This investigation will use this ability in a different way. Studies have shown that a category can create a grade structure of prototypical items. We will take a graded structure generated by a category, and see if we can recreate the category based on presenting prototypical examples in a variety of ways. Five different sampling techniques will be used to determine which one is the best for category reconstruction. Since the items themselves have bits of information about the category, the number of samples presented will also be manipulated to determine if this is a factor in determining the category. The independent variables investigated were: sampling technique, and prompt conditions. In determining the effect of the independent variables on matching a category, the independent variables were also considered as mediating variables of each other. The method of opportunistic sampling was used for the surveys. The main participants were undergraduate 3rd year students taking a MSci 311 course at the University of Waterloo. Results indicate that there was no statistical significance. Fluctuations in significance levels indicate some random findings. Participants are not discriminating the samples or prompts which were given. This research is a contribution to this field because little research has been conducted in this area and implications are drawn for future research on the saliency of a category or attribute that can vary by context or knowledge

Names and name-binding are useful concepts in the theory and practice of formal systems. In this thesis we study them in the context of dependent type theory. We propose a novel dependent type theory with primitives for the explicit handling of names. As the main application, we consider programming and reasoning with abstract syntax involving variable binders. Gabbay and Pitts have shown that Fraenkel Mostowski (FM) set theory models a notion of name using which informal work with names and binding can be made precise. They have given a number of useful constructions for working with names and binding, such as a syntax-independent notion of freshness that formalises when two objects do not share names, and a freshness quantifier that simplifies working with names and binding. Related to FM set theory, a number of systems for working with names have been given, among them are the first-order Nominal Logic, the higher-order logic FM-HOL, the Theory of Contexts as well as the programming language FreshML. In this thesis we study how dependent type theory can be extended with primitives for working with names and binding. Our choice of primitives is different from that in FM set theory. In FM set theory the fundamental primitive for working with names is swapping. All other concepts such as \alpha-equivalence classes and binding are constructed from it. For dependent type theory, however, this approach of constructing everything from swapping is not ideal, since it requires us to make strong assumptions on the type theory. For instance, the construction of \alpha-equivalence classes from swapping appears to require quotient types. Our approach is to treat constructions such as \alpha-equivalence classes and name-binding directly, turning them into primitives of the type theory. To do this, it is convenient to take freshness rather than swapping as the fundamental primitive. Based on the close correspondence between type theories and categories, we approach the design of the dependent type theory by studying the categorical structure underlying FM set theory. We start from a monoidal structure capturing freshness. By analogy with the definition of simple dependent sums \Sigma and products \Pi from the cartesian product, we define monoidal dependent sums \Sigma * and products \Pi * from the monoidal structure. For the type of names N, we have an isomorphism \Sigma *_N\iso\Pi *_N generalising the freshness quantifier. We show that this structure includes \alpha-equivalence classes, name binding, unique choice of fresh names as well as the freshness quantifier. In addition to the set theoretic model corresponding to FM set theory, we also give a realizability model of this structure. The semantic structure leads us to a bunched type theory having both a dependent additive context structure and a non-dependent multiplicative context structure. This type theory generalises the simply-typed \alpha\lambda-calculus of O'Hearn and Pym in the additive direction. It includes novel monoidal products \Pi * and sums \Sigma * as well as hidden-name types H for working with names and binding. We give examples for the use of the type theory for programming and reasoning with abstract syntax involving binders. We show that abstract syntax can be handled both in the style of FM set theory and in the style of Weak Higher Order Abstract Syntax. Moreover, these two styles of working with abstract syntax can be mixed, which has interesting applications such as the derivation of a term for the unique choice of new names.

In this thesis we give an exposition of the notion of category and the Baire category theorem as a set theoretical method for proving existence. The category method was introduced by René Baire to describe the functions that can be represented by a limit of a sequence of continuous real functions. Baire used the term functions of the first class to denote these functions. The usage of the Baire category theorem and the category method will be illustrated by some of its numerous applications in real and functional analysis. Since the usefulness, and generality, of the category method becomes fully apparent in Banach spaces, the applications provided have been restricted to these spaces. To some extent, basic concepts of metric topology will be revised, as the Baire category theorem is formulated and proved by these concepts. In addition to the Baire category theorem, we will give proof of equivalence between different versions of the theorem. Explicit examples, of first class functions will be presented, and we shall state a theorem, due to Baire, providing a necessary condition on the set of points of continuity for any function of the first class.

<p>In this thesis we give an exposition of the notion of <em>category </em>and the <em>Baire category theorem </em>as a set theoretical method for proving existence. The category method was introduced by René Baire to describe the functions that can be represented by a limit of a sequence of continuous real functions. Baire used the term <em>functions of the first class </em>to denote these functions.</p><p>The usage of the Baire category theorem and the category method will be illustrated by some of its numerous applications in real and functional analysis. Since the usefulness, and generality, of the category method becomes fully apparent in Banach spaces, the applications provided have been restricted to these spaces.</p><p>To some extent, basic concepts of metric topology will be revised, as the Baire category theorem is formulated and proved by these concepts. In addition to the Baire category theorem, we will give proof of equivalence between different versions of the theorem.</p><p>Explicit examples, of first class functions will be presented, and we shall state a theorem, due to Baire, providing a necessary condition on the set of points of continuity for any function of the first class.</p><p> </p>

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Categorization is a fundamental process that underlies much of cognition. People form categories that allow them to generalize to and make inferences about novel objects and events. Current accounts of category learning suggest that there are two systems for learning categories, an explicit rule-based system that depends on frontal-striatal loops and working memory, and a procedural system that learns implicitly and depends on the tail of the caudate nucleus and occipital regions. In the present thesis, I propose that an additional declarative category learning system exists that is recruited to learn categories that are associated with multiple conjunctive and explicit, but not strictly rule-based, representations. The basis of the declarative category learning system is then tested in several behavioral and physiological recording experiments. The first issue that is examined in relation to the declarative category learning system is how subjects’ ability to encode stimuli affects their ability to form new flexible conjunctive representations. I provide evidence consistent with the idea that there are two ways to encode stimuli in category learning, either as a conjunction of individual parts or as holistic images. Forming part-based representations is found to be especially critical for forming new conjunctive representations for exceptions in brief single session experiments. A second question is how emotional processes interact with the declarative category learning system. Numerous lines of evidence suggest that emotional processes strongly affect learning and behavior. In a study using skin conductance, I find that anticipatory emotions (i.e., emotions present before a behavioral response) show a pattern consistent with orienting attention to behaviorally significant or potentially novel events. A final fMRI project ties together hypotheses about anticipatory emotions and encoding to their neural basis and provides a test of the predicted mapping of the declarative category learning system to the brain. By relating quantitative predictions from SUSTAIN, a model that shares relationships to the medial temporal lobes (MTL) and declarative category learning system, to fMRI data, I find clusters in an MTL-midbrain-PFC network that show patterns of activation consistent with recognizing exception items and updating these representations in response to error or surprise. / text

The relational data model uses set theory to provide a formal background, thus ensuring a rigorous mathematical data model with support for manipulation. Newer generation database models are based on the object-oriented paradigm, and so fall short of having such a formal background, especially in some of the more complex data manipulation areas. We use category theory to provide a formalism for object databases, in particular the object-relational model. Our model is known as the Product Model. This thesis will describe our formal model for the key aspects of object databases. In particular, we will examine how the Product Model deals with three of the most important problems inherent in object databases, those of queries, closure and views. As well as this, we investigate the more common database concepts, such as keys, relationships and aggregation. We will illustrate the feasibility of this model, by producing a prototype implementation using PIFDM. PIFDM is a semantic data model database system based on the functional model of Shipman, with object-oriented extensions.

This work constructs and compares different kinds of categorified cohomology of a locally contractible topological space X. Fix a commutative ring k of characteristic 0 and also denote by k the differential graded category with a single object and endomorphisms k. In the Morita model structure k is weakly equivalent to the category of perfect chain complexes over k. We define and compute derived global sections of the constant presheaf k considered as a presheaf of dg-categories with the Morita model structure. If k is a field this is done by showing there exists a suitable local model structure on presheaves of dg-categories and explicitly sheafifying constant presheaves. We call this categorified Cech cohomology Morita cohomology and show that it can be computed as a homotopy limit over a good (hyper)cover of the space X. We then prove a strictification result for dg-categories and deduce that under mild assumptions on X Morita cohomology is equivalent to the category of homotopy locally constant sheaves of k-complexes on X. We also show categorified Cech cohomology is equivalent to a category of ∞-local systems, which can be interpreted as categorified singular cohomology. We define this category in terms of the cotensor action of simplicial sets on the category of dg-categories. We then show ∞-local systems are equivalent to the category of dg-representations of chains on the loop space of X and find an explicit method of computation if X is a CW complex. We conclude with a number of examples.

Ontologies are a formalization of a particular domain through a collection of axioms founded, usually, in Description Logic. Within its structure, the knowledge in the axioms contain semantic information of the domain and that fact has motivated the development of methods that capture such knowledge and, therefore, can perform different tasks such as prediction and similarity computation. Under the same motivation, we present a new method to capture semantic information from an ontology. We explore the logical component of the ontologies and their theoretical connections with their counterparts in Category Theory, as Category Theory develops a structural representation of mathematical systems and the structures found there have strong relationships with Logic founded in the so-called Curry-Howard-Lambek isomorphism. In this regard, we have developed a method that represents logical axioms as Categorical diagrams and uses the commutativity property of such diagrams as a constraint to generate embeddings of ontology classes in Rn. Furthermore, as a contribution in terms of software tools, we developed mOWL: Machine Learning Library With Ontologies. mOWL is a software library that incorporates methods in the state of the art, usually in Machine Learning, which utilizes ontologies as background knowledge. We rely on mOWL to implement the proposed method and compare it with the existing ones.

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