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Lecture# Introduction to Categories: Products and Coproducts

Description

This lecture introduces the theory of categories by generalizing the Cartesian product and disjoint union of sets to any category. The concepts of product and coproduct play a crucial role in category theory, providing essential properties for constructing objects and morphisms.

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Related concepts (95)

Related lectures (52)

Coproduct

In , the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic to the , which means the definition is the same as the product but with all arrows reversed.

Section (category theory)

In , a branch of mathematics, a section is a right inverse of some morphism. , a retraction is a left inverse of some morphism. In other words, if and are morphisms whose composition is the identity morphism on , then is a section of , and is a retraction of . Every section is a monomorphism (every morphism with a left inverse is left-cancellative), and every retraction is an epimorphism (every morphism with a right inverse is right-cancellative). In algebra, sections are also called split monomorphisms and retractions are also called split epimorphisms.

Biproduct

In and its applications to mathematics, a biproduct of a finite collection of , in a with zero objects, is both a and a coproduct. In a the notions of product and coproduct coincide for finite collections of objects. The biproduct is a generalization of finite direct sums of modules. Let C be a with zero morphisms. Given a finite (possibly empty) collection of objects A1, ...

Dual (category theory)

In , a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the Cop. Given a statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms, a corresponding dual statement is obtained regarding the opposite category Cop. Duality, as such, is the assertion that truth is invariant under this operation on statements.

Group (mathematics)

In mathematics, a group is a non-empty set with an operation that satisfies the following constraints: the operation is associative, has an identity element, and every element of the set has an inverse element. Many mathematical structures are groups endowed with other properties. For example, the integers with the addition operation is an infinite group, which is generated by a single element called 1 (these properties characterize the integers in a unique way).

Covers the concept of group cohomology, focusing on chain complexes, cochain complexes, cup products, and group rings.

Introduces Cartesian product and induction for proofs using integers and sets.

Explores the topological Künneth Theorem, emphasizing commutativity and homotopy equivalence in chain complexes.

Explores active learning in Group Theory, focusing on products, coproducts, adjunctions, and natural transformations.

Explores the properties of p-subgroups of Sylow in a group.

Après une introduction à la théorie des catégories, nous appliquerons la théorie générale au cas particulier des groupes, ce qui nous permettra de bien mettre en perspective des notions telles que quo