Cartesian differential categories revisited

We revisit the definition of Cartesian differential categories, showing that a slightly more general version is useful for a number of reasons. As one application, we show that these general differential categories are comonadic over Cartesian categories, so that every Cartesian category has an associated cofree differential category. We also work out the corresponding results when the categories involved have restriction structure, and show that these categories are closed under splitting restriction idempotents.Comment: 17 page

Nuclear and Trace Ideals in Tensored *-Categories

We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored *-categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called ``probabilistic relations''. The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. We introduce the notion of nuclear ideal to analyze these classes of morphisms. In compact closed categories, we see that all morphisms are nuclear, and in the category of Hilbert spaces, the nuclear morphisms are the Hilbert-Schmidt maps. We also introduce two new examples of tensored *-categories, in which integration plays the role of composition. In the first, morphisms are a special class of distributions, which we call tame distributions. We also introduce a category of probabilistic relations which was the original motivating example. Finally, we extend the recent work of Joyal, Street and Verity on traced monoidal categories to this setting by introducing the notion of a trace ideal. For a given symmetric monoidal category, it is not generally the case that arbitrary endomorphisms can be assigned a trace. However, we can find ideals in the category on which a trace can be defined satisfying equations analogous to those of Joyal, Street and Verity. We establish a close correspondence between nuclear ideals and trace ideals in a tensored *-category, suggested by the correspondence between Hilbert-Schmidt operators and trace operators on a Hilbert space.Comment: 43 pages, Revised versio

Learning Theory and the Evolutionary Analogy

In this article, past comparisons of learning and evolution as analogous processes are discussed and some inaccuracies and omissions in those discussions are pointed out. The evolutionary analogy is examined for its ability to suggest solutions to five fundamental theoretical issues about learning - superstitions, why a reinforcer has the effect it does, the relationship among various procedures yielding learning, the relevance of the matching law to the problem of what reinforces an avoidance response, and whether behavioral and cognitive views of learning can be reconciled. In each case it is argued that the analogy is instructive

Music teachers' stories about fostering the development of creative attributes in elementary students

In recent history, the desire for an internationally competitive economy has driven education policy and reform. Creativity has become a signifier for innovative growth in global economies. As a result, creators of recent international education policies expect teachers to develop creative attributes in students. Tensions can exist when writers of educational policy encourage teachers to develop creative attributes in students while enacting other policies that subject teachers to high stakes accountability. The purpose of this study was to explore how two of my colleagues and I engaged students in creative activities while we were implementing the National Core Arts Standards for the first time in elementary music classrooms. Using Burnard’s adaptation of Csikszentmihalyi’s systems model of creativity as a conceptual framework and narrative inquiry as the approach, I sought to determine (a) what choices we made to provide creative opportunities for students when implementing the National Core Arts Standards in the elementary music classroom, (b) what choices we made to allow students to engage in risk-taking and flexible thinking with agency and relevancy, and (c) how we provided students with opportunities to engage in collaboration and use technology while engaging in creative activities. My colleagues and I shared that we engaged all grade levels in agency and flexible thinking in nearly all the lessons we taught throughout the year, and provided many lessons relevant to students’ lives outside of school, other subjects, and other music lessons. We found that we provided many opportunities for students to collaborate with other students in the classroom. However, we realized that due to the many roles we were involved with at our sites, we did not have the resources necessary to provide opportunities for students to collaborate with people outside of the classroom either in person or through technology. We also felt that we did not have time to plan to provide creative opportunities for students while they were preparing for their grade-level performances. To provide these opportunities for students, teachers may need to emphasize the benefits of the creative process to parents, administrators, and students, and devote planning time to developing collaborative lessons

The evolutionary approach to history:sociocultural phylogenetics

There are three forms of modern Darwinian evolutionism in the social sciences and humanities: the gene-based biological, the social learning-based sociocultural, and gene–culture coevolution dealing with their interaction. This chapter focuses on cultural or sociocultural evolution. It begins with a discussion of the Darwinian-inspired evolutionary approach to history. It then outlines modern evolutionary phylogenetic methods borrowed from biology but now used extensively in the social sciences and humanities. The chapter provides examples of how language trees may be inferred; phylogenetic comparative methods that use language trees to answer questions about aspects of geographical, social, political, cultural, or economic organization; and phylogenetic investigations of material culture and traditions. It is concluded that culture does indeed “descend with modification.”</p