ordered by preexes, as the counterpart of the usual families of conngurations ordered by subsets. On these objects we deene two types of morphism: event and order morphisms, resulting in categories FPos and FPos v. We then show the following: { Families of posets, in contrast to families of conngurations, are always prime algebraic; in fact the category FPos v is eq...

This is an account for the combinatorially minded reader of various categories of directed and undirected graphs, and their analogies with the category of sets. As an application, the endomorphisms of a graph are in this context not only composable, giving a monoid structure, but also have a notion of adjacency, so that the set of endomorphisms is both a monoid and a graph. We extend Shrimpton’...

This paper proposes a formal cognitive framework for problem solving based on category theory. We introduce cognitive categories, which are categories with exactly one morphism between any two objects. Objects in these categories are interpreted as states and morphisms as transformations between states. Moreover, cognitive problems are reduced to the specification of two objects in a cognitive ...

The paper is devoted to the question is the Cartesian product X × P of a compact Hausdorff space X and a polyhedron P a product in the strong shape category SSh of topological spaces. The question consists of two parts. The existence part, which asks whether, for a topological space Z, for a strong shape morphism F : Z → X and a homotopy class of mappings [g] : Z → P , there exists a strong sha...

1 Basic Properties Definition 1. Let X be a scheme. We denote the category of sheaves of OX -modules by OXMod or Mod(X). The full subcategories of qausi-coherent and coherent modules are denoted by Qco(X) and Coh(X) respectively. Mod(X) is a grothendieck abelian category, and it follows from (AC,Lemma 39) and (H, II 5.7) that Qco(X) is an abelian subcategory of Mod(X). If X is noetherian, then ...

In this note, we outline the general development of a theory of symmetric homology of algebras, an analog of cyclic homology where the cyclic groups are replaced by symmetric groups. This theory is developed using the framework of crossed simplicial groups and the homological algebra of module-valued functors. The symmetric homology of group algebras is related to stable homotopy theory. Two sp...

A restriction category is an abstract formulation for a category of partial maps, defined in terms of certain specified idempotents called the restriction idempotents. All categories of partial maps are restriction categories; conversely, a restriction category is a category of partial maps if and only if the restriction idempotents split. Restriction categories facilitate reasoning about parti...

We present in this paper a systematic study on how to morph a well-trained neural network to a new one so that its network function can be completely preserved. We define this as network morphism in this research. After morphing a parent network, the child network is expected to inherit the knowledge from its parent network and also has the potential to continue growing into a more powerful one...

A restriction category is an abstract formulation for a category of partial maps, defined in terms of certain specified idempotents called the restriction idempotents. All categories of partial maps are restriction categories; conversely, a restriction category is a category of partial maps if and only if the restriction idempotents split. Restriction categories facilitate reasoning about parti...

We define and study a lift of the Boardman-Vogt tensor product of operads to bimodules over operads. Introduction Let Op denote the category of symmetric operads in the monoidal category S of simplicial sets. The Boardman-Vogt tensor product [3] −⊗− : Op× Op→ Op, which endows the category Op with a symmetric monoidal structure, codifies interchanging algebraic structures. For all P,Q ∈ Op, a (P...