This survey serves to introduce sheaves and some basic properties of sheaves from a categorical perspective. The objective is that if one has a basic understanding of category theory that this approach to defining sheaves is more intuitive than the usual approach of a typical Algebraic Geometry text. We

This paper explores the topology of planning with uncertainty in discrete spaces. The paper defines strategy complex as the collection of all plans for accomplishing all tasks specified by goal states in a finite discrete graph. Transitions in the graph may be nondeterministic or stochastic. One key result is that a system can attain any state in its graph despite control uncertainty if and only if its strategy complex is homotopic to a sphere of dimension two less than the number of states in the graph.

Abstract Complex hierarchical structures composed of simple nanoscale building blocks form the basis of most biological materials. Here, we demonstrate how analogies between seemingly different fields enable the understanding of general principles by which functional properties in hierarchical systems emerge, similar to an analogy learning process. Specifically, natural hierarchical materials like spider silk exhibit properties comparable to classical music in terms of their hierarchical structure and function. As a comparative tool, here, we apply hierarchical ontology logs that follow a rigorous mathematical formulation based on category theory to provide an insightful system representation by expressing knowledge in a conceptual map. We

Classical and Connectionist theories of cognitive architecture “explain ” systematicity, whereby the capacity for some cognitive behaviors is intrinsically linked to the capacity for others, as a consequence of syntactically and functionally combinatorial representations, respectively. However, both theories depend on ad hoc assumptions to exclude specific architectures—grammars, or Connectionist networks—that do not account for systematicity. By analogy with the Ptolemaic (i.e., geocentric) theory of planetary motion, although either theory can be made to be consistent with the data, both nonetheless fail to explain it (Aizawa, 2003b). Category theory provides an alternative explanation based on the formal concept of adjunction, which consists of a pair of structure preserving maps, called functors. A functor generalizes the notion of a map between representational states to include a map between state transformations (processes). In a formal sense, systematicity is a necessary consequence of a “higher-order ” theory of cognitive architecture, in contrast to the “first-order ” theories derived from Classicism or Connectionism. Category theory offers a re-conceptualization for cognitive science, analogous to the one that Copernicus provided for astronomy, where representational states are no longer the center of the cognitive universe—replaced by the relationships between the maps that transform them.

Abstract. In this article, the author proposes another way to define the completion of a metric space, which is different from the classical one via the dense property, and prove the equivalence between two definitions. This definition is based on considerations from category theory, and can be generalized to arbitrary categories. 1.

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Abstract. This chapter explores the topology of planning with uncertainty in discrete spaces. The chapter defines the strategy complex of a finite discrete graph as the collection of all plans for accomplishing all tasks specified by goal states in the graph. Transitions in the graph may be nondeterministic or stochastic. One key result is that a system can attain any state in its graph despite control uncertainty if and only if its strategy complex is homotopic to a sphere of dimension two less than the number of states in the graph. 1. Planning with Uncertainty in Robotics The goal of Robotics is to animate the inanimate, so as to endow machines with the ability to act purposefully in the world. Roboticists, working in the subfield of planning, create software by which robots reason about future outcomes of potential actions. Using such planning software, robots combine individual actions into collections that together accomplish particular tasks in the world [29, 30]. Two fundamental and intertwined issues confound this seemingly straightforward approach. One is world complexity, the other is uncertainty.