scholarly journals THE GRADED CENTER OF A TRIANGULATED CATEGORY

2016 ◽  
Vol 102 (1) ◽  
pp. 74-95
Author(s):  
JON F. CARLSON ◽  
PETER WEBB
Keyword(s):  
Group Algebra ◽  
Special Kind ◽  
Module Category ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Stable Module Category ◽  
Natural Transformations ◽  
Stable Module ◽  
Serre Functor

With applications in mind to the representations and cohomology of block algebras, we examine elements of the graded center of a triangulated category when the category has a Serre functor. These are natural transformations from the identity functor to powers of the shift functor that commute with the shift functor. We show that such natural transformations that have support in a single shift orbit of indecomposable objects are necessarily of a kind previously constructed by Linckelmann. Under further conditions, when the support is contained in only finitely many shift orbits, sums of transformations of this special kind account for all possibilities. Allowing infinitely many shift orbits in the support, we construct elements of the graded center of the stable module category of a tame group algebra of a kind that cannot occur with wild block algebras. We use functorial methods extensively in the proof, developing some of this theory in the context of triangulated categories.

scholarly journals A Category Theoretic Approach to Metaphor Comprehension: Theory of Indeterminate Natural Transformation

2020 ◽  
Author(s):  
Miho Fuyama ◽  
Hayato Saigo ◽  
Tatsuji Takahashi
Keyword(s):  
Category Theory ◽  
Natural Transformation ◽  
Mathematical Structure ◽  
Theoretic Approach ◽  
Metaphor Comprehension ◽  
Central Notion ◽  
Meaning Creation ◽  
Natural Transformations ◽  
Categories And Functors ◽  
The Relationship

We propose the theory of indeterminate natural transformation (TINT) to investigate the dynamical creation of meaning as an association relationship between images, focusing on metaphor comprehension as an example. TINT models meaning creation as a type of stochastic process based on mathematical structure and defined by association relationships, such as morphisms in category theory, to represent the indeterminate nature of structure-structure interactions between the systems of image meanings. Such interactions are formulated in terms of the so-called coslice categories and functors as structure-preserving correspondences between them. The relationship between such functors is “indeterminate natural transformation”, the central notion in TINT, which models the creation of meanings in a precise manner. For instance, metaphor comprehension is modeled by the construction of indeterminate natural transformations from a canonically defined functor, which we call the base-of-metaphor functor.

scholarly journals On naturality of some construction of connections

2020 ◽  
Vol 74 (1) ◽  
pp. 57
Author(s):  
Jan Kurek ◽  
Włodzimierz Mikulski
Keyword(s):  
Linear Connection ◽  
Bundle Functor ◽  
Fibred Manifold ◽  
Natural Transformations ◽  
General Connection

Let \(F\) be a bundle functor on the category of all fibred manifolds and fibred maps. Let \(\Gamma\) be a general connection in a fibred manifold \(\mathrm{pr}:Y\to M\) and \(\nabla\) be a classical linear connection on \(M\). We prove that the  well-known general connection \(\mathcal{F}(\Gamma,\nabla)\) in \(FY\to M\) is canonical with respect to fibred maps and with respect to natural transformations of bundle functors.

Functors, Transformations, and Modifications

2021 ◽  
pp. 147-202
Author(s):  
Niles Johnson ◽  
Donald Yau
Keyword(s):  
Natural Transformations ◽  
Yoneda Lemma

This chapter discusses functors, transformations, and modifications that are bicategorical analogs of functors and natural transformations. The main concepts covered are lax functors, lax transformations, modifications, and icons. A section is devoted to representable pseudofunctors, representable transformations, and representable modifications, which will be used in the Bicategorical Yoneda Lemma.

Category theory

2020 ◽  
pp. 19-38
Author(s):  
Ash Asudeh ◽  
Gianluca Giorgolo
Keyword(s):  
Natural Language ◽  
Category Theory ◽  
Categorial Grammar ◽  
Natural Language Semantics ◽  
Fundamental Notion ◽  
Language Semantics ◽  
Natural Transformations ◽  
Glue Semantics

This chapter aims to introduce sufficient category theory to enable a formal understanding of the rest of the book. It first introduces the fundamental notion of a category. It then introduces functors, which are maps between categories. Next it introduces natural transformations, which are natural ways of mapping between functors. The stage is then set to at last introduces monads, which are defined in terms of functors and natural transformations. The last part of the chapter provides a compositional calculus with monads for natural language semantics (in other words, a logic for working with monads) and then relates the compositional calculus to Glue Semantics and to a very simple categorial grammar for parsing. The chapter ends with some exercises to aid understanding.

scholarly journals Eventually dendric shift spaces

2020 ◽  
pp. 1-26
Author(s):  
FRANCESCO DOLCE ◽  
DOMINIQUE PERRIN
Keyword(s):  
Discrete Geometry ◽  
Block Coding ◽  
Shift Spaces ◽  
New Class ◽  
Fixed Length ◽  
Natural Transformations ◽  
Number Of Classes

We define a new class of shift spaces which contains a number of classes of interest, like Sturmian shifts used in discrete geometry. We show that this class is closed under two natural transformations. The first one is called conjugacy and is obtained by sliding block coding. The second one is called the complete bifix decoding, and typically includes codings by non-overlapping blocks of fixed length.

A Calculus for (Meta)Models and Transformations

2014 ◽  
Vol 24 (05) ◽  
pp. 715-730 ◽  
Author(s):  
Laurent Thiry ◽  
Michel Hassenforder
Keyword(s):  
Category Theory ◽  
Point Of View ◽  
Modeling Languages ◽  
Formal Representation ◽  
Equational Reasoning ◽  
Natural Transformations ◽  
Formal Point

This paper proposes a formal representation of modeling languages based on category theory. These languages are generally described by "metamodels", i.e. structures composed by classes and relations, and related by "transformations". Thus, this paper studies how the key categorical concepts such as functors and relations between functors (called natural transformations) can be used for equational reasoning about modeling artifacts (models, metamodels, transformations). As a result, this paper proposes a formal point of view of models usable to specify/prove equivalence between models or transformations (with an application to refactoring).

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