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.

The brief introductory chapter attempts to motivate the project by pointing to (a) the intuitive appeal and importance of the notion of an object (that is, a “paradigmatic” individual), and (b) the need – for the sake of progress in at least two important debates in ontology – to replace this notion with a series of related notions of individuals of different sorts. Section One of Chapter Two aims to accomplish two primary tasks. The first is to clarify the intensions of three often employed but ambiguous categorical terms: ‘individual’, ‘particular’ and ‘object’, with emphasis on the third, which is often taken to be of particular philosophical significance. I carry out this clarificatory task by weighing various positions in the literature and arguing for explications of each notion that are maximally economical and neutral, that is, explications which (a) overlap as little as possible with other important ontological notions and (b) do not require us to take a stand on any apparently intractable (but not directly relevant) debates (e.g. on the problem of realism vs. nominalism about universals). The second task of 2.1 is to delineate the various ontological distinctions that will be turned, in Chapter Four, into the “dimensions” of which the ontological framework I will be advocating there is composed. The delineation of these distinctions takes place naturally in the course of attempting to characterize the notion of an object (an intrinsically unified, independent concrete particular) and to distinguish it from the notions of an individual and a particular, in spite of the fact that objects are both individuals and particulars. In the second section of Chapter Two I illustrate the centrality of the notion of an object in Ontology by showing how that notion figures in the debate over the existence of artifacts. I argue that progress in this debate has been hindered by the way it has been framed, and that seeing the issue as concerning not whether artifacts exist but whether artifacts are objects (in the sense outlined in 2.1) enables us to better appreciate and accommodate the different perspectives of the debate’s participants. At the same time, this way of dissolving the dispute makes clear that existence is not limited to entities that fall under the relevant concept of an object, foreshadowing the pluralistic ontological framework to be developed in Chapter Four. Chapter Three pronounces on a second debate in ontology, in which three positions concerning the correct ontological assay of the class of intrinsically unified independent concrete particulars (objects) are in competition with each other. My conclusion is that none of the three positions succeeds, since each faces fairly serious difficulties. I suggest that the (or at least one major) root of our inability to locate the correct ontological assay is the inclination to treat all ontologically significant entities as objects in the indicated sense, and the corresponding inclination to attempt to give an ontological assay that covers all objects, neglecting important differences between distinct types of individuals. Chapter Four begins by displaying in greater detail the considerations (canvassed very briefly in the introductory chapter) that make the notion of an object appear to be indispensible. However, the results of the second section of Chapter Two and of the entirety of Chapter Three have already shown two areas in which the notion of an object tends to lead to confusion. So a tension emerges between the prima facie necessity of the notion and the reasons we have found for thinking that this notion either is itself problematic or at least tends to cause problems for other issues in Ontology. The remainder of Chapter Four consists in explaining my strategy for moving forward. Briefly, this strategy involves replacing the notion of an object with a series of concepts applicable to individuals of various types. Each of the components belonging to a given “individual-concept” is drawn from one or another side of one of the ontological distinctions that together form an overall ontological framework, and which components are involved is a matter to be determined by examining the conceptual demands imposed by the various practices (explanatory or otherwise) which we engage in, that require us to appeal to individuals of the type in question. The resulting “pluralistic” ontological framework provides a way of situating and relating types of individuals that both avoids the confusions that the single general concept of an object leads to, and is capable of indicating the varying degrees of “ontological robustness” or “object-like-ness” of any given type of individual. I conclude by suggesting how the framework I am advocating can be elaborated on and put to use in further research.

The brief introductory chapter attempts to motivate the project by pointing to (a) the intuitive appeal and importance of the notion of an object (that is, a “paradigmatic” individual), and (b) the need – for the sake of progress in at least two important debates in ontology – to replace this notion with a series of related notions of individuals of different sorts. Section One of Chapter Two aims to accomplish two primary tasks. The first is to clarify the intensions of three often employed but ambiguous categorical terms: ‘individual’, ‘particular’ and ‘object’, with emphasis on the third, which is often taken to be of particular philosophical significance. I carry out this clarificatory task by weighing various positions in the literature and arguing for explications of each notion that are maximally economical and neutral, that is, explications which (a) overlap as little as possible with other important ontological notions and (b) do not require us to take a stand on any apparently intractable (but not directly relevant) debates (e.g. on the problem of realism vs. nominalism about universals). The second task of 2.1 is to delineate the various ontological distinctions that will be turned, in Chapter Four, into the “dimensions” of which the ontological framework I will be advocating there is composed. The delineation of these distinctions takes place naturally in the course of attempting to characterize the notion of an object (an intrinsically unified, independent concrete particular) and to distinguish it from the notions of an individual and a particular, in spite of the fact that objects are both individuals and particulars. In the second section of Chapter Two I illustrate the centrality of the notion of an object in Ontology by showing how that notion figures in the debate over the existence of artifacts. I argue that progress in this debate has been hindered by the way it has been framed, and that seeing the issue as concerning not whether artifacts exist but whether artifacts are objects (in the sense outlined in 2.1) enables us to better appreciate and accommodate the different perspectives of the debate’s participants. At the same time, this way of dissolving the dispute makes clear that existence is not limited to entities that fall under the relevant concept of an object, foreshadowing the pluralistic ontological framework to be developed in Chapter Four. Chapter Three pronounces on a second debate in ontology, in which three positions concerning the correct ontological assay of the class of intrinsically unified independent concrete particulars (objects) are in competition with each other. My conclusion is that none of the three positions succeeds, since each faces fairly serious difficulties. I suggest that the (or at least one major) root of our inability to locate the correct ontological assay is the inclination to treat all ontologically significant entities as objects in the indicated sense, and the corresponding inclination to attempt to give an ontological assay that covers all objects, neglecting important differences between distinct types of individuals. Chapter Four begins by displaying in greater detail the considerations (canvassed very briefly in the introductory chapter) that make the notion of an object appear to be indispensible. However, the results of the second section of Chapter Two and of the entirety of Chapter Three have already shown two areas in which the notion of an object tends to lead to confusion. So a tension emerges between the prima facie necessity of the notion and the reasons we have found for thinking that this notion either is itself problematic or at least tends to cause problems for other issues in Ontology. The remainder of Chapter Four consists in explaining my strategy for moving forward. Briefly, this strategy involves replacing the notion of an object with a series of concepts applicable to individuals of various types. Each of the components belonging to a given “individual-concept” is drawn from one or another side of one of the ontological distinctions that together form an overall ontological framework, and which components are involved is a matter to be determined by examining the conceptual demands imposed by the various practices (explanatory or otherwise) which we engage in, that require us to appeal to individuals of the type in question. The resulting “pluralistic” ontological framework provides a way of situating and relating types of individuals that both avoids the confusions that the single general concept of an object leads to, and is capable of indicating the varying degrees of “ontological robustness” or “object-like-ness” of any given type of individual. I conclude by suggesting how the framework I am advocating can be elaborated on and put to use in further research.

This thesis introduces the infinitesimal symmetries of Dixmier-Douady gerbes over smooth manifolds. The collection of these symmetries are the counterpart for gerbes of the Lie algebra of circle invariant vector fields on principal circle bundles, and are intimately related to connective structures and curvings. We prove that these symmetries possess a Lie 2-algebra structure, and relate them to equivariant gerbes via a "differentiation functor". We also explain the relationship between the infinitesimal symmetries of gerbes and other mathematical structures including Courant algebroids and the String Lie 2-algebra. / text

In 1997, Drs. G. R. Blakley and I. Borosh published two papers whose stated purpose was to present a general formulation of the notion of a code that depends only upon a code's structure and not its functionality. In doing so, they created a further generalization--the idea of a precode. Recently, Drs. Blakley, Borosh, and A. Klappenecker have worked on interpreting the structures and results in these pioneering papers within the framework of category theory. The purpose of this dissertation is to further the above work. In particular, we seek to accomplish the following tasks within the ``general theory of codes.' 1. Rewrite the original two papers in terms of the alternate representations of precodes as bipartite digraphs and Boolean matrices. 2. Count various types of bipartite graphs up to isomorphism, and count various classes of codes and precodes up to isomorphism. 3. Identify many of the classical objects and morphisms from category theory within the categories of codes and precodes. 4. Describe the various ways of constructing a code from a precode by ``splitting' the precode. Identify important properties of these constructions and their interrelationship. Discuss the properties of the constructed codes with regard to the factorization of homomorphisms through them, and discuss their relationship to the code constructed from the precode by ``smashing.' 5. Define a parametrization of a precode and give constructions of various parametrizations of a given precode, including a ``minimal' parametrization. 6. Use the computer algebra system, Maple, to represent and display a precode and its companion, opposite, smash, split, bald-split, and various parametrizations. Implement the formulae developed for counting bipartite graphs and precodes up to isomorphism.

Herein we develop category-theoretic tools for understanding network-style diagrammatic languages. The archetypal network-style diagrammatic language is that of electric circuits; other examples include signal flow graphs, Markov processes, automata, Petri nets, chemical reaction networks, and so on. The key feature is that the language is comprised of a number of components with multiple (input/output) terminals, each possibly labelled with some type, that may then be connected together along these terminals to form a larger network. The components form hyperedges between labelled vertices, and so a diagram in this language forms a hypergraph. We formalise the compositional structure by introducing the notion of a hypergraph category. Network-style diagrammatic languages and their semantics thus form hypergraph categories, and semantic interpretation gives a hypergraph functor. The first part of this thesis develops the theory of hypergraph categories. In particular, we introduce the tools of decorated cospans and corelations. Decorated cospans allow straightforward construction of hypergraph categories from diagrammatic languages: the inputs, outputs, and their composition are modelled by the cospans, while the 'decorations' specify the components themselves. Not all hypergraph categories can be constructed, however, through decorated cospans. Decorated corelations are a more powerful version that permits construction of all hypergraph categories and hypergraph functors. These are often useful for constructing the semantic categories of diagrammatic languages and functors from diagrams to the semantics. To illustrate these principles, the second part of this thesis details applications to linear time-invariant dynamical systems and passive linear networks.

In the paper \cite{FHS12}, the authors announce the discovery of an invariant for Grothendieck toposes which they call the isotropy group of a topos. Roughly speaking, the isotropy group of a topos carries algebraic data in a way reminiscent of how the subobject classifier carries spatial data. Much as we like to compute invariants of spaces in algebraic topology, we would like to have tools to calculate invariants of toposes in category theory. More precisely, we wish to be in possession of theorems which tell us how to go about computing (higher) isotropy groups of various toposes. As it turns out, computation of isotropy groups in toposes can often be reduced to questions at the level of small categories and it is therefore interesting to try and see how isotropy behaves with respect to standard constructions on categories. We aim to provide a summary of progress made towards this goal, including results on various commutation properties of higher isotropy quotients with colimits and the way isotropy quotients interact with categories collaged together via certain nice kinds of profunctors. The latter should be thought of as an analogy for the Seifert-van Kampen theorem, which allows computation of fundamental groups of spaces in terms of fundamental groups of smaller subspaces.

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We build a model structure from the simple point of departure of a structured interval in a monoidal category — more generally, a structured cylinder and a structured co-cylinder in a category.

The purpose of this paper is to examine some basic topics in category theory. A category consists of a class of mathematical objects along with a morphism class having an associative composition. The paper is divided into two chapters. Chapter I deals with intrinsic properties of categories. Various "sub-objects" and properties of morphisms are defined and examples are given. Chapter II deals with morphisms between categories called functors and the natural transformations between functors. Special types of functors are defined and examples are given.