Comma category

In mathematics, a comma category (a special case being a slice category) is a construction in . It provides another way of looking at morphisms: instead of simply relating objects of a to one another, morphisms become objects in their own right. This notion was introduced in 1963 by F. W. Lawvere (Lawvere, 1963 p. 36), although the technique did not become generally known until many years later. Several mathematical concepts can be treated as comma categories. Comma categories also guarantee the existence of some s and colimits. The name comes from the notation originally used by Lawvere, which involved the comma punctuation mark. The name persists even though standard notation has changed, since the use of a comma as an operator is potentially confusing, and even Lawvere dislikes the uninformative term "comma category" (Lawvere, 1963 p. 13). The most general comma category construction involves two functors with the same codomain. Often one of these will have domain 1 (the one-object one-morphism category). Some accounts of category theory consider only these special cases, but the term comma category is actually much more general. Suppose that , , and are categories, and and (for source and target) are functors: We can form the comma category as follows: The objects are all triples with an object in , an object in , and a morphism in . The morphisms from to are all pairs where and are morphisms in and respectively, such that the following diagram commutes: Morphisms are composed by taking to be , whenever the latter expression is defined. The identity morphism on an object is . Overcategory The first special case occurs when , the functor is the identity functor, and (the category with one object and one morphism). Then for some object in . In this case, the comma category is written , and is often called the slice category over or the category of objects over . The objects can be simplified to pairs , where . Sometimes, is denoted by .

À propos de ce résultat

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Publications associées (15)

Personnes associées (3)

Unités associées (2)

Concepts associés (13)

Cours associés (1)

Séances de cours associées (17)

MATH-436: Homotopical algebra

This course will provide an introduction to model category theory, which is an abstract framework for generalizing homotopy theory beyond topological spaces and continuous maps. We will study numerous

We extend the group-theoretic notion of conditional flatness for a localization functor to any pointed category, and investigate it in the context of homological categories and of semi-abelian categories. In the presence of functorial fiberwise localizatio ...

CAMBRIDGE UNIV PRESS2023

Hopf algebras and Hopf-Galois extensions in infinity-categories

Aras Ergus

In this thesis, we study interactions between algebraic and coalgebraic structures in infinity-categories (more precisely, in the quasicategorical model of (infinity, 1)-categories). We define a notion of a Hopf algebra H in an E-2-monoidal infinity-catego ...

EPFL2022

Dinosaurs in transition? A conceptual exploration of local incumbents in the swiss and German energy transition

The understanding of incumbents' behaviour in sustainability transitions in the energy sector is gaining increasing scholarly attention. However, two key structural characteristics of many incumbents in the energy sector are hardly taken into account: they ...

2019

Diagonal functor

In , a branch of mathematics, the diagonal functor is given by , which maps as well as morphisms. This functor can be employed to give a succinct alternate description of the product of objects within the : a product is a universal arrow from to . The arrow comprises the projection maps. More generally, given a , one may construct the , the objects of which are called . For each object in , there is a constant diagram that maps every object in to and every morphism in to .

Espace pointé

En topologie, un espace pointé est un espace topologique dont on spécifie un point particulier comme étant le point de base. Formellement, il s'agit donc d'un couple (E, x) pour lequel x est un élément de E. Une application pointée entre deux espaces pointés est une application continue préservant les points de base. Les espaces pointés sont les objets d'une catégorie, notée parfois Top, dont les morphismes sont les applications pointées. Cette catégorie admet le point comme objet nul.

Ensemble pointé

En mathématiques, un ensemble pointé est un ensemble avec un élément distingué , qui est appelé le point de base. Les morphismes d'ensembles pointés (applications pointées) sont les applications qui envoient un point de base sur un autre, i.e. une application telle que . On note habituellement Les ensembles pointés peuvent être regardés comme une structure algébrique simple. Au sens de l'algèbre universelle, ce sont des structures munies d'une opération d'arité zéro qui conserve le point de base.

Théorie de l'homotopie des complexes de chaînes MATH-436: Homotopical algebra

Explore la théorie de l'homotopie des complexes de chaîne, en se concentrant sur les rétractions et les structures de catégorie de modèle.

Critique de Simplicial Sets MATH-436: Homotopical algebra

Couvre les limites, les colimits, les simplices standard, les espaces cartographiques, les limites, les cornes et les épines.

Théorie de la catégorie: Introduction

Couvre les bases des catégories et des functeurs, explorant les propriétés, la composition et l'unicité dans la théorie des catégories.