Basic ∞-Category Theory

2022 ◽  
pp. 1-4
Keyword(s):  
Category Theory ◽  
Basic Category

2-Dimensional Categories

2021 ◽  
Author(s):  
Niles Johnson ◽  
Donald Yau
Keyword(s):  
Tensor Product ◽  
Category Theory ◽  
Basic Category ◽  
Double Categories ◽  
Grothendieck Construction ◽  
Monoidal Bicategories ◽  
Coherence Theorem ◽  
First Time ◽  
Yoneda Lemma

2-Dimensional Categories provides an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories; pasting diagrams; lax functors; 2-/bilimits; the Duskin nerve; the 2-nerve; internal adjunctions; monads in bicategories; 2-monads; biequivalences; the Bicategorical Yoneda Lemma; and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double categories. Completely detailed proofs of several fundamental but hard-to-find results are presented for the first time. With exercises and plenty of motivation and explanation, this book is useful for both beginners and experts.

scholarly journals Fuzzy Analogues of Sets and Functions Can Be Uniquely Determined from the Corresponding Ordered Category: A Theorem

2017 ◽  
Author(s):  
Christian Servin ◽  
Gerardo Muela ◽  
Vladik Kreinovich
Keyword(s):  
Fuzzy Sets ◽  
Category Theory ◽  
Fuzzy Relations ◽  
Basic Category ◽  
Ordered Category ◽  
Modern Mathematics ◽  
Universe Of Discourse

In modern mathematics, many concepts and ideas are described in terms of category theory. From this viewpoint, it is desirable to analyze what can be determined if, instead of the basic category of sets, we consider a similar category of fuzzy sets. In this paper, we describe a natural fuzzy analog of the category of sets and functions, and we show that, in this category, fuzzy relations (a natural fuzzy analogue of functions) can be determined in category terms -- of course, modulo 1-1 mapping of the corresponding universe of discourse and 1-1 re-scaling of fuzzy degrees.

scholarly journals Category Theory In Geography?

2015 ◽  
Vol 34 (4) ◽  
pp. 61-68
Author(s):  
Sandra L. Arlinghaus ◽  
Joseph Kerski
Keyword(s):  
Research Program ◽  
Category Theory ◽  
Basic Category ◽  
Gis Software

Abstract Is mathematical category theory a unifying tool for geography? Here we look at a few basic category theoretical ideas and interpret them in geographic example. We also offer links to indicate how category theory has been used as such in other disciplines. Finally, we announce the direction of our research program on this topic as a way to facilitate the learning, and maintenance of learning, of GIS software – and in the spirit of Quaestiones Geographicae, invite debate, comment, and contribution to this program in spatial mathematics.

scholarly journals Models for Concurrency

1994 ◽  
Vol 1 (12) ◽  
Author(s):  
Glynn Winskel ◽  
Mogens Nielsen
Keyword(s):  
Petri Nets ◽  
Parallel Computation ◽  
Category Theory ◽  
Parallel Composition ◽  
Transition Systems ◽  
Basic Category ◽  
Event Structures ◽  
Causal Independence ◽  
Use Category ◽  
Mazurkiewicz Traces

This report surveys a range of models for parallel computation to include interleaving models like transition systems, synchronisation trees and languages (often called Hoare traces in this context), and models like Petri nets, asynchronous transition systems, event structures, pomsets and Mazurkiewicz traces where concurrency is represented more explicitly by a form of causal independence. The presentation is unified by casting the models in a category-theoretic framework. One aim is to use category theory to provide abstract characterisations of constructions like parallel composition valid throughout a range of different models and to provide formal means for translating between different models.<br /> <br />A knowledge of basic category theory is assumed, up to an acquaintance with the notion of adjunction.

scholarly journals Models for Concurrency

1993 ◽  
Vol 22 (463) ◽  
Author(s):  
Glynn Winskel ◽  
Mogens Nielsen
Keyword(s):  
Category Theory ◽  
Parallel Composition ◽  
Transition Systems ◽  
Basic Category ◽  
Final Version ◽  
Oxford University ◽  
Event Structures ◽  
Causal Independence ◽  
Use Category ◽  
Mazurkiewicz Traces

<p>Revised version of DAIMI PB-429</p><p> </p><p>This is, we believe, the final version of a chapter for the Handbook of Logic and the Foundations of Computer Science, vol. IV, Oxford University Press.</p><p>It surveys a range of models for parallel computation to include interleaving models like transition systems, synchronisation trees and languages (often called Hoare traces in this context), and models like Petri nets, asynchronous transition systems, event structures, pomsets and Mazurkiewicz traces where concurrency is represented more explicitly by a form of causal independence.</p><p>The presentation is unified by casting the models in a category-theoretic framework. One aim is to use category theory to provide abstract characterisations of constructions like parallel composition valid throughout a range of different models and to provide formal means for translating between different models. A knowledge of basic category theory is assumed, up to an acquaintance with the notion of adjunction.</p>

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