2013/01/22 by Octavio Malherbe, Malherbe, Octavio
Computer Science · Mathematics · #Advanced Algebra and Logic #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.CT
paper · pdf · doi:10.48550/arxiv.1301.5087
Ph.D. thesis, University of Ottawa, September 2010
arxiv created 2013/01/22 · openalex publication_date 2013/01/22 · arxiv updated 2013/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This dissertation has two main parts. The first part deals with questions relating to Haghverdi and Scott's notion of partially traced categories. The main result is a representation theorem for such categories: we prove that every partially traced category can be faithfully embedded in a totally traced category. Also conversely, every monoidal subcategory of a totally traced category is partially traced, so this characterizes the partially traced categories completely. The main technique we use is based on Freyd's paracategories, along with a partial version of Joyal, Street, and Verity's Int construction. Along the way, we discuss some new examples of partially traced categories, mostly arising in the context of quantum computation. The second part deals with the construction of categorical models of higher-order quantum computation. We construct a concrete semantic model of Selinger and Valiron's quantum lambda calculus, which has been an open problem until now. We do this by considering presheaf categories over appropriate base categories arising from first-order quantum computation. The main technical ingredients are Day's convolution theory and Kelly and Freyd's notion of continuity of functors. We first give an abstract description of the properties required of the base categories for the model construction to work; then exhibit a specific example of base categories satisfying these properties.