Categories: History, basic and developments (original) (raw)

A generalization of certain homological functors

Annali di Matematica Pura ed Applicata, 1966

By a natural process of relativization groups Extn(A, ¢p), rcn(A , ¢~) are defined in any abelian category with sufficient injectives and projec~ives. Functoriality in ¢~ i spro. ved and excision properties are established; and the groups are shown to behave well under suspension. The technique involves an interplay of different mapping-cone constructions 1.-Introduction. In their study of the spectral sequences which arise in algebraic topology [3,4]. Eo~A~ and HILTO~ were concerned with the demonstration of the importance of relativizing the relevant functors. That is they put emphasis on the fact that, while the primary interest of algebraic topology may reside in the eohomology or homotopy groups of spaces the techniques of algebraic topology require that these functors be also defined for maps. Precisely they showed that a spectral sequence always arises when a connected sequence of fanciers is applied to a faetorization of a morphism f,

Categorical, Homological, and Homotopical Properties of Algebraic Objects

Journal of Mathematical Sciences, 2017

This monograph is based on the doctoral dissertation of the author defended in the Iv. Javakhishvili Tbilisi State University in 2006. It begins by developing internal category and internal category cohomology theories (equivalently, for crossed modules) in categories of groups with operations. Further, the author presents properties of actions in categories of interest, in particular, the existence of an actor in specific algebraic categories. Moreover, the reader will be introduced to a new type of algebras called noncommutative Leibniz-Poisson algebras, with their properties and cohomology theory and the relationship of new cohomologies with well-known cohomologies of underlying associative and Leibniz algebras. The author defines and studies the category of groups with an action on itself and solves two problems of J.-L. Loday. Homotopical and categorical properties of chain functors category are also examined.

Category Theory I: A gentle prologue

2024

[March 28, 2024 version] An update to the early chapters of my earlier Gentle Introduction notes, requiring only modest mathematical background. There are chapters on categories, and on constructions like products, pullbacks, exponentials that can occur in different categories. There is also a first encounter with functors. Part II continues the story, talking about natural transformations between functors, the Yoneda lemma and adjunctions and we take an introductory look at the idea of an elementary topos This book Part I is available as a cheap print-on-demand paperback from mid April, but I think of it as a beta version, still work in progress and all comments are still most welcome.

Remarks on topologically algebraic functors

Cahiers de Topologie et Géométrie Différentielle Catégoriques, 1979

Remarks on topologically algebraic functors Cahiers de topologie et géométrie différentielle catégoriques, tome 20, n o 2 (1979), p. 155-177 http://www.numdam.org/item?id=CTGDC\_1979\_\_20\_2\_155\_0 © Andrée C. Ehresmann et les auteurs, 1979, tous droits réservés. L'accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Category theory lecture notes

2001

These notes, developed over a period of six years, were written for an eighteen lectures course in category theory. Although heavily based on Mac Lane's Categories for the Working Mathematician, the course was designed to be self-contained, drawing most of the examples from category theory itself.

The Category of Functors

Lecture Notes in Physics, 2013

We define the category Func Î with functors F:D F¨S COTT (D F´C PO) as objects and pairs (f:D F¨DG ,ú:FÿGºf) as morphisms (ú is a natural transformation). We show that this category is closed under the common domain theoretical operations +,ÿ,÷ and ¨. The category Func Î is an O-category Tennent 1993].