2-Dimensional Categories

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

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 Gray Monoidal Product of Double Categories

The category of double categories and double functors is equipped with a symmetric closed monoidal structure. For any double category $${\mathbb {A}}$$A, the corresponding internal hom functor "Equation missing" sends a double category $${\mathbb {B}}$$B to the double category whose 0-cells are the double functors $${\mathbb {A}} \rightarrow {\mathbb {B}}$$A→B, whose horizontal and vertical 1-cells are the horizontal and vertical pseudo transformations, respectively, and whose 2-cells are the modifications. Some well-known functors of practical significance are checked to be compatible with this monoidal structure.

LAYANAN KONSELING ATLET: PENGABDIAN MASYARAKAT UNTUK PERSATUAN BULUTANGKIS JAYA RAYA

The background of this community service conducted by Department of Psychology Universitas Pembangunan Jaya (PSI UPJ) and Jaya Raya Badminton Club (PB Jaya Raya) is as follow. For optimum achievement, athletes should have excellent physical, technical and psychological skills. This applies as well in badminton, since this sport requires training as early as in adolescence. The objective of this counselling for athlete program is to develop nine mental skills, comprising from attitude to concentration. Method used is action research - from focus group discussion, interviews, group counselling, observation during competition and individual counselling. This activity involves technical and physical coaches, teachers, boarding school manager and 12-18 years-old male and female athlete from single, double and mixed-double categories; as well as lectures and students. This collaboration is made possible under the Memorandum of Understanding between two institutions - both are part of the family of Pembangunan Jaya Group. From March to September 2017, this community service results in a counselling program, 3 articles of Info Bintaro and Buletin Konsorsium Psikologi Ilmiah Nusantara electronic media, 1 publication in Widyakala UPJ journal of Vol 4 No 1 2017 and 1 Counselling for Athletes Module as ISBN-registered book publication and copyrights.

Cyclic multicategories, multivariable adjunctions and mates

A multivariable adjunction is the generalisation of the notion of a 2-variable adjunction, the classical example being the hom/tensor/cotensor trio of functors, ton+ 1 functors ofnvariables. In the presence of multivariable adjunctions, natural transformations between certain composites built from multivariable functors have “dual” forms. We refer to corresponding natural transformations as multivariable or parametrised mates, generalising the mates correspondence for ordinary adjunctions, which enables one to pass between natural transformations involving left adjoints to those involving right adjoints. A central problem is how to express the naturality (or functoriality) of the parametrised mates, giving a precise characterization of the dualities so-encoded.We present the notion of “cyclic double multicategory” as a structure in which to organise multivariable adjunctions and mates. While the standard mates correspondence is described using an isomorphism of double categories, the multivariable version requires the framework of “double multicategories”. Moreover, we show that the analogous isomorphisms of double multicategories give a cyclic action on the multimaps, yielding the notion of “cyclic double multicategory”. The work is motivated by and applied to Riehl's approach to algebraic monoidal model categories.