Diagramme (théorie des catégories)

In , a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms that also need indexing. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a function from a fixed index set to the class of sets. A diagram is a collection of objects and morphisms, indexed by a fixed category; equivalently, a functor from a fixed index category to some category. The universal functor of a diagram is the diagonal functor; its right adjoint is the of the diagram and its left adjoint is the colimit. The natural transformation from the diagonal functor to some arbitrary diagram is called a . Formally, a diagram of type J in a C is a (covariant) functor The category J is called the index category or the scheme of the diagram D; the functor is sometimes called a J-shaped diagram. The actual objects and morphisms in J are largely irrelevant; only the way in which they are interrelated matters. The diagram D is thought of as indexing a collection of objects and morphisms in C patterned on J. Although, technically, there is no difference between an individual diagram and a functor or between a scheme and a category, the change in terminology reflects a change in perspective, just as in the set theoretic case: one fixes the index category, and allows the functor (and, secondarily, the target category) to vary. One is most often interested in the case where the scheme J is a or even finite category. A diagram is said to be small or finite whenever J is. A morphism of diagrams of type J in a category C is a natural transformation between functors. One can then interpret the category of diagrams of type J in C as the CJ, and a diagram is then an object in this category. Given any object A in C, one has the constant diagram, which is the diagram that maps all objects in J to A, and all morphisms of J to the identity morphism on A.

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Publications associées (3)

Concepts associés (28)

Cours associés (2)

Séances de cours associées (35)

Diagramme (théorie des catégories)

En théorie des catégories, un diagramme est une collection d'objets et de flèches d'une catégorie donnée. En principe, un diagramme n'est pas un objet mathématique mais seulement une figure, destinée à faciliter la lecture d'un raisonnement. En pratique, on se sert souvent des diagrammes comme de symboles abréviateurs, qui évitent de nommer tous les objets et les flèches que l'on veut considérer; on dit souvent que "considérons le diagramme ci-dessus" au lieu de dire par exemple dans la catégorie des ensembles: "considérons quatre ensembles et une application de dans .

Cone (category theory)

In , a branch of mathematics, the cone of a functor is an abstract notion used to define the of that functor. Cones make other appearances in category theory as well. Let F : J → C be a in C. Formally, a diagram is nothing more than a functor from J to C. The change in terminology reflects the fact that we think of F as indexing a family of and morphisms in C. The J is thought of as an "index category". One should consider this in analogy with the concept of an indexed family of objects in set theory.

Somme amalgamée

vignette|Diagramme commutatif traduisant la propriété universelle de la somme amalgamée. En mathématiques, la somme amalgamée est une opération entre deux ensembles constituant les espaces d'arrivée de deux applications définies sur un même troisième ensemble. Le résultat satisfait une propriété universelle de factorisation de diagrammes, duale de celle du produit fibré et qui peut être valable dans d'autres catégories que celle des ensembles, comme celle des groupes.

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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

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Singular cohomology is defined by dualizing the singular chain complex for spaces. We will study its basic properties, see how it acquires a multiplicative structure and becomes a graded commutative a

Injective And Projective Model Structures On Enriched Diagram Categories

Lyne Moser

In the enriched setting, the notions of injective and projective model structures on a category of enriched diagrams also make sense. In this paper, we prove the existence of these model structures on enriched diagram categories under local presentability, accessibility, and "acyclicity" conditions, using the methods of lifting model structures from an adjunction introduced by Garner, Hess, Kedziorek, Riehl, and Shipley.

2019

, ,

We prove existence results à la Jeff Smith for left-induced model category structures, of which the injective model structure on a diagram category is an important example. We further develop the notions of fibrant generation and Postnikov presentation from work of the second author, which are dual to a weak form of cofibrant generation and cellular presentation. As examples, for k a field and H a differential graded Hopf algebra over k, we produce a left-induced model structure on augmented H-comodule algebras and show that the category of bounded below chain complexes of finite-dimensional k-vector spaces has a Postnikov presentation. To conclude, we investigate the fibrant generation of (generalized) Reedy categories. In passing, we also consider cofibrant generation, cellular presentation, and the small object argument for Reedy diagrams.

2015

Left-Induced Model Structures and Diagram Categories

Kathryn Hess Bellwald, Varvara Karpova, Magdalena Kedziorek

We prove existence results a la Jeff Smith for left-induced model category structures, of which the injective model structure on a diagram category is an important example. We further develop the notions of fibrant generation and Postnikov presentation from [9], which are dual to a weak form of cofibrant generation and cellular presentation. As examples, for k a field and H a differential graded Hopf algebra over k, we produce a left-induced model structure on augmented H-comodule algebras and show that the category of bounded below chain complexes of finite-dimensional k-vector spaces has a Postnikov presentation. To conclude, we investigate the fibrant generation of (generalized) Reedy categories. In passing, we also consider cofibrant generation, cellular presentation, and the small object argument for Reedy diagrams.

Amer Mathematical Soc2015

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