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.

Further 2-Dimensional Categorical Structures

2021 ◽  
pp. 513-574
Author(s):  
Niles Johnson ◽  
Donald Yau
Keyword(s):  
Tensor Product ◽  
Double Categories ◽  
Monoidal Bicategories

In this chapter, further 2-dimensional categorical structures are presented and discussed. These include monoidal bicategories, as one-object tricategories, along with braided monoidal bicategories, sylleptic monoidal bicategories, and symmetric monoidal bicategories. The rest of the chapter discusses the Gray tensor product on 2-categories, Gray monoids, double categories, and monoidal double categories.

The Yoneda Lemma and Coherence

2021 ◽  
pp. 305-330
Author(s):  
Niles Johnson ◽  
Donald Yau
Keyword(s):  
Coherence Theorem ◽  
Local Equivalence ◽  
Yoneda Lemma

In this chapter, the Yoneda Lemma and the Coherence Theorem for bicategories are stated and proved. The chapter discusses the bicategorical Yoneda pseudofunctor, a bicategorical version of the Yoneda embedding for a bicategory, which is a local equivalence, and the Bicategorical Yoneda Lemma. A consequence of the Bicategorical Whitehead Theorem and the Bicategorical Yoneda Embedding is the Bicategorical Coherence Theorem, which states that every bicategory is biequivalent to a 2-category.

scholarly journals LIFTING THEOREMS FOR TENSOR FUNCTORS ON MODULE CATEGORIES

2011 ◽  
Vol 10 (01) ◽  
pp. 129-155 ◽  
Author(s):  
ROBERT WISBAUER
Keyword(s):  
General Theory ◽  
Tensor Product ◽  
Category Theory ◽  
Systematic Approach ◽  
Wreath Products ◽  
Baxter Equation ◽  
General Category ◽  
Module Theory ◽  
Elementary Module ◽  
Special Cases

Any (co)ring R is an endofunctor with (co)multiplication on the category of abelian groups. These notions were generalized to monads and comonads on arbitrary categories. Starting around 1970 with papers by Beck, Barr and others a rich theory of the interplay between such endofunctors was elaborated based on distributive laws between them and Applegate's lifting theorem of functors between categories to related (co)module categories. Curiously enough some of these results were not noticed by researchers in module theory and thus notions like entwining structures and smash products between algebras and coalgebras were introduced (in the nineties) without being aware that these are special cases of the more general theory. The purpose of this survey is to explain several of these notions and recent results from general category theory in the language of elementary module theory focusing on functors between module categories given by tensoring with a bimodule. This provides a simple and systematic approach to smash products, wreath products, corings and rings over corings (C-rings). We also highlight the relevance of the Yang–Baxter equation for the structures on the threefold tensor product of algebras or coalgebras (see 3.6).

scholarly journals Enriched category as a model of qualia structure based on similarity judgements

2021 ◽  
Author(s):  
Naotsugu Tsuchiya ◽  
Steven Phillips ◽  
Hayato Saigo
Keyword(s):  
Category Theory ◽  
Sorites Paradox ◽  
Perceived Similarity ◽  
Enriched Category ◽  
Enriched Categories ◽  
Important Extension ◽  
Similarity Relationships ◽  
Yoneda Lemma ◽  
Similarity Judgements ◽  
Qualia Structure

Qualitative relationships between two instances of conscious experiences can be quantified through the perceived similarity. Previously, we proposed that by defining similarity relationships as arrows and conscious experiences as objects, we can define a category of qualia in the context of category theory. However, the example qualia categories we proposed were highly idealized and limited to cases where perceived similarity is binary: either present or absent without any gradation. When similarity is graded, a situation can arise where A0 is similar to A1, A1 is similar to A2, and so on, yet A0 is not similar to An, which is called the Sorites paradox. Here, we introduce enriched category theory to address this situation. Enriched categories generalize the concept of a relation between objects as a directed arrow (or morphism) in ordinary category theory to a more flexible notion, such as a measure of distance. As an alternative relation, here we propose a graded measure of perceived dissimilarity between the two objects. These measures combine in a way that addresses the Sorites paradox; even if the dissimilarity between Ai and Ai+1 is small for i = 0 … n, hence perceived as similar, the dissimilarity between A0 and An can be large, hence perceived as different. In this way, we show how dissimilarity-enriched categories of qualia resolve the Sorites paradox. We claim that enriched categories accommodate various types of conscious experiences. An important extension of this claim is the application of the Yoneda lemma in enriched category; we can characterize a quale through a collection of relationships between the quale and the other qualia up to an (enriched) isomorphism.

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 土谷・西郷「圏論による意識の理解」へのコメント

2019 ◽  
Author(s):  
Miho Fuyama
Keyword(s):  
Category Theory ◽  
Extensive Review ◽  
Consciousness Studies ◽  
Yoneda Lemma ◽  
The Way

Tsuchiya and Saigo (2019) proposed the way of understanding consciousness based on category theory. In this commentary, I discussed four points about their article: 1. Validity of their treatment of the definition and model of consciousness, 2. Verifiability and novelty of their proposal using Yoneda lemma,3. Lack of extensive review of consciousness studies, and 4. Readers' images and the unity of image.

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