scholarly journals Object-Free Definition of Categories

2013 ◽  
Vol 21 (3) ◽  
pp. 193-205
Author(s):  
Marco Riccardi
Keyword(s):  
Binary Relation ◽  
Category Theory ◽  
Final Part ◽  
One To One ◽  
Definition Of

Summary Category theory was formalized in Mizar with two different approaches [7], [18] that correspond to those most commonly used [16], [5]. Since there is a one-to-one correspondence between objects and identity morphisms, some authors have used an approach that does not refer to objects as elements of the theory, and are usually indicated as object-free category [1] or as arrowsonly category [16]. In this article is proposed a new definition of an object-free category, introducing the two properties: left composable and right composable, and a simplification of the notation through a symbol, a binary relation between morphisms, that indicates whether the composition is defined. In the final part we define two functions that allow to switch from the two definitions, with and without objects, and it is shown that their composition produces isomorphic categories.

scholarly journals Unification of Graphs and Relations in Mizar

2020 ◽  
Vol 28 (2) ◽  
pp. 173-186
Author(s):  
Sebastian Koch
Keyword(s):  
Binary Relation ◽  
The Other ◽  
Other Hand ◽  
System 2 ◽  
Definition Of

Summary A (di)graph without parallel edges can simply be represented by a binary relation of the vertices and on the other hand, any binary relation can be expressed as such a graph. In this article, this correspondence is formalized in the Mizar system [2], based on the formalization of graphs in [6] and relations in [11], [12]. Notably, a new definition of createGraph will be given, taking only a non empty set V and a binary relation E ⊆ V × V to create a (di)graph without parallel edges, which will provide to be very useful in future articles.

scholarly journals Handling i narratologien, litteraturen og livet

2006 ◽  
Vol 34 (101) ◽  
pp. 122-139
Author(s):  
Thor Grünbaum
Keyword(s):  
The Other ◽  
Final Part ◽  
Communicative Function ◽  
Other Hand ◽  
The One ◽  
Definition Of

Action in Narratology, Literature, and LifeIn this article I argue that the representation of simple, bodily action has the function of endowing the narrative sequence with a visualizing power: It makes the narrated scenes or situations ready for visualization by the reader or listener. By virtue of this visualizing power or disposition, these narrated actions disrupt the theoretical divisions, on the one hand, between the narrated story and the narrating discourse, and on the other hand, between plot-narratology and discourse-narratology. As narrated actions they seem to belong to the domain of plot-narratology, but in so far as they serve an important visualizing function, these narrated actions have a communicative function and as such they can be said to belong to the domain of discourse-narratology. In a first part of the article, I argue that a certain type of plot-narratology, due to its retrospective epistemology and abstract definition of action, is unable to conceive of this visualizing function. In a second part, I argue that discourse-narratology fares no better since the visualizing function is independent of voice and focalization. In a final part, I sketch a possible account of the visualizing function of simple actions in narratives.

scholarly journals Categories as Mathematical Models

2018 ◽  
Author(s):  
David I. Spivak
Keyword(s):  
Dynamical Systems ◽  
Mathematical Modelling ◽  
Mathematical Models ◽  
Category Theory ◽  
Vector Spaces ◽  
Mathematical Objects ◽  
Working Definition ◽  
Modelling Framework ◽  
Categorical Models ◽  
Definition Of

Category theory is presented as a mathematical modelling framework that highlights the relationships between objects, rather than the objects in themselves. A working definition of model is given, and several examples of mathematical objects, such as vector spaces, groups, and dynamical systems, are considered as categorical models.

Grid movies

2014 ◽  
Vol 23 (08) ◽  
pp. 1450038 ◽  
Author(s):  
Matthew Graham
Keyword(s):  
Planar Grid ◽  
Grid Algorithm ◽  
One To One ◽  
Definition Of

We present a grid diagram analogue of Carter, Rieger and Saito's smooth movie theorem. Specifically, we give definitions for grid movies, grid movie isotopies and present a definition of grid planar isotopy as a particular subset of the grid diagram moves: stabilization, destabilization and commutation. We show that grid planar isotopy classes are in one-to-one correspondence with smooth planar isotopy classes by using a new planar grid algorithm that takes a smooth knot diagram to a grid diagram. We then present generalizations of both the smooth and grid movie theorems that apply to surfaces with boundary.

Textbook on Land Law

2018 ◽  
Author(s):  
Judith-Anne MacKenzie ◽  
Aruna Nair
Keyword(s):  
Real Life ◽  
Third Party ◽  
Final Part ◽  
Land Law ◽  
The Real ◽  
The Family ◽  
Family Home ◽  
Definition Of ◽  
Straightforward Approach

Course-focused and comprehensive, Textbook on Land Law continues to provide an interesting, accessible, and original account of contemporary land law. The seventeenth edition builds upon the book’s unique and straightforward approach. Using a fictional case study to illustrate the key principles of land law, the chapters demonstrate the real-life applications of a subject students often find very abstract, while clarifying complex areas and common points of confusion. The book consists of seven parts. Part I provides an introduction to estates and interests in land. Part II looks at the acquisition of estates in land. Part III considers the two legal estates of freehold and leasehold, and in particular looks in detail at the obligations in a leasehold estate, their enforcement and remedies for their breach. Part IV looks at trusts and proprietary estoppel. Part V is about licences and a review of the law relating to the family home. The next part considers third party rights, including mortgages, and the final part concludes with a consideration of the definition of ‘land’.

Textbook on Land Law

2020 ◽  
Author(s):  
Judith-Anne MacKenzie ◽  
Aruna Nair
Keyword(s):  
Real Life ◽  
Third Party ◽  
Final Part ◽  
Land Law ◽  
The Real ◽  
The Family ◽  
Family Home ◽  
Definition Of ◽  
Straightforward Approach

Course-focused and comprehensive, Textbook on Land Law continues to provide an interesting, accessible, and original account of contemporary land law. The eighteenth edition builds upon the book’s unique and straightforward approach. Using a fictional case study to illustrate the key principles of land law, the chapters demonstrate the real-life applications of a subject students often find very abstract, while clarifying complex areas and common points of confusion. The book consists of seven parts. Part I provides an introduction to estates and interests in land. Part II looks at the acquisition of estates in land. Part III considers the two legal estates of freehold and leasehold, and in particular looks in detail at the obligations in a leasehold estate, their enforcement and remedies for their breach. Part IV looks at trusts and proprietary estoppel. Part V is about licences and a review of the law relating to the family home. The next part considers third party rights, including mortgages, and the final part concludes with a consideration of the definition of ‘land’.

SPECIFICATION OF SOFTWARE ARCHITECTURE

2000 ◽  
Vol 10 (01) ◽  
pp. 69-95 ◽  
Author(s):  
MARK J. GERKEN
Keyword(s):  
Software Architecture ◽  
Category Theory ◽  
Formal Definition ◽  
The Past ◽  
Algebraic Specifications ◽  
Software Specifications ◽  
Architecture Theory ◽  
Definition Of ◽  
Semantic Compatibility ◽  
Use Category

Over the past several years, software architecture representation and analysis has become an active area of research. However, most approaches to software architecture representation and analysis have been informal. We postulate that through formality, the term "architecture" can be precisely defined and important properties of systems, such as semantic compatibility between connected entities, can be investigated with precision. In this paper, we use category theory and algebraic specifications to develop a formal definition of architecture and show how architecture theory can be used in the construction of software specifications.

scholarly journals Univalent categories and the Rezk completion

2015 ◽  
Vol 25 (5) ◽  
pp. 1010-1039 ◽  
Author(s):  
BENEDIKT AHRENS ◽  
KRZYSZTOF KAPULKIN ◽  
MICHAEL SHULMAN
Keyword(s):  
Category Theory ◽  
Type Theory ◽  
Dependent Type Theory ◽  
Equivalence Of Categories ◽  
Segal Spaces ◽  
Definition Of ◽  
Categorical Semantics ◽  
Univalent Foundations ◽  
Dependent Type

We develop category theory within Univalent Foundations, which is a foundational system for mathematics based on a homotopical interpretation of dependent type theory. In this system, we propose a definition of ‘category’ for which equality and equivalence of categories agree. Such categories satisfy a version of the univalence axiom, saying that the type of isomorphisms between any two objects is equivalent to the identity type between these objects; we call them ‘saturated’ or ‘univalent’ categories. Moreover, we show that any category is weakly equivalent to a univalent one in a universal way. In homotopical and higher-categorical semantics, this construction corresponds to a truncated version of the Rezk completion for Segal spaces, and also to the stack completion of a prestack.

scholarly journals Diagrammatic logic applied to a parameterisation process

2010 ◽  
Vol 20 (4) ◽  
pp. 639-654 ◽  
Author(s):  
CÉSAR DOMÍNGUEZ ◽  
DOMINIQUE DUVAL
Keyword(s):  
Category Theory ◽  
Formal Parameter ◽  
Actual Parameter ◽  
Definition Of ◽  
The Given ◽  
Parameter Passing

This paper provides an abstract definition of a class of logics, called diagrammatic logics, together with a definition of morphisms and 2-morphisms between them. The definition of the 2-category of diagrammatic logics relies on category theory, mainly on adjunction, categories of fractions and limit sketches. This framework is applied to the formalisation of a parameterisation process. This process, which consists of adding a formal parameter to some operations in a given specification, is presented as a morphism of logics. Then the parameter passing process for recovering a model of the given specification from a model of the parameterised specification and an actual parameter is shown to be a 2-morphism of logics.

scholarly journals MOTIVE AS THE HIGHEST FORM OF NEEDFUL AROUSAL

2019 ◽  
Vol 29 (3) ◽  
pp. 276-285
Author(s):  
Yu.T. Glazunov
Keyword(s):  
Decision Maker ◽  
Binary Relation ◽  
Mental Activity ◽  
Mathematical Approach ◽  
Prognostic Information ◽  
Pragmatic Information ◽  
Binary Relationship ◽  
Motivational Process ◽  
Definition Of ◽  
Types Of Information

The definition of motive is considered. The research is based on the information and mathematical approach. The method is based on comparing two types of information available to the decision-maker. Firstly, it is the prognostic information on all means necessary for successful achievement of the goal; secondly, it is the pragmatic information on the means available for this purpose actually. Information has a numerical measure, and thus can be mathematized. Motivation is defined as mental activity to set a goal, formulate a motive and create a program of actions aimed at meeting the need. In the process of motivation the need passes through the stages of a hidden, objectified, specified and actualized need. Ultimately, it becomes an active need. These stages correspond to a needful arousal in the form of attraction, impulse, desire and intention, respectively. Intention turns into a motive that becomes a needful arousal that accompanies an active (motivated) need. The result of motivation is an integral formation called a binary relationship "motive-goal". It is shown that motive does not exist as a separate mental essence. It can be considered only within the framework of this binary relation. The analysis of the motivational process leads to the following definition: motive is the highest form of needful arousal that meets the actual need.

Sign in / Sign up

Export Citation Format

Share Document